1 Introduction
We complete the classification of all saturated fusion systems supported on Sylow p-subgroups of rank
$2$
groups of Lie type in characteristic p, unifying several results already in the literature. The rationale behind this undertaking comes from earlier results concerning fusion systems supported on Sylow p-subgroups of
$\mathrm {PSL}_3(p)$
,
$\mathrm {G}_2(p)$
and
$\mathrm {Sp}_4(p^n)$
which exposed exotic fusion systems which were previously unknown. Our main result below indicates these are the only instances when a Sylow p-subgroup of a rank
$2$
group of Lie type can support an exotic fusion system.
Main Theorem. Suppose that
$\mathcal {F}$
is a saturated fusion system on a p-group S, where S is isomorphic to a Sylow p-subgroup of some rank
$2$
simple group of Lie type in characteristic
$p$
. Then
$\mathcal {F}$
is known. Moreover, if
$\mathcal {F}$
is exotic, then
$O_p(\mathcal {F})=\{1\}$
and one of the following holds:
-
1.
$S\cong 7^{1+2}_+$
and
$\mathcal {F}$
is described in [Reference Ruiz and Viruel26]; or -
2. S is isomorphic to a Sylow
$7$
-subgroup of
$\mathrm {G}_2(7)$
and
$\mathcal {F}$
is described in [Reference Parker and Semeraro22]; or -
3. S is isomorphic to a Sylow p-subgroup of
$\mathrm {Sp}_4(p^n)$
for p an odd prime and
$n\in \mathbb {N}$
.
We include the ‘nonsimple’ versions of the rank
$2$
groups of Lie type, as well as their derived subgroups, that is,
$\mathrm {PSp}_4(2), \mathrm {G}_2(2), {}^2\mathrm {F}_4(2)$
and their derived subgroups. We note that our work provides evidence in support of a conjecture made by Parker and Semeraro [Reference Parker and Semeraro23, Conjecture 2] on the rigidity of saturated fusion systems supported on Sylow p-subgroups of groups of Lie type in defining characteristic.
The Main Theorem is a culmination of various results published over several years. The origins of these endeavors are in a paper of Ruiz and Viruel [Reference Ruiz and Viruel26], which classified all saturated fusion systems supported on an extraspecial group of order
$p^3$
and exponent p (a Sylow p-subgroup of
$\mathrm {PSL}_3(p)$
). As alluded to above, they uncovered three exotic fusion systems supported on
$7^{1+2}_+$
. The generic case of fusion systems supported on a Sylow p-subgroup of
$\mathrm {PSL}_3(p^n)$
for
$n\in \mathbb {N}$
was examined in the thesis of Clelland [Reference Clelland6], yielding no further exotic examples.
Next came the paper of Parker and Semeraro [Reference Parker and Semeraro22] on fusion systems supported on a Sylow p-subgroup of
$\mathrm {G}_2(p)$
, again for p odd. They uncovered 27 new exotic fusion systems, all supported on a Sylow
$7$
-subgroup of
$\mathrm {G}_2(7)$
. The next entry is the effort of Henke and Shpectorov in classifying fusion systems on a Sylow p-subgroup of
$\mathrm {Sp}_4(p^n)$
. There are many exotic fusion systems supported on these p-groups, all but two of which are subsystems of the exotic fusion systems unearthed in [Reference Clelland and Parker7], and the outliers are certain exceptional configurations arising when
$p^n=9$
. The outcomes of this work are, as of yet, unpublished.
The case of Sylow p-subgroups of
$\mathrm {PSU}_4(p)$
, for p an odd prime, was handled disjointly in two papers: the case
$p=3$
dealt with by Baccanelli, Franchi and Mainardis [Reference Baccanelli, Franchi and Mainardis4] and the cases
$p\geqslant 5$
analyzed by Moragues–Moncho [Reference Moncho18]. No exotic fusion systems were found here. The final entry, up to the present paper, was work of the author classifying saturated fusion systems on Sylow p-subgroups of
$\mathrm {G}_2(p^n)$
and
$\mathrm {PSU}_4(p^n)$
for any prime p and any
$n\in \mathbb {N}$
, subsuming several of the previously described cases. The methodology here was slightly different than what was employed in previous works and more explicitly relied on certain uniqueness results pertaining to amalgams and weak BN-pairs (in the sense of [Reference Delgado and Stellmacher8, Hypothesis A]) where Greenbook is the text of Delgado-Stellmacher “Groups and Graphs: New Results and Methods” arising in the context of fusion systems.
We complete this program with the determination of the remaining cases: saturated fusion systems on Sylow p-subgroups of
${}^2\mathrm {F}_4(2^n)$
,
${}^3\mathrm {D}_4(p^n)$
or
$\mathrm {PSU}_5(p^n)$
.
Theorem A. Suppose that
$\mathcal {F}$
is a saturated fusion system on a p-group S, where S is isomorphic to a Sylow p-subgroup of
${}^2\mathrm {F}_4(2^n)$
,
${}^3\mathrm {D}_4(p^n)$
or
$\mathrm {PSU}_5(p^n)$
and
$p=2$
in the first case. Then
$\mathcal {F}$
is realized by a known finite group. Moreover, if
$O_p(\mathcal {F})=\{1\}$
, then
$\mathcal {F}$
is isomorphic to the p-fusion category of an almost simple group G with
$F^*(G)\cong {}^2\mathrm {F}_4(2^n)$
,
${}^3\mathrm {D}_4(p^n)$
or
$\mathrm {PSU}_5(p^n)$
, respectively.
This is proved as Proposition 3.1, Theorem 3.15, Theorem 5.19 and Theorem 4.10.
Given a fixed p-group, or a fixed family of p-groups for which one wants to classify all saturated fusion systems supported, the approach generally breaks down into two steps: Determine all the potential essential subgroups of
$\mathcal {F}$
as well as their automizers, and then show that
$\mathcal {F}$
is a known fusion system using certain uniqueness results. This is the approach adopted in this paper. For our work, the uniqueness results mainly stem from the uniqueness of weak BN-pairs of rank
$2$
, as shown in [Reference Delgado and Stellmacher8], and recognizing certain groups with strongly p-embedded subgroups and their associated
$\mathrm {GF}(p)$
-modules. Consequently, the majority of the effort in proving Theorem A is in determining a complete list of potential essential subgroups. None of the techniques employed for this are especially new and for the most part are adaptations of arguments from standard references. We do provide several structural results about Sylow p-subgroups of rank
$2$
simple groups of Lie type which may find use elsewhere.
We use this opportunity to indicate exactly where, in this paper and across all the works needed to prove the Main Theorem, there is any reliance on the classification of the finite simple groups. As a starting point, at the time of writing there is no known generic way to distinguish an exotic fusion system without invoking the classification, and so whenever a system is shown to be exotic, the classification is used. For the determination of automizers in fusion systems on a Sylow p-subgroup of
$\mathrm {PSU}_4(p^n)$
in [Reference van Beek27], the classification is used to identify the group
$\mathrm {PSL}_2(p^{2n})$
, acting on an essential subgroup E as a natural
$\Omega _4^-(p^n)$
-module. In determining the possible fusion systems supported on a Sylow p-subgroup of
$\mathrm {Sp}_4(p^n)$
, whenever
$J(S)$
is an essential subgroup of the fusion system,
$\mathrm {Aut}_{\mathcal {F}}(J(S))$
acts irreducibly on
$J(S)$
which is elementary abelian of order
$p^{3n}$
. The classification is used here to determine
$\mathrm {Aut}_{\mathcal {F}}(J(S))$
from this information. In the current work, we rely on the classification to determine the automizer of a certain essential subgroup of a Sylow p-subgroup of
${}^3\mathrm {D}_4(p^n)$
. The boils down to determining the group
$\mathrm {SL}_2(p^{3n})$
(embedded in
$\mathrm {Sp}_8(p^n)$
) acting on a triality module, for p an odd prime.
2 Preliminaries: groups and fusion systems
In this first section, we set the scene for the proof of the Main Theorem, assembling conventions and providing standard results regarding fusion systems. For this reason, there is a large overlap with various other works, for example, [Reference Aschbacher, Kessar and Oliver2] and [Reference van Beek27].
Definition 2.1. Let S be a finite p-group. A fusion system
$\mathcal {F}$
on S is a category with object set
$\mathrm {Ob}(\mathcal {F}):=\{Q: Q\le S\}$
and whose morphism set satisfies the following properties for
$P, Q\le S$
:
-
•
$\mathrm {Hom}_S(P, Q)\subseteq \mathrm {Mor}_{\mathcal {F}}(P,Q)\subseteq \mathrm {Inj}(P,Q)$
, and -
• each
$\phi \in \mathrm {Mor}_{\mathcal {F}}(P,Q)$
is the composite of an
$\mathcal {F}$
-isomorphism followed by an inclusion,
where
$\mathrm {Inj}(P,Q)$
denotes injective homomorphisms between P and Q. To motivate the group analogy, we write
$\mathrm {Hom}_{\mathcal {F}}(P,Q):=\mathrm {Mor}_{\mathcal {F}}(P,Q)$
and
$\mathrm {Aut}_{\mathcal {F}}(P):=\mathrm {Hom}_{\mathcal {F}}(P,P)$
.
Example 2.2. Let G be a finite group with
$S\in \mathrm {Syl}_p(G)$
. The p-fusion category of G on S, written
$\mathcal {F}_S(G)$
, is the category with object set
$\mathrm {Ob}(\mathcal {F}_S(G)):= \{Q: Q\le S\}$
and for
$P,Q\le S$
,
$\mathrm {Mor}_{\mathcal {F}_S(G)}(P,Q):=\mathrm {Hom}_G(P,Q)$
, where
$\mathrm {Hom}_G(P,Q)$
denotes maps induced by conjugation by elements of G. That is, all morphisms in the category are induced by conjugation by elements of G.
We will often appeal to [Reference Aschbacher, Kessar and Oliver2] for standard results and terminology regarding fusion systems. However, we use this section to emphasize certain definitions and properties which are pivotal in the communication of this work. Henceforth, we generically let
$\mathcal {F}$
be a fusion system on a finite p-group S.
We say that
$\mathcal {H}$
is a subsystem of
$\mathcal {F}$
, written
$\mathcal {H}\le \mathcal {F}$
, on a p-group T if
$T\le S$
,
$\mathcal {H}\subseteq \mathcal {F}$
as categories and
$\mathcal {H}$
is itself a fusion system. Two subgroups of S are said to be
$\mathcal {F}$
-conjugate if they are isomorphic as objects in
$\mathcal {F}$
. We write
$Q^{\mathcal {F}}$
for the set of all
$\mathcal {F}$
-conjugates of Q.
We say a fusion system is realizable if there exists a finite group G with
$S\in \mathrm {Syl}_p(G)$
and
$\mathcal {F}=\mathcal {F}_S(G)$
, as in Example 2.2. Otherwise, the fusion system is said to be exotic.
Now, for some conventions more concentrated on the objects of
$\mathcal {F}$
: the subgroups of S.
Definition 2.3. Let
$\mathcal {F}$
be a fusion system on a p-group S, and let
$Q\le S$
. Say that Q is
-
• fully
$\mathcal {F}$
-normalized if
$|N_S(Q)|\ge |N_S(P)|$
for all
$P\in Q^{\mathcal {F}}$
; -
• fully
$\mathcal {F}$
-centralized if
$|C_S(Q)|\ge |C_S(P)|$
for all
$P\in Q^{\mathcal {F}}$
; -
• fully
$\mathcal {F}$
-automized if
$\mathrm {Aut}_S(Q)\in \mathrm {Syl}_p(\mathrm {Aut}_{\mathcal {F}}(Q))$
; -
• receptive in
$\mathcal {F}$
if for each
$P\le S$
and each
$\phi \in \mathrm {Iso}_{\mathcal {F}}(P,Q)$
, setting there is
$$\begin{align*}N_{\phi}=\{g\in N_S(P) : {}^{\phi}c_g\in\mathrm{Aut}_S(Q)\},\end{align*}$$
$\overline {\phi }\in \mathrm {Hom}_{\mathcal {F}}(N_{\phi }, S)$
such that
$\overline {\phi }|_P = \phi $
;
-
• S-centric if
$C_S(Q)=Z(Q)$
and
$\mathcal {F}$
-centric if P is S-centric for all
$P\in Q^{\mathcal {F}}$
; -
• S-radical if
$O_p(\mathrm {Out}(Q))\cap \mathrm {Out}_S(Q)=\{1\}$
; and -
•
$\mathcal {F}$
-radical if
$O_p(\mathrm {Out}_{\mathcal {F}}(Q))=\{1\}$
.
Definition 2.4. Let
$\mathcal {F}$
be a fusion system on a p-group S. Then
$\mathcal {F}$
is saturated if the following conditions hold:
-
1. Every fully
$\mathcal {F}$
-normalized subgroup is also fully
$\mathcal {F}$
-centralized and fully
$\mathcal {F}$
-automized. -
2. Every fully
$\mathcal {F}$
-centralized subgroup is receptive in
$\mathcal {F}$
.
By a theorem of Puig [Reference Puig25], the fusion category of a finite group
$\mathcal {F}_S(G)$
is a saturated fusion system. Throughout this work, we will focus only on saturated fusion systems, although several of the results we provide should be applicable more generally.
Definition 2.5. Let
$\mathcal {F}$
be a fusion system on a p-group S and
$Q\le S$
. Say that Q is normal in
$\mathcal {F}$
(or
$\mathcal {F}$
-normal) if
$Q\trianglelefteq S$
and for all
$P,R\le S$
and
$\phi \in \mathrm {Hom}_{\mathcal {F}}(P,R)$
,
$\phi $
extends to a morphism
$\overline {\phi }\in \mathrm {Hom}_{\mathcal {F}}(PQ,RQ)$
such that
$\overline {\phi }(Q)=Q$
.
It may be checked that the product of
$\mathcal {F}$
-normal subgroups is itself
$\mathcal {F}$
-normal. Thus, we may talk about the largest
$\mathcal {F}$
-normal subgroup of
$\mathcal {F}$
which we denote
$O_p(\mathcal {F})$
(and occasionally refer to as the p-core of
$\mathcal {F}$
). Further, it follows immediately from the saturation axioms that any subgroup normal in S is fully normalized and fully centralized.
Definition 2.6. Let
$\mathcal {F}$
be a fusion system on a p-group S, and let Q be a subgroup. The normalizer fusion subsystem of Q, denoted
$N_{\mathcal {F}}(Q)$
, is the largest subsystem of
$\mathcal {F}$
, supported on
$N_S(Q)$
, in which Q is normal.
Theorem 2.7. Let
$\mathcal {F}$
be a saturated fusion system on a p-group S. If
$Q\le S$
is fully
$\mathcal {F}$
-normalized, then
$N_{\mathcal {F}}(Q)$
is saturated.
Proof. See [Reference Aschbacher, Kessar and Oliver2, Theorem I.5.5].
Theorem 2.8 (Model Theorem).
Let
$\mathcal {F}$
be a saturated fusion system on a p-group S. Fix
$Q\le S$
which is
$\mathcal {F}$
-centric and normal in
$\mathcal {F}$
. Then the following hold:
-
1. There is a model for
$\mathcal {F}$
. That is, there is a finite group G with
$S\in \mathrm {Syl}_p(G)$
,
$F^*(G)=O_p(G)$
and
$\mathcal {F}=\mathcal {F}_S(G)$
. -
2. If
$G_1$
and
$G_2$
are two models for
$\mathcal {F}$
, then there is an isomorphism
$\phi : G_1\to G_2$
such that
$\phi |_S = \mathrm {Id}_S$
. -
3. For any finite group G with
$S\in \mathrm {Syl}_p(G)$
,
$F^*(G)=Q$
and
$\mathrm {Aut}_G(Q)=\mathrm {Aut}_{\mathcal {F}}(Q)$
, there is
$\beta \in \mathrm {Aut}(S)$
such that
$\beta |_Q = \mathrm {Id}_Q$
and
$\mathcal {F}_S(G) =\mathcal {F}^\beta $
. Thus, there is a model for
$\mathcal {F}$
which is isomorphic to G.
Proof. See [Reference Aschbacher, Kessar and Oliver2, Theorem I.4.9].
We now come to the strategy to prove the main theorem. As remarked in the introduction, the boils down to two steps: determining the essential subgroups of
$\mathcal {F}$
and the morphism attached to them and then determining
$\mathcal {F}$
from certain uniqueness results.
Definition 2.9.
E is
$\mathcal {F}$
-essential if E is an
$\mathcal {F}$
-centric, fully
$\mathcal {F}$
-normalized subgroup of S such that
$\mathrm {Out}_{\mathcal {F}}(E)$
contains a strongly p-embedded subgroup. If the fusion system
$\mathcal {F}$
is clear from context, we will drop the prefix and refer to E as essential. We write
$\mathcal {E}(\mathcal {F})$
for the set of essential subgroups of a fusion system
$\mathcal {F}$
.
Directly from the definition, it is clear that
$N_{\mathcal {F}}(E)$
is a saturated fusion system with
$O_p(N_{\mathcal {F}}(E))=E$
an
$N_{\mathcal {F}}(E)$
-centric subgroup. Hence, by Theorem 2.8 there is a finite group G with
$E=O_p(G)$
,
$C_G(E)\le E$
and
$N_{\mathcal {F}}(E)=\mathcal {F}_S(G)$
.
The following theorem justifies the first step of our strategy of proof of Theorem A.
Theorem 2.10 (Alperin–Goldschmidt fusion theorem).
Let
$\mathcal {F}$
be a saturated fusion system on a p-group S. Then
$$\begin{align*}\mathcal{F}=\langle \mathrm{Aut}_{\mathcal{F}}(Q) \mid Q\,\, \text{is essential or}\,\, Q=S \rangle.\end{align*}$$
Proof. See [Reference Aschbacher, Kessar and Oliver2, Theorem I.3.5].
In this work, we operate in a situation where S is prescribed but
$\mathcal {F}$
is not necessarily known. Thus, we need techniques for detecting whether subgroups of S can be essential subgroups of an arbitrary saturated fusion systems supported on S.
Lemma 2.11. Let S be a finite p-group. Then
$C_{\mathrm {Aut}(S)}(S/\Phi (S))$
is a normal p-subgroup of
$\mathrm {Aut}(S)$
.
Proof. This is due to Burnside; see [Reference Gorenstein10, Theorem 5.1.4].
Note that for subgroup
$P<Q\le N_S(P)$
with Q centralizing
$P/\Phi (P)$
, P is not S-radical so not
$\mathcal {F}$
-radical and, hence, never essential in any saturated fusion systems supported on S. Actually, this result will often not be enough, and we will require a variation more suited to our application. For this, we need to consider certain
$\mathcal {F}$
-characteristic chains of E.
Definition 2.12. Let
$\mathcal {F}$
be a fusion system on a p-group S and
$P\le Q\le S$
. Say that P is
$\mathcal {F}$
-characteristic in Q if
$\mathrm {Aut}_{\mathcal {F}}(Q)\le N_{\mathrm {Aut}(Q)}(P)$
.
Lemma 2.13. Let
$\mathcal {F}$
be a saturated fusion system on a p-group S,
$E\in \mathcal {E}(\mathcal {F})$
and
$A=\mathrm {Aut}_{\mathcal {F}}(E)$
. Set
$\Phi (E)=E_0 \trianglelefteq E_1 \trianglelefteq E_2 \trianglelefteq \dots \trianglelefteq E_m=E$
such that, for all
$0 \le i \le m$
,
$E_i\alpha = E_i$
for each
$\alpha \in A$
. If
$Q\le S$
is such that
$[Q, E_i]\le E_{i-1}$
for all
$1 \le i \le m$
, then
$Q\le E$
.
Proof. Let
$\mathcal {F}$
, S, E and Q be as described in the lemma. Then
$N_{QE}(E)$
embeds in
$\mathrm {Aut}(E)$
as
$\mathrm {Aut}_{QE}(E)$
and we can arrange that
$N_{QE}(E)$
, and hence
$\mathrm {Aut}_{QE}(E)$
, centralizes (a refinement of)
$E_0 \trianglelefteq E_1 \trianglelefteq E_2 \trianglelefteq \dots \trianglelefteq E_m$
. Setting
$A:=\langle \mathrm {Aut}_{QE}^{\mathrm {Aut}_{\mathcal {F}}(E)}\rangle $
, we have by [Reference Gorenstein10, (I.5.3.3)] that A is a p-group. Since
$A\trianglelefteq \mathrm {Aut}_{\mathcal {F}}(E)$
,
$A\le O_p(\mathrm {Aut}_{\mathcal {F}}(E))=\mathrm {Inn}(E)$
since E is
$\mathcal {F}$
-radical. Hence,
$\mathrm {Aut}_{QE}(E)\le \mathrm {Inn}(E)$
and since E is
$\mathcal {F}$
-centric, we infer that
$N_{QE}(E)\le E$
and
$N_{QE}(E)=E$
. Then
$E=QE$
and
$Q\le E$
, as required.
In order to fully apply Lemma 2.13, we need some characteristic subgroups to work with. In our application, these tend to be members of the lower or upper central series of a candidate essential subgroup E, and their centralizers in E. However, there are other fairly natural characteristic subgroups which will appear later in our analysis which we document below. We start with a subgroup inspired by one defined in [Reference Glauberman and Solomon11].
Definition 2.14. Let S be a finite p-group, and let
$\mathbb {W}(S)$
be the set of elementary abelian normal subgroups A of S satisfying the following property: Whenever
$[A, s, s]=\{1\}$
, we have that
$[A, s]=\{1\}$
for any
$s\in S$
. Set
$W(S)=\langle \mathbb {W}(S)\rangle $
.
Lemma 2.15. Let S be a finite p-group. Then
-
1.
$W(S)$
is a characteristic subgroup of S; -
2.
$W(S)$
is the unique largest member of the set
$\mathbb {W}(S)$
with respect to inclusion; -
3.
$\Omega (Z(S))\le W(S)$
; and -
4. if
$W(S)\le T\le S$
, then
$W(S)\le W(T)$
.
Proof. Let
$A\in \mathbb {W}(S)$
, and let
$\phi $
an automorphism of S. Then
$A\phi $
is an elementary abelian normal subgroup of S. For any
$s\in S$
with
$[A\phi , s, s]=\{1\}$
, we have that
$[A\phi , s, s]=[A, s\phi ^{-1}, s\phi ^{-1}]\phi =\{1\}$
. Since
$A\in \mathbb {W}(S)$
, we deduce that
$[A, s\phi ^{-1}]=\{1\}$
from which it follows that
$[A\phi , s]=\{1\}$
, and so
$A\phi \in \mathbb {W}(S)$
and
$W(S)$
is characteristic in S, proving (1).
For
$A, B\in \mathbb {W}(S)$
, we have that
$[A, B, B]\le [A\cap B, B]=\{1\}$
so that
$[A, B]=\{1\}$
. Hence, every member of
$\mathbb {W}(S)$
is abelian and commutes with every other member of
$\mathbb {W}(S)$
and we infer that
$W(S)$
is abelian. Then
$W(S)$
is generated by elements of order p, and so
$W(S)$
is an elementary abelian normal subgroup of S. Now, for
$s\in S$
with
$[W(S), s, s]=\{1\}$
we have that
$[A, s, s]=[A, s]=\{1\}$
for all
$A\in \mathbb {W}(S)$
. Since
$W(S)=\langle \mathbb {W}(S)\rangle $
, we must have that
$[W(S), s]=\{1\}$
and
$W(S)\in \mathbb {W}(S)$
. Hence, 2 holds.
Since
$\Omega (Z(S))$
is an elementary abelian normal subgroup of S, with
$[\Omega (Z(S)),s]=\{1\}$
for all
$s\in S$
,
$\Omega (Z(S))\in \mathbb {W}(S)$
, and 3 holds. Let
$T\le S$
with
$W(S)\le T\le S$
. Then
$W(S)$
is an elementary abelian normal subgroup of T and for any
$t\in T$
,
$t\in S$
, and so
$[W(S), t, t]=\{1\}$
implies that
$[W(S), t]=\{1\}$
and
$W(S)\in \mathbb {W}(T)$
. Since
$\mathbb {W}(T)$
has a unique maximal element,
$W(S)\le W(T)$
and 4 holds.
Throughout, we use the ‘elementary’ version of the Thompson subgroup.
Definition 2.16. Let S be a finite p-group. Set
$\mathcal {A}(S)$
to be the set of all elementary abelian subgroups of S of maximal rank. Then the Thompson subgroup of S is defined as
$J(S):=\langle A \mid A\in \mathcal {A}(S)\rangle $
.
Related to the Thompson subgroup, for a p-group S we define
$\mathcal {A}_{\trianglelefteq }(S):=\{A\in \mathcal {A}(S) \mid A\trianglelefteq S\}$
and
$J_{\trianglelefteq }(S):=\langle A \mid A\in \mathcal {A}_{\trianglelefteq }(S)\rangle $
. Of course, there is the question of whether the set
$\mathcal {A}_{\trianglelefteq }(S)$
is nonempty, the answer to which is generally no. As a consequence, the group
$J_{\trianglelefteq }(S)$
may not be well defined. However, in our application, whenever we exploit arguments involving
$J_{\trianglelefteq }(S)$
, we will have already demonstrated that
$\mathcal {A}_{\trianglelefteq }(S)$
is nonempty. It is clear that
$J_{\trianglelefteq }(S)$
is a characteristic subgroup of S.
The following lemma is specific only to the case
$p=2$
and relies on a nice property of involutions. In this work, it is used only in the analysis of a Sylow
$2$
-subgroup of
${}^2\mathrm {F}_4(2^n)$
.
Lemma 2.17. Let S be a
$2$
-group, and suppose that there is
$A,B\in \mathcal {A}(S)$
with
$AB=S$
. If
$C_B(a)=A\cap B$
for every
$a\in A\setminus (A\cap B)$
, then
$\mathcal {A}(S)=\{A, B\}$
and every involution of S lies in
$A\cup B$
.
Proof. Aiming for a contradiction, let
$C\in \mathcal {A}(S)$
with
$A\ne C\ne B$
and choose c an involution in S. Then
$c=ab$
for some
$a\in A$
and
$b\in B$
and
$1=c^2=(ab)^2=[a, b]$
. Hence, either
$a\in A\cap B$
so that
$c\in B$
, or
$a\not \in B$
and
$b\in C_B(a)=A\cap B$
so that
$c=ab\in A$
. Thus,
$c\in A\cup B$
and we now have that
$C=(C\cap A)(C\cap B)$
.
Let
$c\in C\cap A$
with
$c\not \in B$
. Then
$c\in A\setminus (A\cap B)$
so that
$C_B(c)=A\cap B$
. But
$C\cap B\le C_B(c)$
so that
$C\cap B\le A$
and
$C\le A$
. Since
$A,C\in \mathcal {A}(S)$
,
$C=A$
, a contradiction. Hence, no such C exists and
$\mathcal {A}(S)=\{A, B\}$
.
As an elementary example, one can take
$S\cong \mathrm {Dih}(8)$
and A and B to be two distinct Klein four subgroups of S.
The Thompson subgroup is intimately related with the notion of failure to factorize modules for finite groups.
Definition 2.18. Let G be a finite group and V a
$\mathrm {GF}(p)$
-module. If there exists
$\{1\}\ne A\le G$
such that
-
1.
$A/C_A(V)$
is an elementary abelian p-group; -
2.
$[V,A]\ne \{1\}$
; and -
3.
$|V/C_V(A)|\leqslant |A/C_A(V)|$
,
then V is a failure to factorize module (abbrev. FF-module) for G and A is an offender on V.
Now follows a selection of results used to identify automizers and the structure of
$\mathrm {Aut}_{\mathcal {F}}(E)$
-chief factors of E, viewed as
$\mathrm {GF}(p)\mathrm {Out}_{\mathcal {F}}(E)$
-modules. We begin with some modules for
$\mathrm {SL}_2(p^n)$
and some ways in which we may recognize them from the actions of
$N_S(E)$
on E. These results will often be used in the remainder of this work without explicit references.
Definition 2.19. A natural
$\mathrm {SL}_2(p^n)$
-module is any irreducible two-dimensional
$\mathrm {GF}(p^n)\mathrm {SL}_2(p^n)$
-module regarded as a
$2n$
-dimension module for
$\mathrm {GF}(p)\mathrm {SL}_2(p^n)$
by restriction.
Lemma 2.20. Suppose
$G\cong \mathrm {SL}_2(p^n)$
,
$S\in \mathrm {Syl}_p(G)$
and V is natural
$\mathrm {SL}_2(p^n)$
-module. Then the following holds:
-
1.
$[V, S,S]=\{1\}$
; -
2.
$|V|=p^{2n}$
and
$|C_V(S)|=p^n$
, and so V is an FF-module; -
3.
$C_V(s)=C_V(S)=[V,S]=[V,s]=[v, S]$
for all
$v\in V\setminus C_V(S)$
and
$1\ne s\in S$
; and -
4.
$V/C_V(S)$
and
$C_V(S)$
are irreducible
$\mathrm {GF}(p)N_G(S)$
-modules upon restriction.
Proof. See [Reference Parker and Rowley20, Lemma 4.6] and [Reference Parker and Rowley21, Lemma 3.13].
Theorem 2.21. Suppose that E is an essential subgroup of a saturated fusion system
$\mathcal {F}$
on a p-group S, and assume that there is an
$\mathrm {Aut}_{\mathcal {F}}(E)$
-invariant subgroups
$U\le V\le E$
such that
$E=C_S(V/U)$
and
$V/U$
is an FF-module for
$G:=\mathrm {Out}_{\mathcal {F}}(E)$
. Then, writing
$L:=O^{p'}(G)$
and
$W:=V/U$
, we have that
$L/C_L(W)\cong \mathrm {SL}_2(p^n)$
,
$C_L(W)$
is a
$p'$
-group and
$W/C_W(O^p(L))$
is a natural
$\mathrm {SL}_2(p^n)$
-module.
Proof. This follows from [Reference Henke12, Theorem 5.6] (c.f. [Reference Henke12, Theorem 1]).
Definition 2.22. Let V be an irreducible two-dimensional
$\mathrm {GF}(p^{2n})$
-module for
$G\cong \mathrm {SL}_2(p^{2n})$
. A natural
$\Omega _4^-(p^n)$
-module for G is any nontrivial irreducible submodule of
$(V\otimes _k V^\tau )_{\mathrm {GF}(p^n)G}$
regarded as a
$\mathrm {GF}(p)G$
-module by restriction, where
$\tau $
is an involutary automorphism of
$\mathrm {GF}(p^{2n})$
. The action of
$g\ \epsilon\ G$
on
$V\tau$
is given by applying
$\tau$
to the entries of the matrix in
$GL_2(p^{2n})$
which is associated to the action of
$g$
on
$V$
.
Lemma 2.23. Let
$G\cong \mathrm {SL}_2(p^{2n})$
,
$S\in \mathrm {Syl}_p(G)$
and V a natural
$\Omega _4^-(p^n)$
-module for G. Then the following hold:
-
1.
$C_G(V)=Z(G)$
; -
2.
$[V, S, S, S]=\{1\}$
; -
3.
$|V|=p^{4n}$
and
$|V/[V,S]|=|C_V(S)|=p^n$
; and -
4.
$V/[V,S]$
and
$C_V(S)$
are irreducible
$\mathrm {GF}(p)N_G(S)$
-modules upon restriction.
Moreover, for
$\{1\}\ne F\le S$
, one of the following occurs:
-
(a)
$[V, F]=[V, S]$
and
$C_{V}(F)=C_{V}(S)$
; -
(b)
$p=2$
,
$[V, F]=C_{V}(F)$
has order
$2^{2n}$
, F is quadratic on V and
$|F|\leqslant 2^n$
; or -
(c) p is odd,
$|[V, F]|=|C_{V}(F)|=p^{2n}$
,
$[V, S]=[V, F]C_{V}(F)$
,
$C_V(S)=C_{[V, F]}(F)$
and
$|F|\leqslant p^n$
.
Proof. See [Reference Parker and Rowley20, Lemma 4.8] and [Reference Parker and Rowley21, Lemmas 3.12, 3.15].
Definition 2.24. Let V be an irreducible two-dimensional
$\mathrm {GF}(p^{3n})$
-module for
$G\cong \mathrm {SL}_2(p^{3n})$
. A triality module for G is any nontrivial irreducible submodule of
$(V\otimes V^{\tau }\otimes V^{\tau ^2})|_{\mathrm {GF}(p^n)G}$
regarded as a
$\mathrm {GF}(p)G$
-module by restriction. Here,
$\tau $
is an automorphism of
$\mathrm {GF}(p^{3n})$
of order
$3$
.
Lemma 2.25. Suppose that
$G\cong \mathrm {SL}_2(p^{3n})$
,
$S\in \mathrm {Syl}_p(G)$
and V is a triality module for G. Then the following hold:
-
1.
$[V, S, S, S, S]=\{1\}$
; -
2.
$|V|=p^{8n}$
,
$|V/[V,S]|=|C_V(S)|=|[V, S, S, S]|=p^n$
and
$|[V,S,S]|=p^{4n}$
; -
3. if p is odd, then
$|V/C_V(s)|=p^{5n}$
, while if
$p=2$
, then
$|V/C_V(s)|=p^{4n}$
, for all
$1\ne s\in S$
; and -
4.
$V/[V,S]$
and
$C_V(S)$
are irreducible
$\mathrm {GF}(p)N_G(S)$
-modules upon restriction.
Proof. See [Reference Parker and Rowley20, Lemma 4.10] and [Reference Parker and Rowley21, Lemmas 3.12, 3.14, 3.16].
Aside from
$\mathrm {SL}_2(p^n)$
, the only other automizers of essential subgroups which will appear later in this work are the other rank
$1$
groups of Lie type in characteristic p. We will never have the deal with the Ree groups
${}^2\mathrm {G}_2(3^n)$
, and so we only describe some modules associated to
$\mathrm {SU}_3(p^n)$
and
$\mathrm {Sz}(2^n)$
.
Definition 2.26. A natural
$\mathrm {SU}_3(p^n)$
-module is the natural module for
$\mathrm {SL}_3(p^{2n})$
regarded as a
$\mathrm {GF}(p)\mathrm {SU}_3(p^n)$
-module by restriction.
Lemma 2.27. Suppose
$G\cong \mathrm {SU}_3(p^n)$
,
$S\in \mathrm {Syl}_p(G)$
and V is a natural module. Then the following hold:
-
1.
$C_V(S)=[V, Z(S)]=[V, S,S]$
is of order
$p^{2n}$
; -
2.
$C_V(Z(S))=[V,S]$
is of order
$p^{4n}$
; and -
3.
$V/[V, S]$
,
$[V, S]/C_V(S)$
and
$C_V(S)$
are irreducible
$\mathrm {GF}(p)N_G(S)$
-modules upon restriction.
Proof. The actions of S,
$Z(S)$
and
$N_G(S)$
on V are described in [Reference Huppert14, Theorem II.10.2].
Definition 2.28. A natural
$\mathrm {Sz}(2^n)$
-module is the natural module for
$\mathrm {Sp}_4(2^n)$
regarded as a
$\mathrm {GF}(2)\mathrm {Sz}(2^n)$
-module by restriction.
Proposition 2.29. Suppose
$G\cong \mathrm {Sz}(2^n)$
,
$S\in \mathrm {Syl}_2(G)$
and V is the natural module. Then the following hold:
-
1.
$[V, S]$
has order
$2^{3n}$
; -
2.
$[V,\Omega (S)]=C_V(\Omega (S))=[V,S,S]$
has order
$2^{2n}$
; -
3.
$C_V(S)=[V,S,\Omega (S)]=[V, \Omega (S), S]=[V,S,S,S]$
has order
$2^n$
; and -
4.
$V/[V,S]$
,
$[V,S]/C_V(\Omega (S))$
,
$C_V(\Omega (S))/C_V(S)$
and
$C_V(S)$
are all irreducible
$\mathrm {GF}(2)N_G(S)$
-modules upon restriction.
Proof. One may calculate in
$\mathrm {Sp}_4(2^n)$
to obtain these results. Generators for S and
$N_G(S)$
as subgroups of
$\mathrm {GL}_4(2^n)$
, as well as many other facts about the
$2$
-modular representation theory of
$\mathrm {Sz}(2^n)$
, are given explicitly in [Reference Martineau17].
With the relevant groups and modules described, we now provide results which detect them. Although the previously quoted Theorem 2.21 is the most prevalent result throughout this work, some others will be required in various places.
Lemma 2.30. Let G be a
$p'$
-central extension of
$\mathrm {PSL}_2(p^n)$
,
$S\in \mathrm {Syl}_p(G)$
and V a faithful irreducible
$\mathrm {GF}(p)G$
-module. If
$|V|<p^{3n}$
, then either
-
1. V is a natural
$\mathrm {SL}_2(p^n)$
-module for
$G\cong \mathrm {SL}_2(p^n)$
; -
2. V is a natural
$\Omega _4^-(p^{n/2})$
-module, n is even, S does not act quadratically on V and
$Z(G)$
acts trivially on V; or -
3. V is a triality module, n is a multiple of
$3$
and S does not act quadratically on V.
Proof. See [Reference Chermak and Delgado5, Lemma 2.6].
Often, upon examining some candidate subgroup E of a saturated fusion system
$\mathcal {F}$
with the aim of showing that it is not essential, we can assume that E is not contained in any other essential subgroup of
$\mathcal {F}$
. We term essential subgroups which are not contained in any other essential subgroup maximally essential.
Definition 2.31. Suppose that
$\mathcal {F}$
is a saturated fusion system on a p-group S. Then
$E\le S$
is maximally essential in
$\mathcal {F}$
if E is essential and, if
$F\le S$
is essential in
$\mathcal {F}$
and
$E\le F$
, then
$E=F$
.
Lemma 2.32. Suppose that E is a maximally essential subgroup of a saturated fusion system
$\mathcal {F}$
which is supported on a p-group S. If
$m_p(\mathrm {Out}_S(E))\geqslant 2$
, then we have that
$O_{p'}(O^{p'}(\mathrm {Out}_{\mathcal {F}}(E)))\le Z(O^{p'}(\mathrm {Out}_{\mathcal {F}}(E)))$
.
Proof. Let
$T\le N_S(E)$
with
$E<T$
. Since E is receptive, for all
$\alpha \in N_{\mathrm {Aut}_{\mathcal {F}}(E)}(\mathrm {Aut}_T(E))$
,
$\alpha $
lifts to a morphism
$\widehat {\alpha }\in \mathrm {Hom}_{\mathcal {F}}(N_\alpha , S)$
with
$N_\alpha \ge T>E$
. Since E is maximally essential, applying the Alperin–Goldschmidt theorem,
$\widehat {\alpha }$
is the restriction to
$N_\alpha $
of a morphism
$\overline {\alpha }\in \mathrm {Aut}_{\mathcal {F}}(S)$
. But then, upon restriction,
$\alpha $
normalizes
$\mathrm {Aut}_S(E)$
, and so
$N_{\mathrm {Aut}_{\mathcal {F}}(E)}(\mathrm {Aut}_T(E))\le N_{\mathrm {Aut}_{\mathcal {F}}(E)}(\mathrm {Aut}_S(E))$
. Thus, we ascertain the inclusion
$N_{\mathrm {Out}_{\mathcal {F}}(E)}(\mathrm {Out}_T(E))\le N_{\mathrm {Out}_{\mathcal {F}}(E)}(\mathrm {Out}_S(E))$
. Since this holds for all
$T\le N_S(E)$
with
$E<T$
, we infer that
$N_{\mathrm {Out}_{\mathcal {F}}(E)}(\mathrm {Out}_S(E))$
is strongly p-embedded in
$\mathrm {Out}_{\mathcal {F}}(E)$
.
Since
$m_p(\mathrm {Out}_S(E))\geqslant 2$
, by coprime action, we have that
$$\begin{align*}O_{p'}(O^{p'}(\mathrm{Out}_{\mathcal{F}}(E)))=\langle C_{O_{p'}(O^{p'}(\mathrm{Out}_{\mathcal{F}}(E)))}(a) \mid a\in \mathrm{Out}_S(E)^\# \rangle.\end{align*}$$
Since
$N_{\mathrm {Out}_{\mathcal {F}}(E)}(\mathrm {Out}_S(E))$
is a strongly p-embedded subgroup of
$\mathrm {Out}_{\mathcal {F}}(E)$
, we conclude that
$O_{p'}(O^{p'}(\mathrm {Out}_{\mathcal {F}}(E)))\le N_{\mathrm {Out}_{\mathcal {F}}(E)}(\mathrm {Out}_S(E))$
. Hence,
$[\mathrm {Out}_S(E), O_{p'}(O^{p'}(\mathrm {Out}_{\mathcal {F}}(E)))]=\{1\}$
and since
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(E))$
is the normal closure in
$\mathrm {Out}_{\mathcal {F}}(E)$
of
$\mathrm {Out}_S(E)$
, we deduce that
$O_{p'}(O^{p'}(\mathrm {Out}_{\mathcal {F}}(E)))$
lies in the center of
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(E))$
, as required.
The following results are independent of the classification of the finite simple groups and provide generic ways in which to identify groups with a strongly p-embedded subgroup.
Proposition 2.33. Suppose that
$\mathcal {F}$
is a saturated fusion system on a
$2$
-group S and
$E\le S$
is maximally essential. If
$m_2(\mathrm {Out}_S(E))\geqslant 2$
, then
$O^{2'}(\mathrm {Out}_{\mathcal {F}}(E))\cong \mathrm {PSL}_2(2^n)$
,
$\mathrm {(P)SU}_3(2^n)$
or
$\mathrm {Sz}(2^n)$
for some
$n>1$
.
Proof. We apply a result of Bender [Reference Bender3] so that, as
$m_2(\mathrm {Out}_S(E))>1$
, we have
$O^{2'}(\mathrm {Out}_{\mathcal {F}}(E))/O_{2'}(O^{2'}(\mathrm {Out}_{\mathcal {F}}(E)))\cong \mathrm {PSL}_2(2^n)$
,
$\mathrm {PSU}_3(2^n)$
or
$\mathrm {Sz}(2^n)$
for some
$n>1$
. Since
$O_{2'}(O^{2'}(\mathrm {Out}_{\mathcal {F}}(E)))\le Z(O^{2'}(\mathrm {Out}_{\mathcal {F}}(E)))$
by Lemma 2.32, using information on Schur multipliers as can be found in [Reference Gorenstein, Lyons and Solomon9, Section 6.1], we have that
$O^{2'}(\mathrm {Out}_{\mathcal {F}}(E))\cong \mathrm {PSL}_2(2^n)$
,
$\mathrm {(P)SU}_3(2^n)$
or
$\mathrm {Sz}(2^n)$
, as required.
Recall that
$X$
is a TI-set in G if
$X\cap X^g=\{1\}$
for
$g\in G\setminus N_G(X)$
.
Proposition 2.34. Suppose that
$\mathcal {F}$
is a saturated fusion system on a p-group S, p is odd,
$E\le S$
is maximally essential and
$\mathrm {Out}_S(E)$
is a TI-set of
$\mathrm {Out}_{\mathcal {F}}(E)$
. If
$m_p(\mathrm {Out}_S(E))\geqslant 2$
and there is
$x\in \mathrm {Out}_S(E)$
with
$[E, x, x]\le \Phi (E)$
, then
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(E))\cong \mathrm {SL}_2(p^{n+1})$
or
$\mathrm {(P)SU}_3(p^n)$
for
$n\geqslant 1$
.
Proof. We apply the main result of [Reference Ho13] from which we deduce that
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(E))/O_{p'}(O^{p'}(\mathrm {Out}_{\mathcal {F}}(E)))\cong \mathrm {PSL}_2(p^{n+1})$
or
$\mathrm {PSU}_3(p^n)$
for
$n\in \mathbb {N}$
. As in Proposition 2.33, Lemma 2.32 yields that
$O_{p'}(O^{p'}(\mathrm {Out}_{\mathcal {F}}(E)))\le Z(O^{p'}(\mathrm {Out}_{\mathcal {F}}(E)))$
and using information about the Schur multipliers of
$\mathrm {PSL}_2(p^{n+1})$
and
$\mathrm {PSU}_3(p^n)$
as can be found in [Reference Gorenstein, Lyons and Solomon9, Section 6.1], we deduce that
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(E))\cong \mathrm {(P)SL}_2(p^{n+1})$
or
$\mathrm {(P)SU}_3(p^n)$
for
$n\geqslant 1$
. Since
$\mathrm {PSL}_2(p^{n+1})$
has abelian or dihedral Sylow
$2$
-subgroups, [Reference Gorenstein10, (I.3.8.4)] provides a contradiction in this case. Hence, the result holds.
A feature of the parabolic subgroups of several small rank groups of Lie type in characteristic p is the appearance of semiextraspecial groups.
Definition 2.35. A p-group Q is semiextraspecial if
$\Phi (Q)=Q'=Z(Q)$
and
$Q/Z$
is extraspecial for Z any maximal subgroup of
$Z(Q)$
. Q is ultraspecial if Q is semiextraspecial and
$|Z(Q)|^2=|Q/Z(Q)|$
.
Lemma 2.36. Suppose that Q is a semiextraspecial p-group. Then the following hold:
-
1.
$|Q/Z(Q)|=p^{2n}$
for some
$n\in \mathbb {N}$
. -
2. For Z a maximal subgroup of
$Z(Q)$
, and n such that
$|Q/Z(Q)|=p^{2n}$
if
$A/Z$
is an abelian subgroup of
$Q/Z$
, then
$|A/Z|\leqslant p^{1+n}$
. Consequently, if
$A\le Q$
is abelian, then
$|A|\leqslant |Z(Q)|p^n$
. -
3. If
$A\le Q$
with
$Z(Q)\le A$
and
$|A|=|Z(Q)|p^n$
, then
$[x, A]=Z(S)$
for any
$x\in Q\setminus A$
.
Proof. See [Reference Parker and Rowley21, Proposition 2.66].
This slew of results will be enough to determine the local actions in saturated fusion system
$\mathcal {F}$
in our setup, so we now turn our attention to determining
$\mathcal {F}$
from this local information. This is the second step in the strategy to determine
$\mathcal {F}$
and concerns the uniqueness arguments for
$\mathcal {F}$
. Fortunately, the hard work for these arguments has been completed elsewhere ([Reference van Beek28, Corollary A] using [Reference Delgado and Stellmacher8]).
Theorem 2.37. Let
$\mathcal {F}$
be a saturated fusion system with exactly two essential subgroups
$E_1, E_2\trianglelefteq S$
such that
$\mathrm {Aut}_{\mathcal {F}}(E_i)$
is a
$\mathcal {K}$
-group for
$i\in \{1,2\}$
. If
$O_p(\mathcal {F})=\{1\}$
, then
$\mathcal {F}$
is known explicitly.
Recall that a
$\mathcal {K}$
-group is a finite group in which every simple section is assumed to be a known finite simple group. For the most part, this
$\mathcal {K}$
-group hypothesis is used to reduce the list of groups with a strongly p-embedded subgroups in the search for automizers of essential subgroups of a given fusion system. For this work, Theorem 2.21, Proposition 2.33 and Proposition 2.34 will generally be enough to determine the form of the automizer and then [Reference Delgado and Stellmacher8] determines the induced amalgam uniquely up to isomorphism. By [Reference van Beek28, Theorem 4.9], this is enough to determine the fusion system up to isomorphism.
The only place where we require this
$\mathcal {K}$
-group hypothesis is in the determination of the automizer of a particular essential subgroup of the p-fusion category of
${}^3\mathrm {D}_4(p^n)$
for p an odd prime. Indeed, here we need to determine subgroups of
$\mathrm {Sp}_8(p^n)$
acting on an eight-dimensional
$\mathrm {GF}(p^n)$
-module V, with a Sylow p-subgroup elementary abelian of order
$p^{3n}$
and acting on V as a Sylow p-subgroup of
$\mathrm {SL}_2(p^{3n})$
acts on a triality module, and with the normalizer of this Sylow p-subgroup strongly p-embedded. As far as the author is aware, there is no result in the literature which characterizes
$\mathrm {SL}_2(p^{3n})$
from this hypothesis without using the classification.
Rather than provide a full list of outcomes in Theorem 2.37, we instead only list the relevant fusion systems for this work in the following three corollaries.
Corollary 2.38. Suppose the hypothesis of Theorem 2.37, and assume that S is isomorphic to a Sylow
$2$
-subgroup of
${}^2\mathrm {F}_4(2^n)$
for some
$n\in \mathbb {N}$
. Then
$\mathcal {F}=\mathcal {F}_S(G)$
, where
$F^*(G)=O^{2'}(G)\cong {}^2\mathrm {F}_4(2^n)$
.
Corollary 2.39. Suppose the hypothesis of Theorem 2.37, and assume that S is isomorphic to a Sylow p-subgroup of
${}^3\mathrm {D}_4(p^n)$
for some
$n\in \mathbb {N}$
. Then
$\mathcal {F}=\mathcal {F}_S(G)$
, where
$F^*(G)=O^{p'}(G)\cong {}^3\mathrm {D}_4(p^n)$
.
Corollary 2.40. Suppose the hypothesis of Theorem 2.37, and assume that S is isomorphic to a Sylow p-subgroup of
$\mathrm {PSU}_5(p^n)$
for some
$n\in \mathbb {N}$
. Then
$\mathcal {F}=\mathcal {F}_S(G)$
, where
$F^*(G)=O^{p'}(G)\cong \mathrm {PSU}_5(p^n)$
.
3 Fusion systems on a Sylow
$2$
-subgroup of
${}^2\mathrm {F}_4(2^n)$
We first deal with the case where S is isomorphic to a Sylow
$2$
-subgroup of
${}^2\mathrm {F}_4(2)$
or
${}^2\mathrm {F}_4(2)'$
. In the case where
$S\in \mathrm {Syl}_2({}^2\mathrm {F}_4(2))$
, we set
$Q_1:=C_S(Z_2(S))$
and
$Q_2:=C_S(Z_3(\Omega (S))/Z(S))$
. The following result is proved using the fusion systems package in MAGMA [Reference Parker and Semeraro23].
Proposition 3.1. Let
$G={}^2\mathrm {F}_4(2)$
and
$S\in \mathrm {Syl}_2(G)$
. If
$\mathcal {F}$
is a saturated fusion system supported on S, then either:
-
1.
$\mathcal {F}=\mathcal {F}_S(S)$
; -
2.
$\mathcal {F}=\mathcal {F}_S(N_G(Q_1))$
, where
$\mathrm {Out}_{\mathcal {F}}(Q_1)\cong \mathrm {Sym}(3)$
; -
3.
$\mathcal {F}=\mathcal {F}_S(N_G(Q_2))$
, where
$\mathrm {Out}_{\mathcal {F}}(Q_2)\cong \mathrm {Sz}(2)$
; or -
4.
$\mathcal {F}=\mathcal {F}_S(G)$
.
Moreover, for each of the above fusion systems, writing
$\mathcal {F}=\mathcal {F}_S(H)$
, we have that
$\mathcal {F}_{T}(H\cap O^2(G))$
is a saturated fusion on
$T:=S\cap O^2(G)$
, and every saturated fusion system on T arises in this fashion.
We remark that
$\mathrm {Out}_{\mathcal {F}_S(N_G(Q_i))}(Q_i)\cong \mathrm {Out}_{\mathcal {F}_T(N_{O^2(G)}(Q_i\cap T))}(Q_i\cap T)$
and
$Q_i\cap T$
has index
$2$
in
$Q_i$
for
$i\in \{1,2\}$
.
We continue now under the assumption that S is isomorphic to a Sylow
$2$
-subgroup of
${}^2\mathrm {F}_4(2^n)$
where
$n>1$
, and write
$q=2^n$
. The reader is referred to [Reference Parrott19] for a description of S in terms of Chevalley commutator formulas. Indeed, in the work which follows, we could have used this in our approach of the analysis of the internal structure of S, but we have elected instead to describe S in terms of its embedding in the two maximal subgroups of
${}^2\mathrm {F}_4(q)$
which contain it.
For
$G:={}^2\mathrm {F}_4(q)$
, we identify
$S\in \mathrm {Syl}_2(G)$
and set
$G_1:=N_G(C_S(Z_2(S)))$
,
$G_2:=N_G(C_S(Z_3(\Omega (S))/Z(S)))$
and
$L_i:=O^{2'}(G_i)$
for
$i\in \{1,2\}$
so that
$L_1$
is of shape
$q^{2+1+2+4+2}:\mathrm {SL}_2(q)$
and
$L_2$
is of shape
$q^{1+4+1+4}:\mathrm {Sz}(q)$
(see [Reference Wilson29, pg. 165]). Note that
$G_1$
and
$G_2$
are the unique maximal parabolic subgroups of G containing
$N_G(S)$
. We write
$Q_i:=O_2(L_i)=O_2(G_i)$
so that
$Q_1=C_S(Z_2(S))$
and
$Q_2=C_S(Z_3(\Omega (S))/Z(S))$
, characteristic subgroups of S. Then set
$V_1:=\Omega (Q_1)$
,
$U_1:=[V_1, Q_1]$
and
$V_2:=Q_2'$
.
We record the following structural properties of
$L_1, L_2$
and S. We will often refer to [Reference Parrott19] for the structure of
$S, Q_1$
and
$Q_2$
, while the chief factor structure of
$L_1$
and
$L_2$
may be extracted from [Reference Wilson29, pg. 165], [Reference Azad, Barry and Seitz1] and [Reference Delgado and Stellmacher8, Chapter 12]. We provide modest proofs of these properties but encourage the reader to work through the details more closely.
Proposition 3.2. The following hold:
-
1.
$|Q_2|=q^{10}$
; -
2.
$\Omega (S/Q_2)=Z(S/Q_2)=\Phi (S/Q_2)$
; -
3.
$|C_{Q_2}(V_2)|=q^6$
and
$Q_2/C_{Q_2}(V_2)$
is a natural module for
$L_2/Q_2\cong \mathrm {Sz}(q)$
; -
4.
$|V_2|=q^5$
,
$V_2\in \mathcal {A}(Q_2)\subseteq \mathcal {A}(S)$
and
$C_{Q_2}(V_2)/V_2$
is centralized by
$L_2$
; -
5.
$Z(S)=Z(Q_2)$
has order q,
$L_2$
centralizes
$Z(Q_2)$
and
$V_2/Z(Q_2)$
is a natural module for
$L_2/Q_2$
; and -
6.
$V_2=\Omega (C_{Q_2}(V_2))=[Q_2, C_{Q_2}(V_2)]$
and
$Z(Q_2)=[Q_2, V_2]=[C_{Q_2}(V_2), C_{Q_2}(V_2)]$
.
Proof. By [Reference Wilson29, pg. 165],
$G_2$
has shape
$q^{1+4+1+4}:(\mathrm {Sz}(q)\times (q-1))$
. Then
$|Q_2|=q^{10}$
and
$Q_2$
is self-centralizing in
$G_2$
so that
$Z(S)\le Z(Q_2)$
. By [Reference Parrott19, pg. 343], we have that
$V_2$
is elementary abelian of order
$q^5$
,
$|C_{Q_2}(V_2)|=q^6$
,
$V_2=\Omega (C_{Q_2}(V_2))$
and
$[Q_2, V_2]=Z(S)$
is of order q. By the commutator formulas in [Reference Parrott19], using the notation therein, we see that
$[\alpha _5(t), \alpha _5(u)]\ne 1$
for some
$t,u\in \mathrm {GF}(q)$
and
$\alpha _5(w)\in C_{Q_2}(V_2)$
for all
$w\in \mathrm {GF}(q)$
. Moreover,
$[\alpha _5(t), \alpha _4(u)]=\alpha _9(tu)$
, where
$\alpha _4(u)\in Q_2\setminus C_{Q_2}(V_2)$
for all
$t, u\in \mathrm {GF}(q)$
, and
$\alpha _9(w)\in V_2/Z(S)$
for
$w\in \mathrm {GF}(q)^\times $
. Hence,
$\{1\}<C_{Q_2}(V_2)'\le Z(S)$
and
$Z(S)\not \ge [Q_2, C_{Q_2}(V_2)]\le V_2$
. Indeed, for M with
$V_2<M\le C_{Q_2}(V_2)$
we have that
${Z(S)\not \ge [Q_2, M]\le V_2}$
.
Since
$L_2/Q_2\cong \mathrm {Sz}(q)$
, we have that
$\Omega (S/Q_2)=Z(S/Q_2)=\Phi (S/Q_2)$
. By [Reference Delgado and Stellmacher8, (5.7)], the minimal degree of a faithful
$\mathrm {Sz}(q)$
representation is
$4n$
, recalling that
$q=2^n$
. It follows that
$V_2/Z(S)$
is an irreducible module for
$L_2/Q_2$
. Since
$V_2\le \Phi (Q_2)$
, we have that
$L_2/Q_2$
acts faithfully on
$Q_2/V_2$
. Since
$|C_{Q_2}(V_2)/V_2|=q$
and
$|Q_2/C_{Q_2}(V_2)|=q^4$
, we see that
$Q_2/C_{Q_2}(V_2)$
is an irreducible module for
$L_2/Q_2$
and
$L_2$
centralizes
$C_{Q_2}(V_2)/V_2$
. Since
$[Q_2, C_{Q_2}(V_2)]\trianglelefteq L_2$
, we have that
$Z(S)[Q_2, C_{Q_2}(V_2)]=V_2$
and since
$[Q_2, V_2]=Z(S)$
, we have that
$[Q_2, C_{Q_2}(V_2)]=V_2$
. Indeed, since any M with
$V_2<M\le C_{Q_2}(V_2)$
is normalized by
$L_2$
, it follows that
$[Q_2, M]=V_2$
from which we conclude that
$Z(Q_2)=C_{V_2}(S)=Z(S)$
. That
$Q_2/V_2$
and
$V_2/Z(Q_2)$
are natural
$\mathrm {Sz}(q)$
-modules follows from the Steinberg tensor product theorem [Reference Gorenstein, Lyons and Solomon9, Corollary 2.8.6] upon extending scalars to
$\mathrm {GF}(q^2)$
. Finally, by [Reference Gorenstein, Lyons and Solomon9, Table 3.3.1], we have that S has
$2$
-rank
$5n$
, and so
$V_2\in \mathcal {A}(Q_2)\subseteq \mathcal {A}(S)$
.
Proposition 3.3. The following hold:
-
1.
$|Q_1|=q^{11}$
; -
2.
$|V_1|=q^9$
and
$Q_1/V_1$
is a natural module for
$L_1/Q_1\cong \mathrm {SL}_2(q)$
; -
3.
$|U_1|=q^5$
,
$U_1\in \mathcal {A}(Q_1)\subset \mathcal {A}(S)$
and
$V_1/U_1$
is an irreducible module of order
$q^4$
for
$L_1/Q_1$
such that
$C_{V_1/U_1}(S)=U_1V_2/U_1$
has order q; -
4.
$|Z(V_1)|=q^3$
and
$U_1/Z(V_1)$
is a natural module for
$L_1/Q_1$
; -
5.
$Z_2(S)=Z(Q_1)$
has order
$q^2$
,
$Z(V_1)/Z(Q_1)$
is centralized by
$L_1$
and
$Z(Q_1)$
is a natural module for
$L_1/Q_1$
; and -
6.
$V_1=Q_1'$
,
$Z(V_1)=V_1'=[U_1, Q_1]$
and
$Z(Q_1)=[Z(V_1), Q_1]=[U_1, V_1]$
.
Proof. By [Reference Wilson29, pg. 165],
$G_1$
has shape
$q^{2+9}:\mathrm {GL}_2(q)$
. Then
$|Q_1|=q^{11}$
and
$Q_1$
is self-centralizing in
$G_1$
. In particular, since
$Z(S)\trianglelefteq G_2$
and
$O_2({}^2\mathrm {F}_4(q))=\{1\}$
, we see that so that
$Z(S)<Z(Q_1)$
. Note that
$[Z_2(S), Q_2]=Z(S)$
, from which we deduce that
$Z_2(S)\le V_2$
. Indeed,
$Z_2(S)/Z(S)=C_{V_2/Z(S)}(S)$
. Since
$V_2=Q_2'\le Q_1$
, we have that
$Z(Q_1)\le C_{Q_2}(V_2)$
. Then
$[Q_1\cap Q_2, Z(Q_1)V_2]\le Z(S)$
and as
$Q_1\cap Q_2\not \le C_{Q_2}(V_2)$
, we have that
$Z(Q_1)\le V_2$
. Then
$Z(Q_1)/Z(S)\le C_{V_2/Z(S)}(S)$
and we deduce that
$Z(Q_1)\le Z_2(S)$
. Since
$V_2/Z(S)$
has the structure of a natural
$\mathrm {Sz}(q)$
-module, we see that
$|Z_2(S)|=q^2$
. Since
$Z(S)<Z(Q_1)$
,
$Z(Q_1)$
contains a noncentral chief factor for
$L_1/Q_1\cong \mathrm {SL}_2(q)$
and as S acts quadratically on
$Z(Q_1)$
, we deduce by Lemma 2.30 that
$Z(Q_1)=Z_2(S)$
is a natural
$\mathrm {SL}_2(q)$
-module for
$L_1/Q_1$
and
$Q_1=C_S(Z_2(S))$
. Note that as
$Z(Q_1)=Z_2(S)\le V_2$
, we see that
$C_{Q_2}(V_2)\le Q_1$
. In particular,
$Z(V_1)\le C_S(V_2)=C_{Q_2}(V_2)$
.
We appeal to the commutator formulas in [Reference Parrott19] to see that
$Q_1$
has class
$5$
, and that
$Q_1'$
has order
$q^9$
and is generated by involutions (
$Q_1$
is referred to as C in [Reference Parrott19, pg. 344]). Observe that as
$L_2/Q_2\cong \mathrm {Sz}(q)$
, we have that
$V_1<Q_1$
. Since
$L_1/Q_1$
acts faithfully on
$Q_1/\Phi (Q_1)$
, and since
$[Q_1, Q_2, Q_2]\le V_2\le V_1$
, by Lemma 2.30 we have that
$[Q_1, Q_1]=\Phi (Q_1)=V_1$
and
$Q_1/V_1$
is a natural
$\mathrm {SL}_2(q)$
-module for
$L_1/Q_1$
. Again, by [Reference Parrott19]
$[Q_1', Q_1]$
is elementary abelian of order
$q^5$
. Further, by the commutator formulas in [Reference Parrott19] we calculate that
$U_1$
is elementary abelian of order
$q^5$
.
Note that
$Z_3(S)\cap V_2$
has order
$q^3$
and, as
$V_2/Z(S)$
has the structure of a natural
$\mathrm {Sz}(q)$
-module, we have that
$[Z_3(S)\cap V_2, Q_1, Q_1]=[Z(Q_1), Q_1]=\{1\}$
. By the three subgroups lemma,
$Z_3(S)\cap V_2\le Z(V_1)$
. Then
$Z(V_1)V_2\trianglelefteq L_2$
and
$Z(V_1)\cap V_2\ge [Z(V_1)V_2, Q_2]\trianglelefteq L_2$
from which we deduce that
$[Z(V_1)V_2, Q_2]=Z(S)$
,
$Z(V_1)\le V_2$
and
$Z(V_1)=Z_3(S)\cap V_2$
has order
$q^3$
. Then
$\{1\}\ne [Q_1, Z(V_1)]\trianglelefteq L_1$
from which deduce that
$[S, Z(V_1)]=[Q_1, Z(V_1)]=Z(Q_1)$
and
$L_1$
centralizes
$Z(V_1)/Z(Q_1)$
. Since
$Q_1/V_1$
is irreducible under the action of
$L_1$
, it follows that
$V_1=C_S(Z(V_1))$
and
$C_{Q_2}(V_2)\le V_1$
.
If
$U_1/Z(V_1)$
does not contain a noncentral chief factor for
$L_1/Q_1$
, then since
$Q_1=V_1[Q_1, O^2(L_1)]$
, we deduce that
$[U_1, Q_1]=[U_1, V_1]$
. But then
$[U_1, Q_1, Q_1]=[U_1, V_1, Q_1]<[U_1, V_1]$
. By the three subgroups lemma,
$[Q_1', U_1]=[V_1, U_1]<[U_1, V_1]$
, absurd. Hence,
$U_1/Z(V_1)$
contains a noncentral chief factor. Since
$U_1=[Q_1, V_1]\le Q_2$
, we have that
$[U_1, Q_2, Q_2]\le Z(S)$
and Lemma 2.30 reveals that
$U_1/Z(V_1)$
is a natural
$\mathrm {SL}_2(q)$
-module. As above, we are able to show that
$[U_1, V_1]<[U_1, Q_1]$
from which we deduce that
$[U_1, V_1]=Z(Q_1)$
and
$[U_1, Q_1]=Z(V_1)$
. Recognizing
$V_1'=[Q_1', V_1]$
and
$[V_1, Q_1]=U_1$
, the three subgroups lemma implies that
$Z(V_1)=V_1'$
. It remains to show that
$V_1/U_1$
is an irreducible module of order
$q^4$
for
$L_1/Q_1$
such that
$C_{V_1/U_1}(S)=U_1V_2/U_1$
has order q.
Once again, we appeal to the commutator formulas in [Reference Parrott19] (where
$V_2=U_8$
,
$U_1=F=\langle V_{12}, \dots , V_9, V_7\rangle $
and
$V_1=\langle V_2, U_5\rangle $
in Parrott’s notation) to see that
$C_{V_1/U_1}(S)=U_1V_2/U_1$
has order q. Write
$\overline {V_1}:=V_1/U_1$
. Note that if
$C_{\overline {V_1}}(L_1)\ne \{1\}$
, then since
${}^2\mathrm {F}_4(q)$
is defined over
$\mathrm {GF}(q)$
, we would have that
$C_{\overline {V_1}}(L_1)=\overline {V_2}$
so that
$V_1U_1\trianglelefteq L_1$
. But
$U_1\le Q_2$
and
$U_1\not \le C_{Q_2}(V_2)$
, and so
$\{1\}\ne [U_1, V_2]=[U_1V_2, U_1, V_2]\le Z(S)$
, a contradiction since
$L_1$
is irreducible on
$Z(Q_1)$
. Hence,
$C_{\overline {V_1}}(L_1)=\{1\}$
and
$\overline {V_1}$
is indecomposable as a
$\mathrm {GF}(p)$
-module for
$L_1/Q_1$
. Let W be the preimage in
$V_1$
of the unique minimal nontrivial
$L_1$
-invariant subgroup of
$\overline {V_1}$
. Then
$L_1$
acts irreducibly on
$\overline {W}$
and
$W=\langle V_2^{L_1}\rangle U_1$
.
Assume that
$W\le Q_2$
. Then
$[W, Q_2]\le V_2$
and we conclude by Theorem 2.21 that
$\overline {W}$
is a natural
$\mathrm {SL}_2(q)$
-module for
$L_1/Q_1$
. Since
$[Q_1, V_1]=U_1$
and
$S=(S\cap O^2(L_1))Q_1$
, we have that
$V_1/W$
contains a noncentral chief factor for
$L_1$
and we deduce again by Theorem 2.21 that
$V_1/W$
is a natural
$\mathrm {SL}_2(q)$
-module for
$L_1/Q_1$
. But then all of
$Q_1/V_1$
,
$V_1/W$
,
$W/U_1$
and
$U_1/Z(V_1)$
have the structure of
$\mathrm {SL}_2(q)$
-modules for
$L_1/Q_1$
, and so an element of
$L_1$
of order
$3$
acts fixed point freely on
$Q_1/Z(V_1)$
. By a classical result of Burnside,
$Q_1/Z(V_1)$
has class at most
$2$
, a contradiction since
$[Q_1', Q_1]=U_1$
. Hence,
$W\not \le Q_2$
. Then
$[W, Q_2]=V_1\cap Q_2$
and since
${}^2\mathrm {F}_4(q)$
is defined over
$\mathrm {GF}(q)$
, we deduce that
$V_1=W(V_1\cap Q_2)=W$
and we conclude that
$\overline {V}$
is irreducible under the action of
$L_1$
.
Proposition 3.4. The following hold:
-
1.
$|S|=q^{12}$
; -
2.
$S'=\Phi (S)=V_1(Q_1\cap Q_2)$
; -
3.
$m_2(S)=5n$
; -
4.
$V_2=Z_3(\Omega (S))$
,
$\Omega (U_1C_{Q_2}(V_2))=U_1V_2$
and
$\Omega (S)=V_1Q_2$
; and -
5. if
$x\in S$
is such that
$x^2=1$
, then
$x\in V_1 \cup Q_2$
.
Proof. That
$|S|=q^{12}$
is clear from the structure of
$L_1$
or
$L_2$
. Note that
$V_1=Q_1'\le S'\le \Phi (S)$
. From the structure of the Sylow
$2$
-subgroups of
$\mathrm {Sz}(q)$
and
$\mathrm {SL}_2(q)$
, we see that
$S'\le \Phi (S)\le Q_1\cap V_1Q_2=V_1(Q_1\cap Q_2)$
by the Dedekind modular law. Since
$Q_1/V_1$
has the structure of a natural
$\mathrm {SL}_2(q)$
-module, we see that
$V_1(Q_1\cap Q_2)=S'=\Phi (S)$
. As observed in Proposition 3.2, we have that S has
$2$
-rank
$5n$
.
Note that
$U_1=[V_1, Q_1]\le Q_2$
since
$V_1Q_2/Q_2=Z(S/Q_2)$
. Moreover,
$U_1$
is elementary abelian of order
$q^5$
and as
$m_2(S)=5n$
,
$U_1\not \le C_{Q_2}(V_2)$
. Hence,
$\Omega (Q_2)\not \le C_{Q_2}(V_2)$
and since
$L_2$
is irreducible on
$Q_2/C_{Q_2}(V_2)$
, we see that
$Q_2=\Omega (Q_2)C_{Q_2}(V_2)$
. Since
$V_2=Z_2(Q_2)$
, it follows that
$Z_2(S)\le V_2$
and
$V_2\le Q_1$
. Since
$V_2$
is elementary abelian, we have that
$V_2\le V_1$
and
$[V_2, V_1]=Z(V_1)$
. Since
$V_2/Z(S)$
has the structure of a natural
$\mathrm {Sz}(q)$
-module, by Proposition 2.29 we see that
$Z(S)[V_2, V_1]$
has order
$q^3$
and we deduce that
$Z(S)[V_2, V_1]=Z(V_1)$
. Since
$L_1$
is irreducible on
$Q_1/V_1$
, we have that
$V_1=C_{Q_1}(Z(V_1))=C_S(Z(V_1))$
and since
$Z(V_1)\le V_2$
, we see that
$C_{Q_2}(V_2)\le V_1$
. Indeed,
$Q_2\le \Omega (Q_2)V_1\le \Omega (S)$
. By the structure of
$S/Q_2$
, we have that
$\Omega (S)=V_1Q_2$
.
Now,
$Z(\Omega (S))=Z(Q_2)=Z(S)$
,
$Z_2(\Omega (S))\le Z_2(Q_2)=V_2$
and by the action of
$V_1$
on
$V_2$
, we see that
$Z(V_1)=Z_2(\Omega (S))$
. Moreover,
$V_2\le Z_3(\Omega (S))$
. Since
$C_{V_1/U_1}(S)=C_{V_1/U_1}(Q_2)=U_1V_2/U_1$
, we have that
$|Z_3(\Omega (S))U_1|\leqslant q^6$
. Finally, since
$U_1/Z(V_1)$
has the structure of a natural
$\mathrm {SL}_2(q)$
-module,
$C_{U_1/Z(V_1)}(Q_2)$
has order q and we conclude that
$|Z_3(\Omega (S))|\leqslant q^5$
and
$V_2=Z_3(\Omega (S))$
. If
$\Omega (U_1C_{Q_2}(V_2))>U_1V_2$
, then since
$\Omega (C_{Q_2}(V_2))=V_2$
, there is
$uc$
with
$u\in U_1\setminus C_{Q_2}(V_2)$
,
$c\in C_{Q_2}(V_2)\setminus V_2$
and
$(uc)^2=1$
. Since
$u=u^{-1}$
,
$c^2\in Z(S)$
and
$c^4=1$
, we see that
$ucu^{-1}c^{-1}c^{-2}=1$
so that
$[u^{-1}, c^{-1}]=c^2\in Z(S)$
. Set
$M_2=\langle c\rangle C_{Q_2}(V_2)$
so that
$V_2<M_2\le C_{Q_2}(V_2)$
and
$M\trianglelefteq L_2$
. Then
$C_{Q_2}(M_2/Z(S))\trianglelefteq Q_2$
from which we deduce
$C_{Q_2}(M_2/Z(S))\le C_{Q_2}(V_2)$
, a contradiction since
$u\not \in C_{Q_2}(V_2)$
. Hence, no such element exists and
$\Omega (U_1C_{Q_2}(V_2))=U_1V_2$
.
Finally, assume that there is
$x\in S$
such that
$x^2=1$
but
$x\not \in V_1 \cup Q_2$
. Write
$x=yv$
for
$y\in Q_2\setminus (Q_2\cap V_1)$
and
$v\in V_1\setminus (V_1\cap Q_2)$
. By the irreducibility of
$V_1/U_1$
under
$L_1$
, it follows that
$v^2\in U_1$
so that
$v^4=1$
. Applying the commutator formulas in [Reference Parrott19], we have that
$y^2\in V_2$
so that
$y^4=1$
. Then
$y^{-2}y^{-1}vyv^{-1}v^{-2}=1$
so that
$[y, v]=y^2v^2\in V_2U_1$
. Since
$Q_2C_{Q_2}(V_2)$
has the structure of a natural
$\mathrm {Sz}(q)$
-module and
$v\not \in Q_2$
, we see by Proposition 2.29 that
$y\in Q_1\cap Q_2$
so that
$x\in Q_1$
. Since
$x^2=1$
, we see that
$x\in V_1$
so that
$y\in V_1$
, a contradiction. Hence, no such x exists and we have shown that if
$x\in S$
is such that
$x^2=1$
, then
$x\in V_1 \cup Q_2$
.
The majority of the work in this section is in reducing the list of candidate essential subgroups of any saturated fusion system supported on S, ultimately showing in Proposition 3.14 that the only possible essentials are
$Q_1$
and
$Q_2$
. This comes as a consequence of a number of lemmas and propositions which follow.
Proposition 3.5. Let
$\mathcal {F}$
be a saturated fusion system supported on S and
$E\in \mathcal {E}(\mathcal {F})$
. Then
$Z(V_1)\le E$
.
Proof. Aiming for a contradiction, assume that
$E\in \mathcal {E}(\mathcal {F})$
with
$Z(V_1)\not \le E$
. We have that
$\Omega (E)\le E\cap \Omega (S)=E\cap V_1Q_2$
so that
$[Z(V_1), \Omega (E)]\le Z(S)\le Z(Q_1)$
. Now, unless
$E\cap Z(Q_1)=Z(S)$
, we have that
$C_S(E\cap Z(Q_1))=Q_1$
so that
$Z(E)\le Q_1$
and
$\Omega (Z(E))\le V_1$
. In this case,
$Z(Q_1)$
centralizes the chain
$\{1\}\trianglelefteq Z(E)\trianglelefteq E$
so that
$Z(Q_1)\le E$
. Then
$Z(V_1)\le N_S(E)$
and
$Z(V_1)$
now normalizes the chain
$\{1\}\trianglelefteq \Omega (Z(E))\trianglelefteq \Omega (E)\trianglelefteq E$
from which we infer that
$Z(V_1)\le E$
.
Therefore, to complete the argument, we may assume that
$E\cap Z(Q_1)=Z(S)$
. Then
$Z(Q_1)$
centralizes
$E/\Omega (Z(E))$
and applying Theorem 2.21, we conclude that
$S=EQ_1=\Omega (Z(E))Q_1$
and
$N_S(E)=EZ(Q_1)$
. But
$\Omega (Z(E))\le V_1Q_2$
by Proposition 3.4 and we have that
$Q_2=(Q_1\cap Q_2)\Omega (Z(E))$
. Since
$Q_2/C_{Q_2}(V_2)$
has the structure of a natural
$\mathrm {Sz}(q)$
-module, we conclude that
$E\le Q_2$
.
Now,
$E\le Q_2$
so that
$V_2\le N_S(E)$
. Since
$N_S(E)=EZ(Q_1)$
, we deduce that
$V_2\cap E$
is elementary of order
$q^4$
. In addition, Theorem 2.21 also yields that
$\Phi (E)\cap Z(S)=\{1\}$
, and so
$[E\cap V_2, E]=\{1\}$
and
$E\cap V_2\le \Omega (Z(E))$
. Hence,
$E\le C_{Q_2}(E\cap V_2)$
from which we deduce that
$EC_{Q_2}(V_2)/C_{Q_2}(V_2)=\Omega (Z(E))C_{Q_2}(V_2)/C_{Q_2}(V_2)$
has order q. Furthermore,
$\{1\}=\mho ^1(E)\cap Z(S)\ge \mho ^1(E\cap C_{Q_2}(V_2))$
from which we deduce that
$E\cap C_{Q_2}(V_2)=E\cap V_2\le \Omega (Z(E))$
. All in, we have that E is elementary abelian of order
$q^5$
. Now, applying Lemma 2.17 we see that that E and
$V_2=(E\cap Q_1)Z(Q_1)$
are the only elementary abelian subgroups of
$N_S(E)$
of order
$q^5$
. But now,
$V_2\le N_S(E)\le Q_2$
so that
$Q_2\le N_S(N_S(E))$
. Indeed,
$Q_2$
acts on the set
$\{E, V_2\}$
and since
$Q_2$
normalizes
$V_2$
we must also have that
$Q_2$
normalizes E, a contradiction. Hence,
$Z(V_1)\le E$
.
Proposition 3.6. Let
$\mathcal {F}$
be a saturated fusion system supported on S and
$E\in \mathcal {E}(\mathcal {F})$
. Then
$E\not \le V_1$
.
Proof. Aiming for a contradiction, suppose that
$E\le V_1$
so that
$Z(V_1)\le \Omega (Z(E))$
. Since
$V_1'=Z(V_1)\le E$
, we have that
$V_1\le N_S(E)$
. Indeed,
$[V_1, E]\le Z(V_1)\le \Omega (Z(E))$
and writing
$A_E:=\langle \mathrm {Aut}_{V_1}(E)^{\mathrm {Aut}_{\mathcal {F}}(E)}\rangle $
,
$A_E$
centralizes
$E/\Omega (Z(E))$
. If
$E<V_1$
, then
$A_E$
contains elements of odd order.
Assume that
$|V_2\cap E|>q^4$
. Then
$\Omega (Z(E))\le \Omega (C_{E}(V_2\cap E))=V_2\cap E$
. Then
$V_2$
centralizes the chain
$\{1\}\trianglelefteq \Omega (Z(E))\le E$
so that
$V_2\le E$
by Lemma 2.13. Hence,
$\Omega (Z(E))\le V_2$
. Now, either
$E\le Q_2$
or
$Z(V_1)\le \Omega (Z(E))\le C_{V_2}(E)=Z(V_1)$
. In the former case, since now
$|V_1/E|>2$
, applying Proposition 2.33 we have that
$O^2(O^{2'}(\mathrm {Aut}_{\mathcal {F}}(E)))\le A_E$
centralizes
$E/\Omega (Z(E))$
and so normalizes
$V_2$
. Moreover,
$Q_2\le N_S(E)$
,
$|Q_2/E|\geqslant q^2$
and
$[Q_2, V_2]=Z(S)$
has order q so that applying Theorem 2.21 and Proposition 2.33, using that
$O^2(O^{2'}(\mathrm {Aut}_{\mathcal {F}}(E)))\le \langle \mathrm {Aut}_{Q_2}(E)^{\mathrm {Aut}_{\mathcal {F}}(E)}\rangle $
,
$O^2(O^{2'}(\mathrm {Aut}_{\mathcal {F}}(E)))$
centralizes
$V_2$
, a contradiction for then
$O^2(O^{2'}(\mathrm {Aut}_{\mathcal {F}}(E)))$
centralizes E. In the latter case,
$V_1$
centralizes the chain
$\{1\}\trianglelefteq Z(V_1)\trianglelefteq E$
, and we deduce that
$V_1=E$
so that
$E\trianglelefteq S$
. Indeed, by [Reference Bender3],
$O^{2'}(\mathrm {Out}_{\mathcal {F}}(E))O_{2'}(\mathrm {Out}_{\mathcal {F}}(E))/O_{2'}(\mathrm {Out}_{\mathcal {F}}(E))\cong \mathrm {SU}_3(q)$
. Since
$Z(Q_1)$
has index q in
$Z(V_1)$
and
$|Q_1/E|=q^2$
,
$O^{2'}(\mathrm {Aut}_{\mathcal {F}}(E))=O^2(O^{2'}(\mathrm {Aut}_{\mathcal {F}}(E)))\mathrm {Inn}(E)$
centralizes
$Z(V_1)$
, a contradiction since
$[\mathrm {Aut}_S(E), Z(V_1)]=[S, Z(V_1)]=Z(Q_1)$
. Hence,
$|V_2\cap E|\leqslant q^4$
.
If
$V_2\cap E=Z(V_1)$
, then
$|V_2E/E|=q^2$
and since
$m_2(E)\leqslant m_2(S)=5n$
, we have that
$Z(V_1)$
has index at most
$q^2$
in
$\Omega (Z(E))$
and is centralized by
$V_2$
. By Theorem 2.21,
$\Omega (Z(E))=\Omega (E)\in \mathcal {A}(E)\subseteq \mathcal {A}(S)$
,
$N_S(E)=EV_2$
and
$\Omega (Z(E))/C_{\Omega (Z(E))}(O^2(\mathrm {Aut}_{\mathcal {F}}(E)))$
is a natural
$\mathrm {SL}_2(q^2)$
-module. In particular, for any
$x\in N_S(E)\setminus E$
,
$C_{\Omega (Z(E))}(x)=Z(V_1)$
. Since
$U_1\not \le E$
, else
$E\cap V_2\ge U_1\cap V_2>Z(V_1)$
, we must have that
$U_1\cap E=Z(V_1)$
and
$N_S(E)=EU_1=EV_2$
. Then
$$ \begin{align*} [V_2, \Omega(Z(E))]&=[N_S(E), \Omega(Z(E))]=[U_1E, \Omega(Z(E))]=[U_1, \Omega(Z(E))]\\ &\le [U_1, V_1]=Z(Q_1). \end{align*} $$
Since
$V_2/Z(S)$
has the structure of a natural
$\mathrm {Sz}(q)$
-module for
$L_2/Q_2$
, we deduce that
$\Omega (Z(E))\le Q_2$
and
$[\Omega (Z(E)), V_2]\le Z(S)$
has order at most q, contradicting the module structure of
$\Omega (Z(E))$
.
Hence,
$V_2\cap E>Z(V_1)$
. Again, since
$V_2/Z(S)$
has the structure of a natural
$\mathrm {Sz}(q)$
-module, we have that
$\Omega (Z(E))\le Q_2$
. Then
$[V_2, \Omega (Z(E))]\le Z(S)$
has order at most q and Theorem 2.21 gives
$|V_2\cap E|=q^4$
. Indeed,
$V_1\le N_S(E)=V_2E\le V_1$
. Then
$Z(S)=[V_2\cap E, E\cap Q_2]\le E'$
so that
$V_2$
centralizes the chain
$\{1\}\trianglelefteq E'\trianglelefteq E'\Omega (Z(E))\trianglelefteq E$
, a contradiction by Lemma 2.13. Hence,
$E\not \le V_1$
.
Lemma 3.7. Let
$\mathcal {F}$
be a saturated fusion system supported on S and
$E\in \mathcal {E}(\mathcal {F})$
. If
$E\le V_1(Q_1\cap Q_2)$
, then
$V_2\le E$
.
Proof. Aiming for a contradiction, suppose that
$E\le V_1(Q_1\cap Q_2)$
and
$V_2\not \le E$
. By Proposition 3.6, we have that
$E\not \le V_1$
. Note that
$\Omega (E)\le E\cap \Omega (Q_1)=E\cap V_1$
,
$Z(V_1)\le \Omega (Z(\Omega (E)))$
and
$\Omega (Z(E))\cap Z(V_1)=Z(Q_1)$
. Set
$B_E:=\langle \mathrm {Aut}_{V_2}(E)^{\mathrm {Aut}_{\mathcal {F}}(E)}\rangle $
. Note that if
$|V_2\cap E|>q^4$
, then as
$V_2\cap E\le \Omega (E)$
,
$Z(V_1)\le \Omega (Z(\Omega (E))\le \Omega (C_E(V_2\cap E))=V_2$
. Hence,
$V_2$
centralizes the chain
$\{1\}\trianglelefteq \Omega (Z(\Omega (E)))\trianglelefteq E$
, a contradiction. Hence,
$|V_2\cap E|\leqslant q^4$
and by Proposition 2.33 since
$q>2$
we have that
$O^2(O^{2'}(\mathrm {Aut}_{\mathcal {F}}(E)))\le B_E$
.
If
$O^2(B_E)$
centralizes
$E/\Omega (Z(E))$
, then it normalizes
$Z(V_1)\Omega (Z(E))$
and so normalizes
$[Z(V_1), E]=Z(S)$
. Indeed, if
$\Omega (Z(E))\le Q_2$
, then as
$[V_2, Q_2]=Z(S)$
, we have that
$O^2(B_E)$
centralizes the chain
$\{1\}\trianglelefteq Z(S)\trianglelefteq \Omega (Z(E))\trianglelefteq E$
, a contradiction. Then
$\Omega (Z(E))\not \le Q_2$
so that
$V_2\cap E=Z(V_1)$
. But now
$V_2$
centralizes
$(\Omega (Z(E))\cap Q_2)/Z(S)$
and since
$|V_2E/E|=q^2$
, Theorem 2.21 implies that
$O^2(B_E)$
centralizes
$\Omega (Z(E))/Z(S)$
, and we arrive at the same contradiction as before. Hence,
$O^2(B_E)$
is nontrivial on
$E/\Omega (Z(E))$
.
Now,
$E\cap Q_2$
has index at most q in E and since
$|V_2E/E|\geqslant q$
,
$Z(S)\le \Omega (Z(E))$
and
$[V_2, Q_2]=Z(S)$
, applying Theorem 2.21 we have that
$|V_2E/E|=q$
,
$N_S(E)=EV_2$
and
$V_1Q_2=EQ_2$
. Unless
$V_2\cap E=[V_2, Q_1]$
, we have that
$Z(V_1)=Z(Q_1)[V_2\cap E, E]$
so that
$C_E(E')=C_E(E'Z(Q_1))\le C_E(Z(V_1))=E\cap V_1$
. Since
$[U_1, E]\le [U_1, Q_1]=Z(V_1)\le C_E(E')\le V_1$
and
$[U_1, V_1]=Z(Q_1)\le \Omega (Z(E))$
, we have that
$U_1$
centralizes the chain
$\Omega (Z(E))\trianglelefteq C_E(E')\trianglelefteq E$
. Since
$O^2(B_E)$
is nontrivial on
$E/\Omega (Z(E))$
and applying Proposition 2.33 since
$|V_2E/E|=q$
, we infer that
$U_1\le E$
. But then
$[V_2, Q_1]\le U_1\le E$
and
$|V_2\cap E|>q^4$
, a contradiction.
Therefore, if
$V_2\not \le E$
, then
$V_2\cap E=[V_2, Q_1]$
and
$\Omega (Z(\Omega (E)))\le \Omega (C_S(V_2\cap E))=\Omega (U_1C_{Q_2}(V_2))=U_1V_2$
. Since
$Z(V_1)\le \Omega (Z(\Omega (E)))\cap V_2$
, if
$\Omega (Z(\Omega (E)))\le V_2$
, then
$V_2$
centralizes the chain
$\{1\}\trianglelefteq \Omega (Z(\Omega (E)))\trianglelefteq E$
, a contradiction since
$V_2\not \le E$
. It follows by Lemma 2.17 that
$\Omega (Z(\Omega (E)))\le U_1$
,
$U_1$
centralizes a chain and
$U_1\le \Omega (E)$
. But then
$[V_2, E]\le Z(V_1)\le \Omega (Z(\Omega (E)))\le U_1$
and
$[V_2, \Omega (Z(\Omega (E)))]\le [V_2, U_1]=Z(S)\le \Omega (Z(E))$
and finally, we have that
$O^2(B_E)$
centralizes
$E/\Omega (Z(E))$
, a contradiction. Hence,
$V_2\le E$
.
Proposition 3.8. Let
$\mathcal {F}$
be a saturated fusion system supported on S and
$E\in \mathcal {E}(\mathcal {F})$
. Then
$E\not \le V_1(Q_1\cap Q_2)$
.
Proof. We assume throughout that
$E\le V_1(Q_1\cap Q_2)$
so that by Lemma 3.7,
$V_2\le E$
. In particular,
$\Omega (Z(E))\le \Omega (C_S(V_2))=V_2$
. If
$E\le Q_2$
, then
$Q_2\le N_S(E)$
and
$[Q_2, \Omega (Z(E))]=Z(S)$
. Applying Theorem 2.21 and Proposition 2.33, either
$E=Q_1\cap Q_2$
or
$\mathrm {Aut}_S(E)\le O^2(\langle \mathrm {Aut}_{Q_2}(E)^{\langle \mathrm {Aut}_{\mathcal {F}}(E)\rangle })\mathrm {Inn}(E)$
centralizes
$\Omega (Z(E))$
. In the latter case, we get that
$Z(S)=[\mathrm {Aut}_{Q_2}(E), Z(Q_1)]\le [\mathrm {Aut}_{Q_2}(E), \Omega (Z(E))]=\{1\}$
, absurd. Hence, if
$E\le Q_2$
, then
$E=Q_1\cap Q_2\trianglelefteq S$
. Then
$S/E$
is a group of exponent strictly greater than
$2$
and has
$V_1Q_2/E\le Z(S/E)$
. However, by Proposition 2.33,
$S/E$
is isomorphic to a Sylow
$2$
-subgroup of
$\mathrm {SL}_2(q)$
,
$\mathrm {Sz}(q)$
or
$\mathrm {SU}_3(q)$
, a contradiction.
Hence, if
$E\le V_1(Q_1\cap Q_2)$
, then
$V_2\le E$
,
$E\not \le Q_2$
and
$\Omega (Z(E))\le C_{V_2}(E)\le Z(V_1)$
. Moreover,
$Z(V_1)\le \Omega (Z(\Omega (E)))\le V_2$
and since
$[U_1, E]\le [U_1, Q_1]=Z(V_1)$
,
$U_1$
centralizes the chain
$\{1\}\trianglelefteq \Omega (Z(E))\trianglelefteq \Omega (Z(\Omega (E)))\trianglelefteq E$
and
$U_1\le E$
. Since
$[E, V_1]\le [Q_1, V_1]=U_1\le E$
, we have that
$V_1\le N_S(E)$
. Then
$U_1V_2\le E$
and we further have that
$Z(V_1)\le \Omega (Z(\Omega (E)))\le [V_2, Q_1]$
and
$[V_2, Q_1, V_1]=Z(Q_1)\le \Omega (Z(E))\le Z(V_1)$
. Hence,
$V_1$
centralizes the chain
$\{1\}\trianglelefteq \Omega (Z(E))\trianglelefteq \Omega (Z(\Omega (E)))\trianglelefteq \Omega (E)\trianglelefteq E$
so that
$V_1\le E$
. Since
$E\not \le V_1$
, we have that
$\Omega (Z(E))=Z(Q_1)$
and
$Z(V_1)=\Omega (Z(\Omega (E)))$
. Then
$U_1$
is the preimage in
$V_1$
of
$Z(V_1/Z(Q_1))$
and since
$V_1=\Omega (E)$
is characteristic in E, so too is
$U_1$
. But now
$Q_1$
centralizes the chain
$\{1\}\trianglelefteq Z(Q_1)\trianglelefteq Z(V_1)\trianglelefteq U_1\trianglelefteq V_1\trianglelefteq E$
, a contradiction since
$Q_1\not \le E$
.
Proposition 3.9. Let
$\mathcal {F}$
be a saturated fusion system supported on S and
$E\in \mathcal {E}(\mathcal {F})$
. If
$E\le Q_1$
, then
$E=Q_1$
.
Proof. By Proposition 3.8, any essential subgroup E of
$\mathcal {F}$
which is contained in
$Q_1$
is not contained in
$V_1(Q_1\cap Q_2)$
. Since
$Q_2/C_{Q_2}(V_2)$
has the structure of a natural
$\mathrm {Sz}(q)$
-module, we deduce that
$\Omega (Z(E))\le \Omega (U_1C_{Q_2}(V_2))=U_1V_2$
. Hence,
$[[V_2, Q_1], \Omega (Z(E))]=\{1\}$
. But now,
$[[V_2, Q_1], E]=Z(V_1)\le \Omega (E)\le V_1$
,
$[[V_2, Q_1], \Omega (E)]\le [[V_2, Q_1], V_1]=Z(Q_1)\le \Omega (Z(E))$
and Lemma 2.13 implies that
$[V_2, Q_1]\le E$
.
Assume that
$Z(Q_1)\not \le E'$
. Since
$E\not \le V_1Q_2$
, we have that
$Z(Q_1)=[[V_2, Q_1], E, E]Z(S)$
, and so we must have that
$Z(S)\not \le E'$
. Hence,
$E\cap Q_2$
centralizes
$[V_2, Q_1]$
so that
$E\cap Q_2\le U_1C_{Q_2}(V_2)$
. If
$V_2\cap E>[V_2, Q_1]$
, then
$\Omega (Z(E))\le \Omega (Z(\Omega (E)))\le \Omega (C_S(V_2\cap E))=V_2$
. Since
$[V_2, \Omega (E)]\le [V_2, V_1]=Z(V_1)\le \Omega (Z(\Omega (E)))$
,
$V_2$
centralizes the chain
$\{1\}\trianglelefteq \Omega (Z(\Omega (E)))\trianglelefteq \Omega (E)\trianglelefteq E$
and
$V_2\le E$
. Indeed, since
$E\not \le V_1Q_2$
,
$Z(Q_1)=\Omega (Z(E))$
. Since
$Z(S)\not \le E'$
, we have that
$[E\cap Q_2, V_2]=\{1\}$
so that
$E\cap Q_2\le C_{Q_2}(V_2)$
. In particular,
$U_1\cap E=[V_2, Q_1]$
. But
$U_1$
centralizes the chain
$\{1\}\trianglelefteq \Omega (Z(E))\trianglelefteq \Omega (E)\trianglelefteq E$
, a contradiction.
Hence, if
$Z(Q_1)\not \le E'$
, then
$[V_2, Q_1]=V_2\cap E$
. Indeed,
$V_2\cap E\le \Omega (E)$
and
$Z(V_1)\le \Omega (Z(\Omega (E)))\le Q_2$
. In particular,
$V_2$
centralizes the chain
$\Omega (Z(E))\trianglelefteq \Omega (Z(\Omega (E)))\trianglelefteq \Omega (E)\trianglelefteq E$
and so
$C_{V_2}(\Omega (Z(E)))=V_2\cap E$
by Lemma 2.13. Since
$[\Omega (Z(E)), V_2]\le Z(S)$
and
$|V_2/V_2\cap E|=q$
, applying Theorem 2.21 we deduce that
$[\Omega (Z(E)), V_2]=Z(S)$
,
$N_S(E)=V_2E$
and
$\Omega (Z(E))\cap V_2$
has index q in
$\Omega (Z(E))$
. Since
$\Omega (Z(E))\le \Omega (U_1C_{Q_2}(V_2))=U_1V_2$
and
$\Omega (Z(E))(V_2\cap E)$
is elementary abelian of order
$q^5$
,
$\Omega (Z(E))\in \mathcal {A}(U_1V_2)$
. By Lemma 2.17,
$U_1=\Omega (Z(E))(V_2\cap E)\le E$
. Since
$[E, V_1]\le [Q_1, V_1]=U_1\le E$
,
$V_1\le N_S(E)$
. But
$V_1\cap Q_2\le N_S(E)\cap Q_2=V_2(E\cap Q_2)\le U_1C_{Q_2}(V_2)<V_1\cap Q_2$
, and we have a contradiction.
Therefore,
$Z(Q_1)\le E'$
and since
$[E, U_1]\le [Q_1, U_1]=Z(V_1)$
and
$[\Omega (E), U_1]\le Z(Q_1)$
, we have by Lemma 2.13 that
$U_1\le E$
. In particular,
$U_1\le \Omega (E)$
and
$Z(V_1)\le \Omega (Z(\Omega (E)))\le U_1$
. Since
$[\Omega (E), V_2]\le [V_1, V_2]=Z(V_1)\le \Omega (Z(\Omega (E)))$
and
$[\Omega (Z(\Omega (E))), V_2]\le [U_1, V_2]=Z(S)\le E'$
, we deduce by Lemma 2.13 that
$V_2\le E$
. As before,
$V_2\le \Omega (E)$
and
$\Omega (Z(\Omega (E)))\le V_2\cap U_1=[V_2, Q_1]$
. Moreover,
$\Omega (Z(E))\le C_{V_2}(E)$
and since
$E\not \le V_1Q_2$
,
$\Omega (Z(E))=Z(Q_1)$
. But now,
$[E, V_1]\le [Q_1, V_1]=U_1\le \Omega (E)$
,
$[\Omega (E), V_1]\le V_1'=Z(V_1)\le \Omega (Z(\Omega (E)))$
,
$[\Omega (Z(\Omega (E))), V_1]\le [[V_2, Q_1], V_1]=Z(Q_1)=\Omega (Z(E))$
, and so Lemma 2.13 implies that
$V_1\le E$
. In fact,
$V_1=\Omega (E)$
and
$Z(V_1)=Z(\Omega (E))$
. Then
$U_1$
is the preimage in
$V_1$
of
$Z(V_1/Z(Q_1))$
and as
$V_1$
and
$Z(Q_1)$
are characteristic in E, so too is
$U_1$
. But then
$Q_1$
centralizes the chain
$\{1\}\trianglelefteq Z(Q_1)\trianglelefteq Z(V_1)\trianglelefteq U_1\trianglelefteq V_1\trianglelefteq E$
and Lemma 2.13 implies that
$E=Q_1$
, as desired.
Proposition 3.10. Suppose that
$E\in \mathcal {E}(\mathcal {F})$
and
$E\le Q_2$
. Then
$E=Q_2$
.
Proof. Suppose first that
$V_2\not \le E$
. Note that if
$E\cap V_2\not \le Z(E)$
, then
$Z(S)=[E, E\cap V_2]\le \Phi (E)$
so that
$[V_2, E]\le \Phi (E)$
, a contradiction by Lemma 2.13. Thus,
$Z(V_1)\le E\cap V_2\le Z(E)$
and we have that
$E\le V_1$
. By Proposition 3.6, this is a contradiction. Hence,
$V_2\le E$
. Indeed,
$\Omega (Z(E))\le \Omega (C_E(V_2))=V_2$
and as
$[Q_2, E]\le Q_2'=V_2\le E$
,
$Q_2\le N_S(E)$
.
We aim to show that
$Z(S)$
is normalized by
$O^{2'}(\mathrm {Aut}_{\mathcal {F}}(E))$
. Firstly, if
$|Q_2/E|<q$
, then it is clear that
$Z(S)=Z(E)$
. Hence,
$m_2(\mathrm {Out}_S(E))\geqslant 2$
and applying Proposition 2.33, we have that
$O^{2'}(\mathrm {Aut}_{\mathcal {F}}(E))=\langle \mathrm {Aut}_{Q_2}(E)^{\mathrm {Aut}_{\mathcal {F}}(E)}\rangle $
. Now,
$[Q_2, \Omega (Z(E))]\le Z(S)$
and we conclude by Theorem 2.21 that either
$|Q_2/E|=q$
or
$O^{2'}(\mathrm {Aut}_{\mathcal {F}}(E))$
is trivial on
$\Omega (Z(E))$
. In the latter case, it is clear that
$Z(S)$
is normalized by
$O^{2'}(\mathrm {Aut}_{\mathcal {F}}(E))$
. In the former case, we must have that
$|Q_2/E|=q$
and
$C_{Q_2}(V_2)\le E$
, else
$\Omega (Z(E))=Z(S)$
and
$O^{2'}(\mathrm {Out}_{\mathcal {F}}(E))$
is trivial on
$\Omega (Z(E))$
. It follows that
$|\Omega (Z(E))|=q^2$
,
$E=C_{Q_2}(\Omega (Z(E)))$
and Theorem 2.21 gives that
$O^{2'}(\mathrm {Out}_{\mathcal {F}}(E))\cong \mathrm {SL}_2(q)$
and
$N_S(E)=Q_2$
. Note that if
$E(V_1\cap Q_2)<Q_2$
, then
$\Omega (Z(E))\le C_{V_2}(E\cap V_1)\le Z(V_1)$
and as
$E=C_{Q_2}(\Omega (Z(E)))$
, we have that
$V_1\cap Q_2\le E$
. But then
$V_1\le N_S(E)$
, a contradiction. Hence,
$Q_2=E(V_1\cap Q_2)$
so that
$V_2=[Q_2, C_{Q_2}(V_2)]=[E, C_{Q_2}(V_2)][V_1\cap Q_2, C_{Q_2}(V_2)]=E'Z(V_1)$
. Since
$|\Omega (Z(E))|=q^2$
, we deduce that
$Z(S)=[E, E']$
is characteristic in E and normalized by
$O^{2'}(\mathrm {Aut}_{\mathcal {F}}(E))$
, and
$O^{2'}(\mathrm {Aut}_{\mathcal {F}}(E))$
acts trivially on
$Z(S)$
.
Hence,
$O^{2'}(\mathrm {Aut}_{\mathcal {F}}(E))$
is trivial on
$Z(S)$
in all cases. Since
$Q_2\le N_S(E)$
and
$\Omega (Z(E))\le V_2$
, we deduce that
$\Omega (Z(E))\le C_{V_2}(Q_2)$
and
$\Omega (Z(E))=Z(S)$
. Since
$L_2$
acts trivially on
$C_{Q_2}(V_2)/V_2$
, for any subgroup A with
$V_2\le A\le C_{Q_2}(V_2)$
,
$C_{Q_2}(A/Z(S))\trianglelefteq L_2$
. Since
$C_{Q_2}(V_2)\le C_{Q_2}(A/Z(S))$
and
$V_2=Z(Q_2/Z(S))$
, if
$V_2<A$
, then we have that
$C_{Q_2}(A/Z(S))=C_{Q_2}(V_2)$
. Hence, either
$V_2$
is the preimage in E of
$Z(E/Z(S))$
or
$E\cap C_{Q_2}(V_2)=V_2$
. Even in this latter case, since
$Q_2$
centralizes
$V_2/Z(S)$
, we must have by Theorem 2.21 that
$|Q_2/E|\leqslant |E/V_2|$
and since
$|Q_2/V_2|=q^5$
, we certainly have that
$|E/V_2|\geqslant q^2$
. One can calculate that
$V_2$
is the preimage in E of
$Z(E/Z(S))$
in this case also. Hence,
$V_2$
is
$O^{2'}(\mathrm {Aut}_{\mathcal {F}}(E))$
-invariant and
$Q_2$
centralizes the chain
$\{1\}\trianglelefteq Z(S)\trianglelefteq V_2\trianglelefteq E$
so that
$E=Q_2$
, as desired.
For the remainder of this section, ultimately aiming for a contradiction, we let E be an essential subgroup of
$\mathcal {F}$
distinct from
$Q_1$
and
$Q_2$
and chosen maximally with respect to this condition. Hence, E is contained in no other essential subgroups of
$\mathcal {F}$
, and so, applying the Alperin–Goldschmidt theorem and using that E is receptive in
$\mathcal {F}$
, for any
$\alpha \in \mathrm {Aut}_{\mathcal {F}}(E)$
which lifts to a subgroup of S strictly larger than E,
$\alpha $
may be lifted to some
$\widehat {\alpha }\in \mathrm {Aut}_{\mathcal {F}}(S)$
.
Lemma 3.11.
$V_2\le E$
.
Proof. Aiming for a contradiction we assume that
$V_2\not \le E$
. Since
$E\not \le Q_1$
, we have that
${Z(S)=[Z(Q_1), E]\le E'}$
. Suppose first that
$E\not \le V_1Q_2$
. Then
$Z(Q_1)=Z(S)[Z(V_1), E]\le E'$
since
$V_2/Z(S)$
has the structure of a natural
$\mathrm {Sz}(q)$
-module. Since
$\Omega (E)E'\le V_1Q_2$
and
$[V_2, Q_1, V_1Q_2]=Z(Q_1)$
, we have that
$[V_2, Q_1]$
centralizes the chain
$E'\trianglelefteq \Omega (E)E'\trianglelefteq E$
and
$[V_2, Q_1]\le E$
. Then
$Z(V_1)=[V_2, Q_1, E]\le E'$
and now
$V_2$
centralizes the chain
$E'\le \Omega (E)E'\trianglelefteq E$
, a contradiction. Hence, if
$V_2\not \le E$
, then
$E\le V_1Q_2$
.
Now,
$[V_2, E]\le Z(V_1)\le E$
so that
$V_2\le N_S(E)$
. Since
$E\not \le Q_1$
,
$Z(S)\le [E, Z(Q_1)]\le E'$
. Then
$E\cap Q_2$
has index at most q in E and as
$[V_2, E\cap Q_2]\le Z(S)\le E'$
, we deduce by Theorem 2.21 that
$|V_2\cap E|\geqslant q^4$
. Set
$A_E:=\langle \mathrm {Aut}_{V_2}(E)^{\mathrm {Aut}_{\mathcal {F}}(E)}\rangle $
.
Suppose that
$|V_2\cap E|=q^4$
so that, by Theorem 2.21, we have
$EQ_2=V_1Q_2$
,
$N_S(E)=EV_2$
and
$O^{2'}(\mathrm {Out}_{\mathcal {F}}(E))\cong \mathrm {SL}_2(q)$
. Assume, in addition, that
$\Omega (Z(E))$
contains an
$\mathrm {Out}_{\mathcal {F}}(E)$
-chief factor. Then
$|\Omega (Z(E))C_{Q_2}(V_2)/C_{Q_2}(V_2)|\geqslant q$
and since
$\Omega (Z(E))(E\cap V_2)$
is elementary abelian,
$|\Omega (Z(E))C_{Q_2}(V_2)/C_{Q_2}(V_2)|=q$
and
$\Omega (Z(E))(E\cap V_2)\in \mathcal {A}(S)$
. Hence,
$\mathcal {A}_{\trianglelefteq }(E)\ne \emptyset $
. Suppose that
$A\in \mathcal {A}_{\trianglelefteq }(E)$
with
$A\not \le Q_1$
. Then
$(A\cap Q_1)Z(Q_1)\in \mathcal {A}(S)$
and
$A\cap Z(Q_1)=Z(S)$
from which we deduce that
$S=AQ_1$
. But then
$C_{V_1/U_1}(A)=U_1V_2/U_1$
and since
$\Omega (Z(E))\le V_1$
, we have that
$\Omega (Z(E))\le U_1V_2$
. In particular,
$\Omega (Z(E))(V_2\cap E)\le U_1V_2$
and by Lemma 2.17 we have that
$\Omega (Z(E))(V_2\cap E)=U_1$
and
$V_2\cap E=[V_2, Q_1]$
. But
$E\not \le Q_1$
and so
$[V_2, Q_1]=Z(V_1)[E, U_1]=Z(V_1)[E, \Omega (Z(E))(V_2\cap E)]\le Z(V_1)$
, a contradiction. Hence,
$J_{\trianglelefteq }(E)$
is characteristic in E and contained in
$\Omega (Q_1)=V_1$
. Then
$[V_2, E]\le Z(V_1)\le Z(J_{\trianglelefteq }(E))$
. Since
$\Omega (Z(E))(V_2\cap E)\in \mathcal {A}_\trianglelefteq (E)$
,
$Z(J_{\trianglelefteq }(E))$
centralizes
$V_2\cap E$
and so is contained in
$Q_2$
. But then
$[V_2, Z(J_{\trianglelefteq }(E))]\le Z(S)\le E'$
, and so by Lemma 2.13, we have a contradiction.
Therefore, if
$|V_2\cap E|=q^4$
, then
$\Omega (Z(E))$
contains no noncentral
$\mathrm {Out}_{\mathcal {F}}(E)$
-chief factors. Since E is maximally essential,
$\mathrm {Aut}_{\mathcal {F}}(E)$
normalizes
$Z(S)$
. On the other hand, if
$|V_2\cap E|>q^4$
, then
$\Omega (Z(E))\le \Omega (C_{Q_2}(V_2\cap E))=V_2$
and
$A_E$
centralizes
$\Omega (Z(E))$
. Hence,
$Z(S)$
is normalized by
$A_E$
in all circumstances. Set A to be the preimage in E of
$Z(E/Z(S))$
so that
$Z(V_1)\le A$
. In particular,
$C_E(A)\le E\cap V_1$
. Since
$|V_2\cap E|\geqslant q^4$
, we have that
$A\le Q_2$
so that
$V_2$
centralizes
$A/Z(S)$
and
$Z(S)$
. Hence, there are elements of
$A_E$
of odd order which centralize A. By the three subgroups lemma, such automorphism also centralize
$E/C_E(A)$
. But now,
$[V_2, C_E(A)]\le [V_2, V_1]=Z(V_1)\le A$
, and so there are odd order elements which centralize
$E/C_E(A)A$
,
$C_E(A)A/A$
,
$A/Z(S)$
and
$Z(S)$
, a contradiction.
Since
$V_2\le E$
,
$V_2\in \mathcal {A}(S)$
and
$V_2\trianglelefteq S$
, we have that
$\emptyset \ne \mathcal {A}_{\trianglelefteq }(E)\subset \mathcal {A}(S)$
. In particular, the group
$J_{\trianglelefteq }(E)$
is well defined.
Lemma 3.12.
$E\le V_1Q_2$
.
Proof. We suppose throughout that
$E\not \le V_1Q_2$
. Since
$V_2\le E$
, we have that
$\Omega (Z(E))\le \Omega (C_S(V_2))=V_2$
. Moreover,
$V_2/Z(S)$
has the structure of a natural
$\mathrm {Sz}(q)$
-module so that
$\Omega (Z(E))\le Z(Q_1)$
. Finally,
$E\not \le Q_1$
so that
$\Omega (Z(E))=Z(S)$
. Let
$A\in \mathcal {A}_{\trianglelefteq }(E)$
. If
$A\not \le Q_1$
, then
$(A\cap Q_1)Z(Q_1)$
is also elementary abelian and
$A\cap Z(Q_1)=Z(S)$
. We deduce that
$S=AQ_1$
. Indeed, we have that
$Q_2=A(Q_1\cap Q_2)$
. Since there is
$x\in E\setminus V_1Q_2$
and
$Q_2/C_{Q_2}(V_2)$
has the structure of a natural
$\mathrm {Sz}(q)$
-module, we deduce that
$Q_2=AC_{Q_2}(V_2)$
. But then
$[A, C_{Q_2}(V_2)]\le A\cap C_{Q_2}(V_2)=Z(S)$
and
$[Q_2, C_{Q_2}(V_2)]=Z(S)$
, a contradiction. Hence, if
$A\in \mathcal {A}_\trianglelefteq (E)$
, then
$A\le \Omega (Q_1)=V_1$
. Then
$Z(V_1)\le \Omega (Z(J_{\trianglelefteq }(E)))\le \Omega (C_E(V_2))=V_2$
. Since
$E\not \le V_1Q_2$
and
$V_2/Z(S)$
has the structure of a natural
$\mathrm {Sz}(q)$
-module, we have that
$Z(Q_1)$
is the intersection of
$\Omega (Z(J_{\trianglelefteq }(E)))$
and the preimage in E of
$Z(E/\Omega (Z(E))$
, and so
$Z(Q_1)$
is characteristic in E.
Now,
$C_E(Z(Q_1))=E\cap Q_1$
is characteristic in E. Then
$[U_1, E]\le E\cap Q_1$
,
$[U_1, E\cap Q_1]\le Z(V_1)\le \Omega (E\cap Q_1)\le E\cap V_1$
,
$[U_1, \Omega (E\cap Q_1)]\le Z(Q_1)$
and
$[U_1, Z(Q_1)]=\{1\}$
so that
$U_1\le E$
. Now,
$U_1V_2\le J_{\trianglelefteq }(E)\le V_1$
and we have that
$Z(V_1)\le Z(J_{\trianglelefteq }(E))\le [V_2, Q_1]$
. Then
$V_1$
centralizes the chain
$\{1\}\trianglelefteq Z(Q_1)\trianglelefteq Z(J_{\trianglelefteq }(E))\trianglelefteq J_{\trianglelefteq }(E)\trianglelefteq E\cap Q_1$
. If
$V_1\not \le E$
, then
$N_{EV_1}(E)>E$
and
$N_{EV_1}(E)$
normalizes the chain
$\{1\}\trianglelefteq \dots \trianglelefteq E\cap Q_1\trianglelefteq E$
, a contradiction. Hence,
$V_1\le E$
. In particular,
$V_1=\Omega (E\cap Q_1)$
is characteristic in E. Now,
$U_1$
is the preimage in
$V_1$
of
$Z(V_1/Z(Q_1))$
and since
$V_1$
and
$Z(Q_1)$
are characteristic in E so too is
$U_1$
. Then
$Q_1$
centralizes the chain
$\{1\}\trianglelefteq Z(Q_1)\trianglelefteq Z(V_1)\trianglelefteq U_1\trianglelefteq V_1\trianglelefteq E\cap Q_1$
and
$Q_1\le E$
. Hence,
$V_1=\Phi (Q_1)\le \Phi (E)\le \Phi (S)=[Q_1, x]\Phi (Q_1)$
for all
$x\in S\setminus Q_1$
. Since
$E\not \le Q_1$
, we have that
$\Phi (E)=\Phi (S)$
and S centralizes
$E/\Phi (E)$
, a final contradiction.
Lemma 3.13.
$V_1\not \le E$
.
Proof. Aiming for a contradiction, let
$V_1\le E$
. Since
$E\ne Q_1$
, we have that
$E\not \le Q_1$
by Proposition 3.9. Let
$A\in \mathcal {A}_{\trianglelefteq }(E)$
and
$A\not \le V_1$
. Then
$A\not \le Q_1$
and since A is abelian,
$A\cap Z(Q_1)=Z(S)$
. Hence,
$(A\cap Q_1)Z(Q_1)\in \mathcal {A}(Q_1)$
and we deduce that
$S=AQ_1$
. Moreover, since
$(A\cap Q_1)Z(Q_1)\in \mathcal {A}(V_1)$
, we have that
$Z(V_1)\le AZ(Q_1)$
so that
$|A\cap Z(V_1)|=q^2$
. Since
$U_1/Z(V_1)$
has the structure of a natural
$\mathrm {SL}_2(q)$
-module for
$L_1/Q_1$
, we have that
$|A\cap U_1|\geqslant |[A, U_1](A\cap Z(V_1))|\geqslant q^3$
. Finally, since
$|[A, V_1]U_1/U_1|>q$
and since
$V_1\le E\le N_S(A)$
, we have that
$|A|>q^5$
, a contradiction since
${m_2(S)=5n}$
. Thus,
$J_{\trianglelefteq }(E)$
is a characteristic subgroup of E contained in
$V_1$
. Then
$Z(V_1)\le Z(J_{\trianglelefteq } (E))\le [V_2, Q_1]$
since
$Z(U_1V_2)=[V_2, Q_2]$
and
$U_1V_2\le J_{\trianglelefteq }(E)$
.
Assume that
$Z(V_1)<Z(J_{\trianglelefteq }(E))$
. Then
$C_S(Z(J_{\trianglelefteq }(E)))=C_E(Z(J_{\trianglelefteq }(E)))=U_1C_{Q_2}(V_2)$
is characteristic in E so that
$U_1V_2=\Omega (C_{Q_2}(V_2))$
is characteristic in E. Then by Lemma 2.17,
$U_1$
and
$V_2$
are the only maximal elementary abelian subgroups of
$U_1V_2$
and since
$U_1, V_2\trianglelefteq S$
, we deduce that both
$U_1$
and
$V_2$
are
$\mathrm {Aut}_{\mathcal {F}}(E)$
-invariant. Then
$Z(S)=[U_1, V_2]$
is
$\mathrm {Aut}_{\mathcal {F}}(E)$
-invariant and
$E\cap Q_2=C_E(V_2/Z(S))$
is also
$\mathrm {Aut}_{\mathcal {F}}(E)$
-invariant. Then
$Q_2$
centralizes the chain
$\{1\}\trianglelefteq Z(S)\trianglelefteq V_2\trianglelefteq E\cap Q_2$
so that
$Q_2\le E$
and
$E=V_1Q_2$
. Then
$Z(S)<[Z(J_{\trianglelefteq }(E)), E]\le Z(Q_1)$
and we get that
$V_1(Q_1\cap Q_2)=C_E([Z(J_{\trianglelefteq }(E)), E])$
is
$\mathrm {Aut}_{\mathcal {F}}(E)$
-invariant. Then
$[Q_1, E]\le V_1(Q_1\cap Q_2)$
and
$[Q_1, V_1(Q_1\cap Q_2)]= V_1\cap Q_2\le \Phi (E)$
and Lemma 2.13 gives a contradiction.
Assume now that
$Z(V_1)=Z(J_{\trianglelefteq }(E))$
so that
$Z(V_1)$
is characteristic in E. Then, as
$V_1<E$
,
$Z(S)=[Z(V_1), E]$
and
$V_1=C_{E}(Z(V_1))$
are also characteristic subgroups of E. Suppose that
$E<V_1Q_2$
. Then
$V_1Q_2$
centralizes
$Z(V_1)/Z(S)$
, and so
$B_E:=\langle \mathrm {Aut}_{V_1Q_2}(E)^{\mathrm {Aut}_{\mathcal {F}}(E)}\rangle $
centralizes
$Z(V_1)/Z(S)$
. Indeed,
$B_E$
normalizes
$Z(Q_1)$
and so normalizes
$E\cap Q_1$
. Furthermore,
$U_1$
is the preimage in
$V_1$
of
$Z(V_1/Z(Q_1))$
so is normalized by
$B_E$
, and then as
$V_1<E\not \le Q_1$
by Proposition 3.9,
$U_1\cap V_2$
is the preimage in
$U_1$
of
$C_{U_1/Z(V_1)}(E)$
, also normalized by
$B_E$
. Then
$V_1Q_2$
centralizes the chain
$\{1\}\trianglelefteq Z(S)\trianglelefteq Z(Q_1)\trianglelefteq U_1\cap V_2\trianglelefteq U_1$
and we deduce that
$O^2(B_E)$
centralizes
$U_1$
. By the three subgroups lemma,
$O^2(B_E)$
centralizes
$E/C_E(U_1)$
and since
$U_1$
is self-centralizing in S, we have that
$O^2(B_E)$
acts trivially on E, a contradiction. Hence,
$E=V_1Q_2$
by Lemma 3.12.
Now,
$U_1V_2$
is the preimage in
$V_1$
of
$C_{V_1/U_1}(S)=C_{V_1/U_1}(E)$
, and so
$C_{V_1/Z(V_1)}(E)=C_{U_1V_2/Z(V_1)}(Q_2)=V_2/Z(V_1)$
so that
$V_2$
is characteristic in E. Furthermore,
$Q_2=C_E(V_2/Z(S))$
is also characteristic in E. Since
$S/E$
is elementary abelian of order
$q>2$
and E is maximally essential, Proposition 2.33 yields that
$O^{2'}(\mathrm {Out}_{\mathcal {F}}(E))\cong \mathrm {SL}_2(q)$
. Let
$G_E$
a model for E and
$L_E:=O^{2'}(G_E)$
so that
$L_E/E\cong \mathrm {SL}_2(q)$
. Then S centralizes
$E/Q_2$
so that
$L_E/Q_2$
is a central extension of
$L_E/E\cong \mathrm {SL}_2(q)$
by
$E/Q_2$
which is elementary abelian of order q. Since
$S/Q_2\cong T\in \mathrm {Syl}_2(\mathrm {Sz}(q))$
, this is a nonsplit extension. However, the Schur multiplier of
$\mathrm {SL}_2(q)$
when
$q=2^n>4$
is trivial, and so we have a contradiction. Hence,
$V_1\not \le E$
.
We now demonstrate that the proposed E does not exist and the only possibilities for the essential subgroups of any saturated fusion system on S are
$Q_1$
and
$Q_2$
. By Proposition 3.9 and Proposition 3.10, we may assume that that
$E\not \le Q_1$
and
$E\not \le Q_2$
and we let E be maximally essential in
$\mathcal {F}$
because of this. Since
$V_2\le E$
by Lemma 3.11 and
$\Omega (C_S(V_2))=V_2$
, we have that
$\Omega (Z(E))\le V_2$
. Since
$E\not \le Q_2$
, we have that
$\Omega (Z(E))\le Z(V_1)$
.
Proposition 3.14. Let
$\mathcal {F}$
be a saturated fusion system supported on S. Then
$\mathcal {E}(\mathcal {F})\subseteq \{Q_1, Q_2\}$
.
Proof. By Proposition 3.9 and Proposition 3.10, aiming for a contradiction, it suffices to let E be an essential subgroup of
$\mathcal {F}$
chosen maximally such that
$E\not \le Q_1$
and
$E\not \le Q_2$
. By Lemma 3.13,
$V_1\not \le E$
. Setting
$A_E:=\langle \mathrm {Aut}_{V_1}(E)^{\mathrm {Aut}_{\mathcal {F}}(E)}\rangle $
,
$A_E$
contains elements of odd order, and since
$\Omega (Z(E))\le Z(V_1)$
,
$A_E$
acts trivially on
$Z(S)$
. Hence,
$A_E$
normalizes
$P_E$
, the preimage in E of
$Z(E/Z(S))$
. Moreover, since
$V_2\le E\not \le Q_2$
, we have that
$Z(V_1)\le P_E\le V_1\cap Q_2$
and
$P_E\cap V_2=Z(V_1)$
. For
$C_E:=C_{A_E}(P_E)$
, by the three subgroups lemma,
$[C_E, E]\le C_E(P_E)\le V_1$
. Then
$[V_1, C_E(P_E)]\le Z(V_1)\le P_E$
from which it follows that
$[C_E, C_E(P_E)]\le P_E$
, and so
$O^2(C_E)$
centralizes E and
$C_E$
is a
$2$
-group. Hence,
$A_E/\mathrm {Inn}(E)$
acts faithfully on
$P_E$
.
Assume first that
$E\cap C_{Q_2}(V_2)=V_2$
, noting that
$m_2(E)=m_2(S)$
. If
$J_{\trianglelefteq }(E)\le Q_1$
, then
$J_{\trianglelefteq }(E)\le V_1$
. Since
$V_2\le J_{\trianglelefteq }(E)$
, we have that
$Z(V_1)\le \Omega (Z(J_{\trianglelefteq }(E))\le V_2$
. Then
$C_{Q_2}(V_2)$
centralizes the chain
$\{1\}\trianglelefteq \Omega (Z(J_{\trianglelefteq }(E)))\trianglelefteq J_{\trianglelefteq }(E)\trianglelefteq E$
, a contradiction. Hence, there is
$A\in \mathcal {A}(E)$
with
$A\trianglelefteq E$
and
$A\not \le Q_1$
. Since
$(A\cap Q_1)Z(Q_1)$
is also elementary abelian and
$A\cap Z(Q_1)=Z(S)$
, we must have that
$S=AQ_1$
. Indeed,
$C_{V_1/U_1}(A)=U_1V_2/U_1$
. Then
$P_EU_1/U_1\le U_1V_2$
so that
$P_E\le U_1V_2$
. By Lemma 2.17,
$\Omega (P_E)\le (P_E\cap V_2)(P_E\cap U_1)$
. Since
$E\not \le Q_1$
,
$P_E\cap U_1\le [V_2, Q_1]$
so that
$\Omega (P_E)\le P_E\cap V_2=Z(V_1)$
. Hence,
$\Omega (P_E)=Z(V_1)$
is centralized by
$A_E$
so that
$C_E(\Omega (P_E))=V_1\cap E$
is also normalized by
$A_E$
. Then
$[V_1, E]\le V_1\cap E$
,
$[V_1, V_1\cap E]\le V_1'=\Omega (P_E)$
, and so
$O^2(A_E)$
is trivial on E, a contradiction.
Assume now that
$V_2<E\cap Q_2\le C_{Q_2}(V_2)$
. Since
$\Omega (E)\le (E\cap V_1)(E\cap Q_2)$
,
$\Omega (E)\le V_1$
. Since
$V_2\le \Omega (E)$
and
$E\not \le Q_2$
, we have that
$\Omega (Z(\Omega (E)))=Z(V_1)$
. But then
$U_1$
centralizes the chain
$\{1\}\trianglelefteq Z(V_1)\trianglelefteq \Omega (E)\le E$
, a contradiction.
Hence,
$V_2<C_{E}(V_2)\trianglelefteq L_2$
and
$E\cap Q_2\not \le C_{Q_2}(V_2)$
. Then
$P_E\le C_{Q_2}(C_E(V_2)/Z(S))\trianglelefteq L_2$
and since
$V_2/Z(S)=Z(Q_2/Z(S))$
, we have that
$P_E\le C_{Q_2}(V_2)$
. Since there are elements in
$E\cap Q_2\setminus C_E(V_2)$
, we conclude that
$P_E\le V_2$
. Thus,
$P_E=P_E\cap V_2=Z(V_1)$
. Indeed,
$E\cap V_1=C_E(P_E)$
is normalized by
$A_E$
. But
$[V_1, E]\le V_1\cap E$
,
$[V_1, V_1\cap E]\le V_1'=P_E$
, and so
$O^2(A_E)$
is trivial on E, a final contradiction. Therefore,
$\mathcal {E}(\mathcal {F})\subseteq \{Q_1, Q_2\}$
.
Finally, we classify all saturated fusion systems supported on S. With the help of Proposition 2.33, we are able to completely determine the local actions in
$\mathcal {F}$
without any reliance on the classification of finite simple groups. We employ Corollary 2.38 which in this usage only relies on [Reference Delgado and Stellmacher8], again independent of the classification.
Theorem 3.15. Let
$\mathcal {F}$
be a saturated fusion system on a Sylow
$2$
-subgroup of
${}^2\mathrm {F}_4(2^n)$
for
$n>1$
. Then either:
-
1.
$\mathcal {F}=\mathcal {F}_S(S: \mathrm {Out}_{\mathcal {F}}(S))$
; -
2.
$\mathcal {F}=\mathcal {F}_S(Q_1: \mathrm {Out}_{\mathcal {F}}(Q_1))$
, where
$O^{2'}(\mathrm {Out}_{\mathcal {F}}(Q_1))\cong \mathrm {SL}_2(2^n)$
; -
3.
$\mathcal {F}=\mathcal {F}_S(Q_2: \mathrm {Out}_{\mathcal {F}}(Q_2))$
, where
$O^{2'}(\mathrm {Out}_{\mathcal {F}}(Q_2))\cong \mathrm {Sz}(2^n)$
; or -
4.
$\mathcal {F}=\mathcal {F}_S(G)$
, where
$F^*(G)=O^{2'}(G)={}^2\mathrm {F}_4(2^n)$
.
Proof. By Proposition 3.14,
$\mathcal {E}(\mathcal {F})\subseteq \{Q_1, Q_2\}$
. If
$\mathcal {F}$
has no essential subgroups, then the Alperin–Goldschmidt theorem implies that
$S\trianglelefteq \mathcal {F}$
and outcome 1 holds.
Suppose that
$Q_1\in \mathcal {E}(\mathcal {F})$
. Then, as
$V_1=\Phi (Q_1)$
and S centralizes
$(Q_1\cap Q_2)\Phi (Q_1)/\Phi (Q_1)$
, an application of Theorem 2.21 yields that
$O^{2'}(\mathrm {Out}_{\mathcal {F}}(Q_1))\cong \mathrm {SL}_2(q)$
. Since
$\mathrm {Out}_S(E)$
is nontrivial on
$Z(Q_1)$
, and acts trivially on
$Z(S)$
, we infer by Theorem 2.21 that
$Z(Q_1)$
is a natural module for
$O^{2'}(\mathrm {Out}_{\mathcal {F}}(Q_1))$
. If
$Q_2$
is not essential, then the Alperin–Goldschmidt theorem yields outcome (2).
Suppose that
$Q_2\in \mathcal {E}(\mathcal {F})$
. Then
$S/Q_2$
is isomorphic to a Sylow
$2$
-subgroup of
$\mathrm {Sz}(q)$
, and so we deduce using Proposition 2.33 that
$O^{2'}(\mathrm {Out}_{\mathcal {F}}(E))\cong \mathrm {Sz}(q)$
. Indeed, since
$|Q_2/C_{Q_2}(V_2)|=|V_2/Z(S)|=q^4$
, it follows that
$Q_2/C_{Q_2}(V_2)$
and
$V_2/Z(S)$
are natural
$\mathrm {Sz}(q)$
-modules for
$O^{2'}(\mathrm {Out}_{\mathcal {F}}(Q_2))$
. If
$Q_1$
is not essential, then outcome 3 holds.
Finally, we claim that whenever
$\mathcal {E}(\mathcal {F})=\{Q_1, Q_2\}$
, we have that
$O_2(\mathcal {F})=\{1\}$
. By [Reference Aschbacher, Kessar and Oliver2, Proposition I.4.5], we have that
$O_2(\mathcal {F})\le Q_1\cap Q_2$
and since
$O^{2'}(\mathrm {Out}_{\mathcal {F}}(Q_2))$
is irreducible on
$Q_2/C_{Q_2}(V_2)$
, we have that
$O_2(\mathcal {F})\le C_{Q_2}(V_2)$
. Then
$\Phi (O_2(\mathcal {F}))$
is a normal subgroup of
$\mathcal {F}$
contained in
$Z(S)$
and since
$O^{2'}(\mathrm {Out}_{\mathcal {F}}(Q_1))$
is irreducible on
$Z(Q_1)$
, we deduce that
$\Phi (O_2(\mathcal {F}))=\{1\}$
and
$O_2(\mathcal {F})$
is elementary abelian. Hence,
$O_2(\mathcal {F})\le \Omega (C_{Q_2}(V_2))=V_2$
. Since
$O^{2'}(\mathrm {Out}_{\mathcal {F}}(Q_2))$
is irreducible on
$V_2/Z(S)$
and
$O^{2'}(\mathrm {Out}_{\mathcal {F}}(Q_1))$
is irreducible on
$Z(Q_1)$
, if
$O_2(\mathcal {F})\ne \{1\}$
, then
$O_2(\mathcal {F})=V_2$
. But then
$C_{Q_2}(V_2)=C_{Q_1}(V_2)$
would be normal in
$\mathcal {F}$
by [Reference Aschbacher, Kessar and Oliver2, Proposition I.4.5], a contradiction. Hence,
$O_2(\mathcal {F})=\{1\}$
. Since
$Q_1$
and
$Q_2$
are characteristic in S, if
$\mathcal {E}(\mathcal {F})=\{Q_1, Q_2\}$
, then
$\mathcal {F}$
satisfies the hypothesis of Corollary 2.38 and outcome 4 holds.
4 Fusion systems on a Sylow p-subgroup of
$\mathrm {PSU}_5(p^n)$
In this section, we classify saturated fusion systems supported on a Sylow p-subgroup of
$\mathrm {PSU}_5(p^n)$
. Throughout, we fix S isomorphic to a Sylow p-subgroup of
$\mathrm {PSU}_5(p^n)$
and write
$q=p^n$
.
For
$G:=\mathrm {PSU}_5(q)$
, we identify
$S\in \mathrm {Syl}_p(G)$
and set
$G_1:=N_G(C_S(Z_2(S)))$
and
$G_2:=N_G(Q)$
, where Q is the preimage in S of
$J(S/Z(S))$
of order
$q^7$
. Let
$L_i:=O^{p'}(G_i)$
for
$i\in \{1,2\}$
so that
$L_1$
is of shape
$q^{4+4}:\mathrm {SL}_2(q^2)$
and
$L_2$
is of shape
$q^{1+6}:\mathrm {SU}_3(q)$
[Reference Wilson29, pg. 67]. Note that
$G_1$
and
$G_2$
are the unique maximal parabolic subgroups of G containing
$N_G(S)$
. We write
$Q_i:=O_p(L_i)=O_p(G_i)$
so that
$Q_1=C_S(Z_2(S))$
and
$Q=Q_2$
, characteristic subgroups of S.
Once again, we require details about S and the structure of chief series of
$L_1$
and
$L_2$
. We garner the necessary information this time from [Reference Wilson29] (although [Reference Azad, Barry and Seitz1] is again applicable), and when combined with the results and techniques in Section 2, we are able to deduce the following well known facts.
Proposition 4.1. The following hold:
-
1.
$|S|=q^{10}$
; -
2.
$S'=\Phi (S)=Z(Q_1)(Q_1\cap Q_2)$
; -
3.
$m_p(S)=4n, 5n$
for
$p=2, odd$
, respectively; -
4.
$Z(S)=Z(Q_2)$
; and -
5.
$Z_2(S)=Z(Q_1)\cap Q_2$
.
Proposition 4.2. The following hold:
-
1.
$|Q_1|=q^8$
; -
2.
$|Z(Q_1)|=q^4$
and
$Q_1/Z(Q_1)$
is a natural module for
$L_1/Q_1\cong \mathrm {SL}_2(q^2)$
; -
3.
$Z(Q_1)\in \mathcal {A}(S)$
if and only if
$p=2$
; -
4.
$Z(Q_1)=\Phi (Q_1)=Q_1'$
is a natural module
$\Omega _4^-(q)$
-module for
$L_1/Q_1\cong \mathrm {SL}_2(q^2)$
; -
5. if
$A\le Z(Q_1)$
with
$|A|>q^2$
, then
$C_S(A)=Q_1$
; -
6.
$Q_2$
is a semiextraspecial group of order
$q^7$
; and -
7.
$Q_2/Z(S)$
is natural module for
$L_2/Q_2\cong \mathrm {SU}_3(q)$
.
We provide some details on the proofs of the above two propositions.
Proof of Proposition 4.1 and Proposition 4.2.
We use the description in [Reference Wilson29, pg. 67] of
$G_1$
and
$G_2$
although the indices are swapped compared to our convention. We have that
$Q_1$
is a special group of order
$q^8$
with
$Z(Q_1)=\Phi (Q_1)=Q_1'$
of order
$q^4$
. Moreover,
$G_1/Q_1\cong \mathrm {GL}_2(q^2)$
so that
$L_1/Q_1\cong \mathrm {SL}_2(q^2)$
. We have that
$Q_2$
is a special group of order
$q^7$
with
$Z(Q_2)=\Phi (Q_2)=Q_2'$
of order q. Moreover,
$G_2/Q_2\cong \mathrm {SU}_3(q)\times \mathrm {GL}_1(q^2)$
so that
$L_1/Q_1\cong \mathrm {SU}_3(q)$
.
The orders of S,
$Q_1$
and
$Q_2$
are obvious. We observe that minimal degree of representations for
$\mathrm {SL}_2(q^2)$
and
$\mathrm {SU}_3(q)$
over
$\mathrm {GF}(p)$
are
$4n$
and
$6n$
, respectively, [Reference Delgado and Stellmacher8, (5.7)]. Since
$Q_1$
is self-centralizing, we have that
$Z(S)<Z(Q_1)$
. Note that
$[S, Q_1]\not \le Z(Q_1)$
for otherwise
$[Z(Q_1), S]=[Q_1', S]=\{1\}$
by the three subgroups lemma. Hence,
$Q_1/Z(Q_1)$
and
$Z(Q_1)$
both contain a noncentral chief factor for
$L_1/Q_1$
from which we deduce that both
$Q_1/Z(Q_1)$
and
$Z(Q_1)$
are irreducible under the action of
$L_1$
. Since
$|Z(S)|=q$
and
$|Z(Q_1)|=q^4$
, we have that
$Z(Q_1)$
is an orthogonal module by Lemma 2.30. By Lemma 2.23, we deduce that
$[Z(Q_1), S]=[Z(Q_1), Q_2)]=Z(Q_1)\cap Q_2$
has order
$q^3$
and
$Z(Q_1)Q_2/Q_2=Z(S/Q_2)$
. Part (5) of Proposition 4.2 follows from Lemma 2.23. Since
$Q_2'=Z(S)\le Z(Q_1)$
, we have that
$S=Q_1Q_2$
is quadratic on
$Q_1/Z(Q_1)$
so that
$Q_1/Z(Q_1)$
is a natural
$\mathrm {SL}_2(q^2)$
-module by Lemma 2.30.
Since
$Q_2$
is self-centralizing,
$Q_2$
contains a noncentral chief factor for
$L_2/Q_2$
from which we deduce that
$Q_2/Z(Q_2)$
is irreducible under the action of
$L_2/Q_2$
. Moreover, it follows that
$L_2$
centralizes
$Z(Q_2)$
and since
$Z(S)\le Z(Q_2)$
we see that
$Z(S)=Z(Q_2)$
. That
$Q_2/Z(Q_2)$
is a natural module follows from the Steinberg tensor product theorem [Reference Gorenstein, Lyons and Solomon9, Corollary 2.8.6] upon extending scalars to
$\mathrm {GF}(q^2)$
. That
$L_2$
centralizes
$Z(Q_2)$
implies
$L_2$
normalizes
$Z(Q_2/Z)$
for all
$Z<Z(Q_2)$
and using that
$L_2$
is irreducible on
$Q_2/Z(Q_2)$
, it follows that
$Q_2$
is semiextraspecial.
From the Sylow p-subgroup structure of
$\mathrm {SL}_2(q^2)$
and
$\mathrm {SU}_3(q)$
, we have that
$S'\le \Phi (S)\le Q_1\cap Z(Q_1)Q_2=Z(Q_1)(Q_1\cap Q_2)$
. Note that
$Z(Q_1)=\Phi (Q_1)=Q_1'\le S'\le \Phi (S)$
and
$Z(Q_1)(Q_1\cap Q_2)=Z(Q_1)[Q_1, S]\le S'$
by Lemma 2.20 since
$Q_1/Z(Q_1)$
has the structure of a natural
$\mathrm {SL}_2(q^2)$
-module. We have that S has p-rank
$4n$
when
$p=2$
and p-rank
$5n$
when p is odd by [Reference Gorenstein, Lyons and Solomon9, Table 3.3.1]. Since S is not quadratic on
$Q_2/Z(Q_2)$
by Lemma 2.27,
$Z_2(S)\le Q_2$
. Furthermore, it follows that
$|Z_2(S)|=q^3$
. Since
$Z(Q_1)\cap Q_2$
has order
$q^3$
, we see by Lemma 2.23 that
$Z_2(S)=Z(Q_1)\cap Q_2$
.
The previous propositions contain all the structural information required to determine a complete list of essentials subgroups of any saturated fusion system
$\mathcal {F}$
supported on S. We establish this list over a series of lemmas and propositions, culminating in Proposition 4.9.
Proposition 4.3. Suppose that
$E\in \mathcal {E}(\mathcal {F})$
and
$E\le Q_1$
. Then
$E=Q_1$
.
Proof. Since
$E\le Q_1$
and E is
$\mathcal {F}$
-centric, we have that
$Z(Q_1)\le E$
. Furthermore, as
$Z(Q_1)=\Phi (Q_1)$
, we have that
$Q_1\le N_S(E)$
. If
$E\le Z(Q_1)(Q_1\cap Q_2)$
, then
$[S, E]\le Z(Q_1)\le Z(E)$
so that
$E\trianglelefteq S$
. But now,
$Z(Q_1)$
has index at most
$q^2$
in E and
$Q_1/E$
has order at least
$q^2$
and Theorem 2.21 yields
$E=Z(Q_1)(Q_1\cap Q_2)$
and
$Q_1=N_S(E)$
, a contradiction. Hence,
$E\not \le Z(Q_1)(Q_1\cap Q_2)$
.
Since
$Z(Q_1)\le E$
we have that
$Z(Q_1)Q_2<EQ_2$
and applying Lemma 2.27, we infer that
$Z(E)\cap Q_2=Z(Q_1)\cap Q_2$
. If
$Z(E)>Z(Q_1)$
, then again applying Lemma 2.27, we have that
$E\cap Q_2=Z(Q_1)\cap Q_2$
so that
$|Q_1/E|\geqslant q^2$
and
$|E/Z(Q_1)|\leqslant q^2$
. In fact,
$|E/Z(E)|<q^2$
and applying Theorem 2.21, we must have that
$E=Z(E)$
,
$|Q_1/E|=q^2=|E/Z(Q_1)|$
and
$Q_1=E(Q_1\cap Q_2)$
. But now, it follows that
$S=EQ_2$
, a contradiction for then
$\{1\}=E'\not \le Q_2$
. Thus,
$Z(E)=Z(Q_1)$
so that
$Q_1$
centralizes the chain
$\{1\}\trianglelefteq Z(E)\trianglelefteq E$
and Lemma 2.13 yields
$E=Q_1$
.
Lemma 4.4. Suppose that
$E\in \mathcal {E}(\mathcal {F})$
with
$E\ne Q_1$
. Then either
-
1.
$\Omega (Z(E))\cap Z(Q_1)=Z(S)$
; or -
2.
$|\Omega (Z(E))\cap Z(Q_1)|=q^2$
.
Moreover, if
$\Omega (Z(E))\not \le Q_1$
, then E is elementary abelian and
$N_S(E)=E(Z(Q_1)\cap Q_2)$
.
Proof. Since
$Q_1<EQ_1\le C_S(\Omega (Z(E))\cap Z(Q_1))$
, applying Lemma 2.23, we either have that
$Z(C_S(\Omega (Z(E))\cap Z(Q_1)))=Z(S)$
, or
$|Z(C_S(\Omega (Z(E))\cap Z(Q_1)))|=q^2$
and
$|EQ_1/Q_1|\leqslant q$
. Since
$Z(C_S(\Omega (Z(E))\cap Z(Q_1)))$
centralizes E which is
$\mathcal {F}$
-centric, we have that
$Z(C_S(\Omega (Z(E))\cap Z(Q_1)))=\Omega (Z(E))\cap Z(Q_1)$
.
Assume that
$\Omega (Z(E))\not \le Q_1$
. Then by Lemma 2.23,
$E\cap Z(Q_1)\le C_{Z(Q_1)}(Z(E))$
has order at most
$q^2$
. Since
$Z(Q_1)\cap Q_2\le N_S(E)$
and
$[Z(Q_1)\cap Q_2, E\cap Q_1]=\{1\}$
, Lemma 2.23 and Theorem 2.21 yield
$|E\cap Z(Q_1)|=q^2$
,
$N_S(E)=E(Z(Q_1)\cap Q_2)$
,
$|N_S(E)/E|=q$
,
$EQ_1/Q_1=\Omega (Z(E))Q_1/Q_1$
has order q and
$\Phi (E)\cap Z(S)=\{1\}$
.
Now,
$E\cap Z(Q_1)$
is centralized by
$\Omega (Z(E))Q_1\ge E$
so that
$E\cap Z(Q_1)=\Omega (Z(E))\cap Z(Q_1)$
. Furthermore, since
$\Omega (Z(E))\not \le Q_1$
, we deduce that
$E\cap Q_1\le Z(Q_1)(Q_1\cap Q_2)$
so that
$E=\Omega (Z(E))(E\cap Z(Q_1)(Q_1\cap Q_2))$
from which it follows that
$E'\le [Q_1\cap Q_2, Q_1\cap Q_2]\cap \Phi (E)=Z(S)\cap \Phi (E)=\{1\}$
and E is abelian. Furthermore, note that
$\mho ^1(E)=\mho ^1(E\cap Z(Q_1)(Q_1\cap Q_2))\le Z(S)\cap \Phi (E)=\{1\}$
. Hence,
$\Phi (E)=\{1\}$
and E is elementary abelian.
With the ultimate aim of demonstrating that
$\mathcal {E}(\mathcal {F})\subseteq \{Q_1, Q_2\}$
we assume that whenever
$E\ne Q_1$
and
$E\not \le Q_2$
, E is maximally essential. Thus, we are free to use Lemma 2.32–Proposition 2.34 throughout.
Proposition 4.5. Suppose that
$E\in \mathcal {E}(\mathcal {F})$
with
$E\ne Q_1$
. Then
$\Omega (Z(E))\le Q_1$
,
$Z(Q_1)\cap Q_2\le E$
and
$Z(Q_1)(Q_1\cap Q_2)\le N_S(E)$
.
Proof. Aiming for a contradiction, assume that
$\Omega (Z(E))\not \le Q_1$
so that by Lemma 4.4, E is elementary abelian and
$N_S(E)=E(Z(Q_1)\cap Q_2)$
. If
$E\le Q_2$
, then as E is elementary abelian and
$Q_2$
is semiextraspecial,
$|E|\leqslant q^4$
and since
$[Q_2, E]\le Q_2'=Z(S)\le E$
,
$Q_2\le N_S(E)$
and
$q=|N_S(E)/E|\geqslant q^3$
, a contradiction. Hence, E is maximally essential in
$\mathcal {F}$
. If
$E\le Z(Q_1)Q_2$
, then
$[Q_1\cap Q_2, E]\le [Q_1\cap Q_2, Z(Q_1)Q_2]=Z(S)\le E$
so that
$Q_1\cap Q_2\le N_S(E)$
. Indeed, since
$|N_S(E)/E|=q$
, we must have that
$|E\cap Q_1|= q^4$
. But then
$(E\cap Q_1)Z(Q_1)=q^6$
and
$(E\cap Q_1)Z(Q_1)$
is elementary abelian, a contradiction since
$m_p(S)\leqslant 5n$
. Hence,
$E\not \le Z(Q_1)Q_2$
and since E is elementary abelian and
$\Omega (S/Q_2)=Z(Q_1)Q_2/Q_2$
when
$p=2$
, we deduce that p is odd. In the application of Theorem 2.21, and using that E is abelian, we have that
$O^{p'}(\mathrm {Aut}_{\mathcal {F}}(E))\cong \mathrm {SL}_2(q)$
and
$E=[E, O^{p'}(\mathrm {Aut}_{\mathcal {F}}(E))]\times C_{E}(O^{p'}(\mathrm {Aut}_{\mathcal {F}}(E)))$
, where
$[E, O^{p'}(\mathrm {Aut}_{\mathcal {F}}(E))]$
is a natural
$\mathrm {SL}_2(q)$
-module.
Let
$\widehat {t}$
be a nontrivial involution in
$Z(O^{p'}(\mathrm {Aut}_{\mathcal {F}}(E)))$
. Then, as E is receptive,
$\widehat {t}$
lifts to subgroup of S strictly containing E, and by the Alperin–Goldschmidt theorem and using that E is maximally essential, we deduce that
$\widehat {t}$
is the restriction to E of some element
$t\in \mathrm {Aut}_{\mathcal {F}}(S)$
. In particular, t normalizes
$Q_1$
and
$Q_2$
. Then t acts isomorphically on
$EQ_1/Q_1\cong E/E\cap Q_1=E/C_E(\mathrm {Aut}_S(E))$
. So t inverts
$EQ_1/Q_1$
. By similar arguments, t inverts
$Z(S)$
and acts trivially on
$(Z(Q_1)\cap Q_2)/Z(S)$
. Let
$Z_t=C_{Z(Q_1)}(t)$
. Then
$[t, Z_t, E]=\{1\}$
and
$[E, Z_t, t]\le [Z(Q_1)\cap Q_2, t]=Z(S)$
. Applying the three subgroups lemma, we have that
$[E, t, Z_t]\le Z(S)$
and since
$[E, t]\not \le Q_1$
, we deduce that
$Z_t\le Z(Q_1)\cap Q_2$
and t inverts
$Z(Q_1)/(Z(Q_1)\cap Q_2)$
.
Replacing
$Z_t$
by
$C_t$
in the previous argument, where
$C_t$
is the preimage in
$Q_1\cap Q_2$
of
$C_{(Q_1\cap Q_2)/Z(S)}(t)$
, since t inverts
$Z(Q_1)Q_2/Q_2=\Phi (S/Q_2)$
, t must act nontrivially on
$S/Z(Q_1)Q_2$
and it follows that t inverts
$(Q_1\cap Q_2)/Z(Q_1)\cap Q_2$
. Furthermore, t acts on
$S/Z(Q_1)Q_2$
as it acts on
$Q_1/Z(Q_1)(Q_1\cap Q_2)$
. Hence,
$[Q_1/Z(Q_1), t]>(Q_1\cap Q_2)/Z(Q_1)$
. Then for D the preimage in
$S/Z(Q_1)$
of
$[S/Z(Q_1)(Q_1\cap Q_2), t]$
, t inverts D so that D is abelian. Since t inverts
$EQ_1/Q_1$
,
$D\not \le Q_1/Z(Q_1)$
. But
$C_{Q_1/Z(Q_1)}(s)=Z(Q_1)(Q_1\cap Q_2)/Z(Q_1)$
for any
$s\in S\setminus Q_1$
since
$Q_1/Z(Q_1)$
is a natural
$\mathrm {SL}_2(q^2)$
-module for
$\mathrm {Out}_{\mathrm {PSU}_5(q)}(Q_1)\cong \mathrm {SL}_2(q^2)$
, and since D is abelian we deduce that
$(D\cap Q_1)/Z(Q_1)=Z(Q_1)(Q_1\cap Q_2)/Z(Q_1)$
and we have a contradiction.
Hence,
$\Omega (Z(E))\le Q_1$
. But now,
$[Z(Q_1)\cap Q_2, E]\le [Z(Q_1)\cap Q_2, S]\le Z(S)\le \Omega (Z(E))$
so that
$Z(Q_1)\cap Q_2$
centralizes the chain
$\{1\}\trianglelefteq \Omega (Z(E))\trianglelefteq E$
and by Lemma 2.13,
$Z(Q_1)\cap Q_2\le E$
. In particular,
$Z(Q_1)(Q_1\cap Q_2)\le N_S(E)$
.
For the following proposition, we recall the definition of
$W(S)$
and its properties, as in Lemma 2.15.
Proposition 4.6. Suppose that
$E\in \mathcal {E}(\mathcal {F})$
. Then
$\Omega (Z(E))\le Z(Q_1)$
.
Proof. Aiming for a contradiction, assume that
$\Omega (Z(E))\not \le Z(Q_1)$
. In particular,
$E\ne Q_1$
so that
$E\not \le Q_1$
by Proposition 4.3. Since
$\Omega (Z(E))\le Q_1$
and
$E\not \le Q_1$
, we must have that
$\Omega (Z(E))\le Z(Q_1)(Q_1\cap Q_2)$
. Furthermore,
$\Omega (Z(E))Z(Q_1)$
is elementary abelian. By [Reference Gorenstein, Lyons and Solomon9, Theorem 3.3.3], when
$p=2$
we have that
$m_2(S)=4n$
and when p is odd,
$m_p(S)=5n$
. Hence, we have a contradiction when
$p=2$
and when p is odd, we deduce that
$|\Omega (Z(E))Z(Q_1)|\leqslant q^5$
. We let p be an odd prime for the remainder of the proof. Note that
$C_S(Z(Q_1)(Q_1\cap Q_2))=Z(Q_1)$
and since
$\Omega (Z(E))\le Q_1$
, we must have that
$Q_1\cap Q_2\not \le E$
.
Assume that
$Z(Q_1)\cap E>Z(Q_1)\cap Q_2$
. Then for any
$x\in E$
such that
$[Z(Q_1)\cap E, x, x]=\{1\}$
, we must have that
$x\in E\cap Q_1$
and
$[Z(Q_1)\cap E, x]=\{1\}$
. Hence,
$Z(Q_1)\cap E\le W(E)$
. Since
$W(E)$
is itself elementary abelian, it acts quadratically on
$Z(Q_1)\cap E$
, and so we deduce that
$W(E)\le Q_1$
. Hence,
$Z(Q_1)$
centralizes the chain
$\{1\}\trianglelefteq W(E)\trianglelefteq E$
and by Lemma 2.13,
$Z(Q_1)\le E$
. Note that
$[W(E), E\cap Q_1]\le Z(Q_1)$
so that
$[W(E), E\cap Q_1, E\cap Q_1]=\{1\}$
and
$W(E)$
centralizes
$E\cap Q_1$
. If
$W(E)\le Z(Q_1)(Q_1\cap Q_2)$
, then as
$\Omega (Z(E))\le W(E)$
, we have that
$Z(Q_1)<Z(Q_1)\Omega (Z(E))\le W(E)\le Z(Q_1)(Q_1\cap Q_2)$
. Moreover,
$[E, Z(Q_1)]\le [E, W(E)]\le Z(Q_1)$
. But then,
$[E, Q_1\cap Q_2]\le Z(Q_1)\cap Q_2\le W(E)$
,
$[W(E), Q_1\cap Q_2]\le Z(S)\le [W(E), E]$
and
$Q_1\cap Q_2$
centralizes
$[W(E), E]$
so that
$Q_1\cap Q_2$
centralizes the chain
$\{1\}\trianglelefteq [W(E), E]\trianglelefteq W(E)\trianglelefteq E$
, a contradiction by Lemma 2.13. Hence, since
$W(E)\le Q_1$
,
$W(E)\not \le Z(Q_1)Q_2$
and
$W(E)\cap Q_2=C_{Q_2}(W(E))=Z(Q_1)\cap Q_2$
. But
$\Omega (Z(E))Z(Q_1)\le W(E)$
, and so
$Z(Q_1)\cap Q_2<\Omega (Z(E))Z(Q_1)\cap Q_2\le W(E)\cap Q_2=Z(Q_1)\cap Q_2$
, a contradiction.
Hence, if
$\Omega (Z(E))\not \le Z(Q_1)$
, then p is odd and
$Z(Q_1)\cap E=Z(Q_1)\cap Q_2$
. Since
$E\not \le Q_1$
, we have that
$Z(S)=[Z(Q_1)\cap Q_2, E]\le \Phi (E)$
. If
$E\le Z(Q_1)Q_2$
, then
$[Q_1\cap Q_2, E]\le Z(S)\le \Phi (E)$
and we get that
$Q_1\cap Q_2\le E$
by Lemma 2.11, a contradiction. Now,
$E\cap Q_1$
centralizes
$Z(E)Z(Q_1)>Z(Q_1)$
and so centralizes
$Z(E)Z(Q_1)\cap Q_2>Z(Q_1)\cap Q_2$
and by Lemma 2.27, we have that
$E\cap Q_1\le Z(Q_1)(Q_1\cap Q_2)$
. Then,
$Z(Q_1)(Q_1\cap Q_2)$
centralizes
$(E\cap Q_1)/\Phi (E)$
which has index at most
$q^2$
in
$E/\Phi (E)$
and by Theorem 2.21,
$Z(Q_1)(Q_1\cap Q_2)\cap E$
has index at most
$q^2$
in
$Z(Q_1)(Q_1\cap Q_2)$
. Indeed, since
$Z(Q_1)\cap E=Z(Q_1)\cap Q_2$
, we deduce that
$|E\cap Q_1\cap Q_2|\geqslant q^4$
. But then, for any
$e\in E\setminus Z(Q_1)Q_2$
,
$|[e, E\cap Q_1\cap Q_2]Z(S)|\geqslant q^2$
, and so
$\Phi (E)\cap Z(Q_1)$
has order at least
$q^2$
. Again, applying Theorem 2.21, we have this time that
$Z(Q_1)(Q_1\cap Q_2)\cap E$
has index at most q in
$Z(Q_1)(Q_1\cap Q_2)$
and
$|N_S(E)/E|\leqslant q$
. Since
$Z(Q_1)\cap E=Z(Q_1)\cap Q_2$
,
$N_S(E)=Z(Q_1)E$
so that
$N_S(E)$
centralizes
$\Omega (Z(E))$
. But now,
$Z(Q_1)(Q_1\cap Q_2)\le N_S(E)$
and we conclude that
$\Omega (Z(E))\le Z(Q_1)$
, a final contradiction.
Lemma 4.7. Suppose that
$E\in \mathcal {E}(\mathcal {F})$
and
$E\ne Q_1$
. If
$\Omega (Z(E))\ne Z(S)$
, then p is odd and
$Z(Q_1)\cap E=Z(Q_1)\cap Q_2$
.
Proof. Suppose throughout that
$|\Omega (Z(E))|=q^2$
. Assume first that
$p=2$
. Then
$J(Q_1)=Z(Q_1)$
and since
$\Omega (C_{Q_1}(J(Q_1)))\le J(Q_1)$
, we have that
$\Omega (Q_1)=Z(Q_1)$
. Now,
$[Z(Q_1), E]=\Omega (Z(E))$
so that by Lemma 2.13,
$Z(Q_1)\le E$
. Let
$A\in \mathcal {A}(E)$
. Then
$A\cap Q_1\le \Omega (C_{Q_1}(J(Q_1)))=Z(Q_1)$
. But now, A centralizes a subgroup of
$Z(Q_1)$
of order at least
$q^3$
and by Lemma 2.23, we have that
$A\le Q_1$
and
$A=Z(Q_1)$
. Hence,
$J(E)=Z(Q_1)$
. Since
$Q_1\cap Q_2$
centralizes the chain
$\{1\}\trianglelefteq J(E)\trianglelefteq E$
by Lemma 2.13,
$Q_1\cap Q_2\le E$
and
$E\trianglelefteq S$
. Then
$Q_1$
centralizes the chain
$\{1\}\trianglelefteq J(E)\trianglelefteq C_E(J(E))=E\cap Q_1\trianglelefteq E$
and
$Q_1\le E$
. Furthermore,
$Q_1$
is a characteristic subgroup of E so that
$Z(Q_1)(Q_1\cap Q_2)=J(E)[E, Q_1]$
is also characteristic in E. Then S centralizes the chain
$J(E)\trianglelefteq J(E)[E, Q_1]\trianglelefteq Q_1\trianglelefteq E$
and since
$J(E)=\Phi (Q_1)\le \Phi (E)$
, we have a contradiction by Lemma 2.13.
Hence, if
$|\Omega (Z(E))|=q^2$
, then p is odd. Aiming for a contradiction, assume that
$Z(Q_1)\cap E>Z(Q_1)\cap Q_2$
. If
$[Z(Q_1)\cap E, x, x]=\{1\}$
for some
$x\in E$
, then by Lemma 2.23 we have that
$x\in Q_1$
so that
$[Z(Q_1)\cap E, x]=\{1\}$
. Hence,
$Z(Q_1)\cap E\le W(E)$
and as
$W(E)$
is elementary abelian, we deduce that
$W(E)\le E\cap Q_1$
. Then
$Z(Q_1)$
centralizes the chain
$\{1\}\trianglelefteq W(E)\trianglelefteq E$
and we have that
$Z(Q_1)\le W(E)$
. If
$W(E)\not \le Z(Q_1)(Q_1\cap Q_2)$
, then by Lemma 2.27, we deduce that
$W(E)\cap Q_2=C_{Q_2}(W(E))=Z(Q_1)\cap Q_2$
and since
$E\cap Q_1\cap Q_2$
acts quadratically on
$W(E)$
, we infer that
$E\cap Q_1\cap Q_2=Z(Q_1)\cap Q_2$
. Then
$[E, Q_1\cap Q_2]\le W(E)$
and
$Q_1\cap Q_2$
centralizes
$Z(Q_1)$
which has index at most q in
$W(E)$
and Theorem 2.21 gives a contradiction.
Therefore,
$Z(Q_1)\le W(E)\le Z(Q_1)(Q_1\cap Q_2)$
so that
$Q_1\cap Q_2$
centralizes the chain
$1\trianglelefteq Z(E)\trianglelefteq W(E)\trianglelefteq E$
,
$Z(Q_1)(Q_1\cap Q_2)\le E$
and
$E\trianglelefteq S$
. Now,
$Z(Q_1)(Q_1\cap Q_2)$
is generated by elementary abelian subgroups of order
$q^5$
so that
$Z(Q_1)(Q_1\cap Q_2)\le J(E)$
. Note that for any
$A\in \mathcal {A}(S)$
,
$A\not \le Q_1$
yields that
$|(A\cap Q_1)Z(Q_1)|=|A\cap Q_1||Z(Q_1)|/|A\cap Z(Q_1)|\geqslant |A||Z(Q_1)|/q^3>|A|$
so that
$J(S)=Q_1$
and
$J(E)\le E\cap Q_1$
. Since
$Z(Q_1)=Z(Z(Q_1)(Q_1\cap Q_2))$
, we have that
$Z(Q_1)=Z(J(E))$
is characteristic in E,
$E\cap Q_1=C_{E}(Q_1)$
is characteristic in E, and
$Q_1$
centralizes the chain
$\{1\}\trianglelefteq Z(Q_1)\trianglelefteq E\cap Q_1\trianglelefteq E$
so that
$Q_1$
is characteristic in E. Then
$Z(Q_1)\cap Q_2=Z(E)[Z(Q_1), E]$
,
$Z(S)=[Z(Q_1)\cap Q_2, E]$
and
$Z(Q_1)(Q_1\cap Q_2)=[E, Q_1]Z(J(E))$
are all characteristic in E and S centralizes the chain
$\{1\}\trianglelefteq Z(S)\trianglelefteq Z(Q_1)\cap Q_2\trianglelefteq Z(Q_1)(Q_1\cap Q_2)\trianglelefteq E$
, a contradiction.
Proposition 4.8. Suppose that
$E\in \mathcal {E}(\mathcal {F})$
. Then
$\Omega (Z(E))=Z(S)$
or
$E=Q_1$
.
Proof. Throughout, we assume that
$\Omega (Z(E))\ne Z(S)$
and
$E\not \le Q_1$
. By Lemma 4.7, we have that
$|\Omega (Z(E))|=q^2$
, p is odd and
$Z(Q_1)\cap E=Z(Q_1)\cap Q_2$
. Then
$Z(Q_1)$
centralizes
$E\cap Q_1$
of index q in E so that by Theorem 2.21,
$N_S(E)=Z(Q_1)E$
and
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(E))\cong \mathrm {SL}_2(q)$
. If
$Q_1\cap Q_2\le E$
, then
$[Q_2, E]\le Q_1\cap Q_2\le E$
and
$Q_2\le N_S(E)$
. But
$Q_2\le N_S(E)=EZ(Q_1)$
centralizes
$\Omega (Z(E))>\Omega (Z(S))$
, a contradiction. Hence,
$Q_1\cap Q_2\not \le E$
. Similarly, if
$E\le Q_2$
, then
$Q_2\le N_S(E)$
and we obtain a similar contradiction. Thus, we have that E is maximally essential and so, using that E is receptive and applying the Alperin–Goldschmidt theorem, any
$p'$
-automorphisms in
$N_{\mathrm {Aut}_{\mathcal {F}}(E)}(\mathrm {Aut}_S(E))$
lift to morphisms in
$\mathrm {Aut}_{\mathcal {F}}(S)$
. Upon restriction to E, all such morphisms normalize
$\mathrm {Aut}_{Q_1\cap Q_2}(E)$
and as
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(E))\cong \mathrm {SL}_2(q)$
,
$N_S(E)=E(Q_1\cap Q_2)$
.
Let
$\widehat {t}$
be a nontrivial involution in
$\mathrm {Aut}_{\mathcal {F}}(E)$
with image in
$Z(O^{p'}(\mathrm {Out}_{\mathcal {F}}(E)))$
. Then,
$\widehat {t}$
is the restriction to E of some element
$t\in \mathrm {Aut}_{\mathcal {F}}(S)$
. Since
$N_S(E)=Z(Q_1)E$
centralizes
$\Omega (Z(E))$
, we deduce that
$O^{p'}(\mathrm {Aut}_{\mathcal {F}}(E))$
acts trivially on
$\Omega (Z(E))$
, and so t acts trivially on
$\Omega (Z(E))$
. Then
$[Z(Q_1), E]\Omega (Z(E))/\Omega (Z(E))=(Z(Q_1)\cap Q_2)\Omega (Z(E))/\Omega (Z(E))$
is of order q inverted by t. For a similar reason,
$E/E\cap Q_1$
is inverted by t and
$(E\cap Q_1)/(Z(Q_1)\cap Q_2)$
is centralized by t. Furthermore, t centralizes
$\mathrm {Aut}_S(E)\cong N_S(E)/E\cong Z(Q_1)/Z(Q_1)\cap E$
and we infer that t centralizes
$Z(Q_1)(E\cap Q_1)/Z(Q_1)\cap Q_2=(N_S(E)\cap Q_1)/Z(Q_1)\cap Q_2$
. Let
$\mathcal {C}$
be the preimage in
$Q_1$
of
$C_{Q_1/Z(Q_1)}(t)$
. Then
$[\mathcal {C}, t, \mathcal {C}]=\{1\}$
and by the three subgroups lemma, t centralizes
$\mathcal {C}'$
. But
$Q_1\cap Q_2\le \mathcal {C}$
and
$[t, Z(Q_1)\cap Q_2]\ne \{1\}$
, and so by Lemma 2.27, we deduce that
$\mathcal {C}\le Z(Q_1)(Q_1\cap Q_2)$
so that
$Q_1/Z(Q_1)(Q_1\cap Q_2)$
is inverted by t. But then,
$[t, Q_1\cap Q_2, Q_1]=\{1\}$
,
$[Q_1, Q_1\cap Q_2, t]=[Z(Q_1)\cap Q_2, t]$
has order q and is normalized by
$Q_1, Q_1\cap Q_2$
and t, and by the three subgroups lemma,
$[Q_1, t, Q_1\cap Q_2]$
has order q. But
$[Q_1, t]\not \le Z(Q_1)(Q_1\cap Q_2)$
and we have a contradiction by Lemma 2.27.
As promised, we now complete the determination of all possible essential subgroups of
$\mathcal {F}$
.
Proposition 4.9. Suppose that
$E\in \mathcal {E}(\mathcal {F})$
. Then
$E\in \{Q_1, Q_2\}$
.
Proof. Ultimately aiming for a contradiction, we assume that
$Q_1\ne E\ne Q_2$
. Since
$\Omega (Z(E))=Z(S)$
by Proposition 4.8, if
$E\le Q_2$
, then Lemma 2.13 implies that
$E=Q_2$
, and so for the remainder of the proof, we assume that
$Q_1\not \ge E\not \le Q_2$
. Let
$Z^E$
be the preimage in E of
$Z(E/\Omega (Z(E)))$
, a characteristic subgroup of E. Then
$Z(Q_1)\cap Q_2\le Z^E$
and
$C_E(Z^E)\le E\cap Q_1$
Assume that
$Z^E\not \le Z(Q_1)Q_2$
so that
$E\cap Q_2=Z(Q_1)\cap Q_2$
by Lemma 2.27. Since
$Q_1\cap Q_2\le N_S(E)$
and
$Q_1\cap Q_2$
centralizes
$E\cap Z(Q_1)Q_2$
modulo
$\Omega (Z(E))$
, applying Theorem 2.21 we have that
$S=EZ(Q_1)Q_2$
. But then
$\Omega (Z(E))\ge [E, Z^E]\not \le Q_2$
, a contradiction.
Hence,
$Z^E\le Z(Q_1)Q_2$
. Then
$[Z^E, Q_1\cap Q_2]\le \Omega (Z(E))$
and
$Q_1\cap Q_2$
centralizes the chain
$\{1\}\trianglelefteq \Omega (Z(E))\trianglelefteq Z^E\trianglelefteq E$
from which we deduce that
$Q_1\cap Q_2\le E$
. Furthermore,
$Q_2$
normalizes E. If
$Z^E\not \le Q_2$
, then
$E\cap Q_2=Q_1\cap Q_2$
by Lemma 2.27 so that
$|Q_2E/E|=q^2$
. We apply Proposition 2.33 when
$p=2$
and use
$[E, Q_2, Q_2]\le \Omega (Z(E))$
and Proposition 2.34 when p is odd so that
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(E))$
is quasisimple. But now,
$Z^E\cap Q_2$
has index at most q in
$Z^E$
and is centralized modulo
$\Omega (Z(E))$
by
$Q_2$
so that, by Theorem 2.21,
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(E))$
acts trivially on
$Z^E$
. Then, it follows that
$Q_2E$
centralizes
$Z^E/\Omega (Z(E))$
and Lemma 2.27 yields a contradiction.
Thus,
$Z^E\le Q_2$
and since
$E\not \le Q_2$
, we have that
$Z^E\le Q_1\cap Q_2$
so that
$Z(Q_1)\cap E\le C_E(Z^E)\le E\cap Q_1$
. Then
$Z(Q_1)$
centralizes the chain
$\{1\}\trianglelefteq C_E(Z^E)\trianglelefteq E$
and we deduce that
$Z(Q_1)\le E$
and
$E\trianglelefteq S$
. Now, if
$E\le Z(Q_1)Q_2$
, then
$Z^E=Q_1\cap Q_2$
and
$Z(Z^E)=Z(Q_1)\cap Q_2$
is characteristic in E. But then,
$Q_1$
centralizes the chain
$\{1\}\trianglelefteq Z(Q_1)\cap Q_2\trianglelefteq Q_1\cap Q_2\trianglelefteq E$
so that
$Q_1\le E$
by Lemma 2.13, a contradiction since
$E\le Z(Q_1)Q_2$
. Hence,
$E\not \le Z(Q_1)Q_2$
and we conclude that
$Z^E=Z(Q_1)\cap Q_2$
and
$C_E(Z^E)=E\cap Q_1$
. Since
$Z(Q_1)(Q_1\cap Q_2)\le E\cap Q_1$
and
$Z(Q_1)=Z(Z(Q_1)(Q_1\cap Q_2))$
, we have that
$Z(Q_1)=Z(C_E(Z^E))$
is characteristic in E and
$Q_1$
centralizes the chain
$\{1\}\trianglelefteq Z(Q_1)\trianglelefteq E\cap Q_1\trianglelefteq E$
. Hence,
$Q_1=E\cap Q_1$
is characteristic in E. Then
$E\not \le Q_1$
so that
$Z(Q_1)[E, Q_1]=Z(Q_1)(Q_1\cap Q_2)$
is also characteristic in E. Finally, S centralizes the chain
$$\begin{align*}\{1\}\trianglelefteq \Omega(Z(E))\trianglelefteq Z^E\trianglelefteq Z(Q_1)\trianglelefteq Z(Q_1)[E, Q_1]\trianglelefteq Q_1\trianglelefteq E\end{align*}$$
and Lemma 2.13 provides a final contradiction.
In the classification of saturated fusion systems supported on S, we apply Corollary 2.40 using Theorem 2.37 when
$Q_1$
and
$Q_2$
are both essential and, as in earlier cases, we remark that this reduces to applying the main result from [Reference Delgado and Stellmacher8]. Moreover, we are able to calculate the local actions in the fusion system without any reliance on the classification of the finite simple groups. In this way, the following theorem independent of any
$\mathcal {K}$
-group hypothesis.
Theorem 4.10. Let
$\mathcal {F}$
be a saturated fusion system on a Sylow p-subgroup of
$\mathrm {PSU}_5(p^n)$
. Then either:
-
1.
$\mathcal {F}=\mathcal {F}_S(S: \mathrm {Out}_{\mathcal {F}}(S))$
; -
2.
$\mathcal {F}=\mathcal {F}_S(Q_1: \mathrm {Out}_{\mathcal {F}}(Q_1))$
, where
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(Q_1))\cong \mathrm {SL}_2(p^n)$
; -
3.
$\mathcal {F}=\mathcal {F}_S(Q_2: \mathrm {Out}_{\mathcal {F}}(Q_2))$
, where
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(Q_2))\cong \mathrm {(P)SU}_3(p^n)$
; or -
4.
$\mathcal {F}=\mathcal {F}_S(G)$
, where
$F^*(G)=O^{p'}(G)=\mathrm {PSU}_5(p^n)$
.
Proof. If
$\mathcal {E}(\mathcal {F})=\emptyset $
, then outcome 1 holds. If
$Q_1\in \mathcal {E}(\mathcal {F})$
, then S centralizes
$Z(Q_1)(Q_1\cap Q_2)$
modulo
$Z(Q_1)=\Phi (Q_1)$
and Theorem 2.21 yields that
$Q_1/\Phi (Q_1)$
is a natural module for
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(Q_1))\cong \mathrm {SL}_2(q^2)$
. Then, as
$\mathrm {Out}_S(Q_1)$
acts nontrivially on
$Z(Q_1)$
, we have that
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(Q_1))$
acts nontrivially and comparing with Lemma 2.30, using that
$\mathrm {Out}_S(Q_1)$
does not act quadratically on
$Z(Q_1)$
, we deduce that
$Z(Q_1)$
is a natural
$\Omega _4^-(q)$
-module for
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(Q_1))\cong \mathrm {SL}_2(q^2)$
, on which
$Z(O^{p'}(\mathrm {Out}_{\mathcal {F}}(Q_1)))$
acts trivially. If
$Q_2\not \in \mathcal {E}(\mathcal {F})$
, then outcome 2 holds.
Hence, we may assume that
$Q_2\in \mathcal {E}(\mathcal {F})$
. Suppose first that
$p=2$
. We apply Proposition 2.33 and use that
$S/Q_2$
is isomorphic to a Sylow
$2$
-subgroup of
$\mathrm {PSU}_3(q)$
to deduce that either
$O^{2'}(\mathrm {Out}_{\mathcal {F}}(Q_2))\cong \mathrm {(P)SU}_3(q)$
, or
$q=2$
and
$\mathrm {Out}_S(Q_2)\cong Q_8$
. In the latter case, we appeal to MAGMA for a list of subgroups of
$\mathrm {GL}_6(2)$
which have a Sylow
$2$
-subgroup isomorphic to
$Q_8$
which acts in the same manner as
$\mathrm {Out}_S(Q_2)$
does on a group of order
$2^6$
. Then
$O^{2'}(\mathrm {Out}_{\mathcal {F}}(Q_2))\cong \mathrm {SU}_3(2)$
in this case too.
Suppose now that p is odd. Then
$Z(Q_1)$
acts quadratically on
$Q_2$
, and so
$\mathrm {Out}_{Z(Q_1)}(Q_2)$
is quadratic on
$Q_2$
. With the aim of applying Proposition 2.34, we need to show that
$\mathrm {Syl}_p(\mathrm {Out}_{\mathcal {F}}(Q_2))$
is a TI-set for
$\mathrm {Out}_{\mathcal {F}}(Q_2)$
. To this end, aiming for a contradiction, assume there is
$1\ne s\in T\cap R$
, where T and R are distinct Sylow p-subgroups of
$\mathrm {Out}_{\mathcal {F}}(Q_2)$
. By Lemma 2.27,
$|C_{Q_2/\Phi (Q_2)}(s)|\in \{q^2, q^4\}$
.
If
$|C_{Q_2/\Phi (Q_2)}(s)|=q^4$
, then
$s\in Z(T)\cap Z(R)$
by Lemma 2.27. Then
$$\begin{align*}C_{Q_2/\Phi(Q_2)}(Z(T))=C_{Q_2/\Phi(Q_2)}(s)=C_{Q_2/\Phi(Q_2)}(Z(R))\end{align*}$$
and
$$\begin{align*}[Q_2/\Phi(Q_2), Z(T)]=[Q_2/\Phi(Q_2), s]=[Q_2/\Phi(Q_2), Z(R)].\end{align*}$$
Thus, by Lemma 2.27,
$\langle T, R\rangle $
centralizes the chain
$$\begin{align*}\{1\}\trianglelefteq [Q_2/\Phi(Q_2), s]\trianglelefteq C_{Q_2/\Phi(Q_2)}(s)\trianglelefteq Q_2/\Phi(Q_2)\end{align*}$$
and Lemma 2.13 yields that
$\langle T, R\rangle $
is a p-group. Since T and R are distinct Sylow p-subgroups, this is a contradiction.
If
$|C_{Q_2/\Phi (Q_2)}(s)|=q^2$
, then
$s\in (T\setminus Z(T))\cap (R\setminus Z(R))$
and
$$\begin{align*}[Q_2/\Phi(Q_2), T]=[Q_2/\Phi(Q_2), s]=[Q_2/\Phi(Q_2), R]\end{align*}$$
by Lemma 2.27. Then
$$\begin{align*}[Q_2/\Phi(Q_2), T, T]=C_{Q_2/\Phi(Q_2)}(s)=[Q_2/\Phi(Q_2), R, R]\end{align*}$$
and
$\langle T,R\rangle $
centralizes the chain
$$\begin{align*}\{1\}\trianglelefteq C_{Q_2/\Phi(Q_2)}(s)\trianglelefteq [Q_2/\Phi(Q_2), s]\trianglelefteq Q_2/\Phi(Q_2)\end{align*}$$
and Lemma 2.13 yields that
$\langle T, R\rangle $
is a p-group. Since T and R are distinct Sylow p-subgroups, this is again a contradiction. Thus,
$\mathrm {Syl}_p(\mathrm {Out}_{\mathcal {F}}(Q_2))$
is a TI-set and as
$\mathrm {Out}_S(Q_2)\cong S/Q_2$
is nonabelian, we conclude from Proposition 2.34 that
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(Q_2))\cong \mathrm {(P)SU}_3(q)$
. If
$\mathcal {E}(\mathcal {F})=\{Q_2\}$
, then outcome 3 holds.
To complete the proof, we are left with the case
$\mathcal {E}(\mathcal {F})=\{Q_1, Q_2\}$
. By [Reference Aschbacher, Kessar and Oliver2, Proposition I.4.5],
$O_p(\mathcal {F})\le Q_1\cap Q_2$
. Since
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(Q_2))$
acts irreducibly on
$Q_2/Z(S)$
, we have that
$O_p(\mathcal {F})\le Z(S)$
. But now,
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(Q_1))$
acts irreducibly on
$Z(Q_1)$
, and we have that
$O_p(\mathcal {F})=\{1\}$
. Then,
$\mathcal {F}$
satisfies the hypothesis of Corollary 2.40, and 4 holds, completing the proof.
5 Fusion systems on a Sylow p-subgroup of
${}^3\mathrm {D}_4(p^n)$
In this final section, we classify saturated fusion systems supported on a Sylow p-subgroup of
${}^3\mathrm {D}_4(p^n)$
, completing the proof of Theorem A and of the Main Theorem. Throughout, we fix S isomorphic to a Sylow p-subgroup of
${}^3\mathrm {D}_4(p^n)$
and write
$q=p^n$
.
For
$G:={}^3\mathrm {D}_4(p^n)$
, we identify
$S\in \mathrm {Syl}_p(G)$
and set
$G_1:=N_G(C_S(Z_2(S)))$
and
$G_2:=N_G(Q)$
, where Q is the preimage in S of
$J(S/Z(S))$
of order
$q^9$
. Let
$L_i:=O^{p'}(G_i)$
for
$i\in \{1,2\}$
so that
$L_1$
is of shape
$q^{2+(1+1+1)+(2+2+2)}:\mathrm {SL}_2(q)$
and
$L_2$
is of shape
$q^{1+8}:\mathrm {SL}_2(q^3)$
(see [Reference Wilson29, pg. 144]). Note that
$G_1$
and
$G_2$
are the unique maximal parabolic subgroups of G containing
$N_G(S)$
. We write
$Q_i:=O_p(L_i)=O_p(G_i)$
so that
$Q_1=C_S(Z_2(S))$
and
$Q=Q_2$
, characteristic subgroups of S.
We appeal to [Reference Wilson29] and [Reference Azad, Barry and Seitz1] for structure of a chief series of
$L_1$
and
$L_2$
, and the structure of certain subgroups of S. We highlight the main features required for this work and provide modest proofs.
Proposition 5.1. The following hold:
-
1.
$|Q_1|=q^{11}$
; -
2.
$|\Phi (Q_1)|=q^5$
,
$\Phi (Q_1)\in \mathcal {A}(S)$
and
$Q_1/\Phi (Q_1)$
is a direct sum of three natural modules for
$L_1/Q_1\cong \mathrm {SL}_2(q)$
; -
3.
$Z_2(S)=Z(Q_1)$
is elementary abelian of order
$q^2$
,
$\Phi (Q_1)/Z(Q_1)$
is centralized by
$L_1$
and
$Z(Q_1)$
is a natural module for
$L_1/Q_1\cong \mathrm {SL}_2(q)$
; -
4.
$\Phi (Q_1)=Q_1'=Z_3(S)$
is elementary abelian of order
$q^5$
and
$Z(Q_1)=[\Phi (Q_1), Q_1]$
; -
5.
$Q_2$
is a semiextraspecial group of order
$q^9$
; and -
6.
$Z(S)=Z(Q_2)$
has order q and
$Q_2/Z(S)$
is a triality module for
$L_2/Q_2\cong \mathrm {SL}_2(q^3)$
.
Proof. We appeal to [Reference Wilson29, pg. 144] and [Reference Korableva16] to see that
$G_2$
has shape
$q^{1+8}:\mathrm {SL}_2(q^3).(q-1)$
and
$G_2$
has exactly two chief factors within
$Q_2$
. Then
$|Q_2|=q^{9}$
and
$Q_2$
is self-centralizing in
$G_2$
. Since
$Q_2$
is nonabelian and
${}^3\mathrm {D}_4(q)$
is defined over
$\mathrm {GF}(q)$
, we have that
$Z(Q_2)=Q_2'=\Phi (Q_2)$
has order q and
$Q_2$
is a special group. Note that
$C_{Q_2}(L_2)$
and
$[Q_2, L_2]$
are invariant under
$G_2$
from which we conclude that
$Z(Q_2)=C_{Q_2}(L_2)$
and
$Q_2=[Q_2, L_2]$
. We apply Lemma 2.20 to see that
$Q_2/Z(Q_2)$
is a triality module for
$L_2/Q_2\cong \mathrm {SL}_2(q^3)$
and
$Z(Q_1)=C_{Q_2}(L_2)=Z(S)$
. Let Z be a maximal subgroup of
$Z(S)$
. Since
$L_2$
centralizes
$Z(S)$
,
$L_2$
normalizes Z and so acts on
$Q_2/Z$
. Since
$L_2$
acts irreducibly on
$Q_2/Z(S)$
, it follows that
$Z(Q_2/Z)=\Phi (Q_2/Z)=Z(S)/Z$
and
$Q_2/Z$
is extraspecial. Hence,
$Q_2$
is semiextraspecial of order
$q^9$
.
Since
$O_p({}^3\mathrm {D}_4(q))=\{1\}$
, we see that
$Z(S)\not \trianglelefteq L_1$
and
$Z(S)<Z(Q_1)$
. By [Reference Wilson29, pg. 144],
$L_1$
is of shape
$q^{2+(1+1+1)+(2+2+2)}:\mathrm {SL}_2(q)$
. Since
$Q_1\cap Q_2$
has index q in
$Q_2$
, from the structure of
$Q_2/Z(S)$
, we have that
$Z(Q_1)\le Q_2$
and
$[S, Z(Q_1)]=[Q_2, Z(Q_1)]\le Z(S)$
so that
$Z(Q_1)\le Z_2(S)$
. Since
$|Z_2(S)|=q^2$
and
$Z(Q_1)$
contains a noncentral chief factor for
$L_1/Q_1\cong \mathrm {SL}_2(q)$
, we have that
$Z(Q_1)=Z_2(S)$
. Again, as
$Q_1\cap Q_2$
has index q in
$Q_2$
and
$Q_2/Z(S)$
is a triality module, Lemma 2.25 yields that
$Z_3(S)\le Z_2(Q_1)\le Q_2$
so that
$Z_3(S)=Z_2(Q_1)$
has order
$q^5$
. Indeed,
$\Phi (Z_2(Q_1))\le Z(S)$
from which we conclude that
$Z_2(Q_1)$
is elementary abelian, and since
$[S, Z_2(Q_1)]=[S, Z_3(S)]=Z_2(S)=Z(Q_1)$
, we have that
$L_1$
centralizes
$Z_2(Q_1)/Z(Q_1)$
.
Observe that
$Z_2(Q_1)=[Q_1\cap Q_2, Q_1]\le Q_1'$
. Then
$[Q_2, Q_1']\le [Q_2, \Phi (Q_1)]\le Z(S)$
and so
$O^p(L_1)$
centralizes
$\Phi (Q_1)/Z(Q_1)$
. Note that
$[O^p(L_1), Q_1]\trianglelefteq G_1$
, and so
$[O^p(L_1), Q_1]Q_2$
is normalized by
$N_{L_2}(S)$
from which we deduce that either
$[O^p(L_1), Q_1]\le Q_2$
or
$S=[O^p(L_1), Q_1]Q_2$
. In the former case, we have that
$[[O^p(L_1), Q_1], Q_2]\le Z(Q_1)$
from which it follows that
$O^p(L_1)$
centralizes
$Q_1/Z(Q_1)$
, a contradiction since
$Z(Q_1)\le Q_1'$
. Hence,
$S=[O^p(L_1), Q_1]Q_2$
. In particular, for X the largest subgroup of
$Q_1$
with
$[O^p(L_1), X]\le Z(Q_1)$
, we have that
$[S, X]\le Z(Q_1)$
and so
$\Phi (Q_1)\le X=Z_2(Q_1)$
. Hence,
$\Phi (Q_1)=Q_1'=Z_3(S)$
. By [Reference Gorenstein, Lyons and Solomon9, Table 3.3.1], the p-rank of
${}^3\mathrm {D}_4(q)$
is
$5n$
and so
$\Phi (Q_1)\in \mathcal {A}(S)$
.
Observe that
$[Q_1, Q_2, Q_2]\le Z(S)\le \Phi (Q_1)$
, and so a result of Niles (see [Reference Parker and Rowley21, Lemma 3.20] for a proof) reveals that
$Q_1/\Phi (Q_1)$
is a direct sum of natural
$\mathrm {SL}_2(q)$
-modules. A consideration of orders yields that
$Q_1/\Phi (Q_1)$
is a direct sum of three natural modules for
$L_1/Q_1\cong \mathrm {SL}_2(q)$
.
Proposition 5.2. The following hold:
-
1.
$|S|=q^{12}$
; -
2.
$S'=\Phi (S)=Q_1\cap Q_2$
; -
3.
$m_p(S)=5n$
; -
4.
$C_S(\Phi (Q_1)/Z(S))=Q_2$
; and -
5. if
$p=2$
and
$x\in S$
is such that
$x^2=1$
, then
$x\in Q_1 \cup Q_2$
.
Proof. The order of S is clear from the structure of either
$G_1$
or
$G_2$
. Since
$S/Q_1$
and
$S/Q_2$
are elementary abelian, we have that
$S'\le \Phi (S)\le Q_1\cap Q_2$
. Since
$Q_1/Q_1'$
is a direct sum of three natural
$\mathrm {SL}_2(q)$
-modules, we have that
$Q_1\cap Q_2=Q_1'[Q_1, Q_2]$
from which we deduce that
$S'=\Phi (S)=Q_1\cap Q_2$
. We have that
$m_p(S)=5n$
by [Reference Gorenstein, Lyons and Solomon9, Table 3.3.1]. Clearly,
$[Q_2, \Phi (Q_1)]\le Z(S)$
. Suppose that
$C_S(\Phi (Q_1)/Z(S))\not \le Q_2$
. Since
$L_2/Q_2\cong \mathrm {SL}_2(q^3)$
and
$C_S(\Phi (Q_1)/Z(S))$
is normalized by
$N_{L_2}(S)$
, we see that
$[S, \Phi (Q_1)]\le Z(S)$
, a contradiction since
$[Q_1, \Phi (Q_1)]=Z(Q_1)$
.
Assume that
$p=2$
, and let
$x\in S$
is such that
$x^2=1$
and
$x\not \in Q_1$
. Write
$x=ty$
, where
$t\in Q_1$
and
$y\in Q_2\setminus (Q_1\cap Q_2)$
. Note that
$\Phi (Q_1)$
is elementary abelian so that
$t^4=1$
, and since
$Q_2$
is special, we see that
$y^2\in Z(S)$
and
$y^4=1$
. Then
$t^{-2}[t, y^{-1}]y^{-2}=t^{-2}t^{-1}yty^{-1}y^{-2}=(ty)^2=x^2=1$
so that
$[t, y^{-1}]=t^2y^2\in \Phi (Q_1)$
. Since
$Q_1/\Phi (Q_1)$
is a direct sum of natural modules, it follows from Lemma 2.20 that
$t\in Q_1\cap Q_2$
and so
$x\in Q_2$
.
We first determine the structure of centralizers of certain p-subgroups of S to facilitate later arguments.
Lemma 5.3. Suppose that
$A\le Q_1$
, and let
$b\in Q_1\setminus \Phi (Q_1)$
. Then
$|C_{\Phi (Q_1)}(A)|\in \{q^2, q^3, q^4, q^5\}$
,
$|C_{\Phi (Q_1)}(b)|=q^4$
and
$|[b, \Phi (Q_1)]|=q$
. Moreover, if
$|C_{\Phi (Q_1)}(A)|=q^2$
, then
$C_{\Phi (Q_1)}(A)=Z(Q_1)$
and if
$|C_{\Phi (Q_1)}(A)|=q^5$
, then
$A\le \Phi (Q_1)$
.
Proof. Let
$b\in Q_1\setminus \Phi (Q_1)$
. Then there is
$x\in L_1$
such that
$b^x\le Q_1\cap Q_2$
. Moreover, since
$Z(Q_1)\le C_{\Phi (Q_1)}(b)\cap C_{\Phi (Q_1}(b^x)$
and
$L_1$
acts trivially on
$\Phi (Q_1)/Z(Q_1)$
, we deduce that
$C_{\Phi (Q_1)}(b)=C_{\Phi (Q_1)}(b^x)$
. Now, the map
$\phi _{b^x}: \Phi (Q_1)\to \Phi (Q_1)$
with
$y\phi _{b^x}=[y, b^x]$
is a homomorphism with kernel
$C_{\Phi (Q_1)}(b^x)$
. Since
$\Phi (Q_1)\in \mathcal {A}(S)$
and
$b^x\le Q_2\setminus \Phi (Q_1)$
, we have by Lemma 2.36 that
$\mathrm {Im}(\phi _{b^x})=[b^x, \Phi (Q_1)]=Z(S)$
has order q so that
$|C_{\Phi (Q_1)}(b^x)|=q^4$
.
Now, let
$A\le Q_1$
so that
$Z(Q_1)\le C_{\Phi (Q_1)}(A)$
. Indeed, if
$|C_{\Phi (Q_1)}(A)|=q^2$
, then
$C_{\Phi (Q_1)}(A)=Z(Q_1)$
. Since
$\Phi (Q_1)$
is abelian and self-centralizing in
$Q_1$
,
$C_{\Phi (Q_1)}(A)=\Phi (Q_1)$
if and only if
$A\le \Phi (Q_1)$
. Hence, to complete the proof, we may assume that there is
$a\in A\setminus (A\cap \Phi (Q_1))$
. Then
$C_{\Phi (Q_1)}(A)\le C_{\Phi (Q_1)}(a)$
and
$C_{\Phi (Q_1)}(a)$
has order
$q^4$
. We claim that for
$a, a'\in A$
either
$C_{\Phi (Q_1)}(a)=C_{\Phi (Q_1)}(a')$
or
$|C_{\Phi (Q_1)}(a)\cap C_{\Phi (Q_1)}(a')|=q^3$
. Let
$a^{\prime x}$
be an
$L_1$
conjugate of
$a'$
in
$Q_1\cap Q_2$
so that
$C_{\Phi (Q_1)}(a')=C_{\Phi (Q_1)}(a^{\prime x})$
. By properties of
$Q_2$
, if
$[C_{\Phi (Q_1)}(a), a^{\prime x}]<Z(S)$
, then
$[C_{\Phi (Q_1)}(a), a^{\prime x}]=\{1\}$
and
$C_{\Phi (Q_1)}(a)=C_{\Phi (Q_1})(a')$
. Otherwise, applying an argument using the commutation homomorphism, we deduce that
$|C_{C_{\Phi (Q_1)}(a)}(a^{\prime x})|=q^3$
and the claim holds. Hence, if
$|C_{\Phi (Q_1)}(A)|<q^4$
, then
$|C_{\Phi (Q_1)}(A)|\leqslant q^3$
. Taking another element
$\widehat {a}$
and an
$L_1$
-conjugate
$\widehat {a}^x\in Q_1\cap Q_2$
and applying the commutation homomorphism to
$C_{\Phi (Q_1)}(a)\cap C_{\Phi (Q_1)}(a')$
, we deduce that
$|C_{\Phi (Q_1)}(a)\cap C_{\Phi (Q_1)}(a')\cap C_{\Phi (Q_1)}(\widehat {a})|\in \{q^2, q^3\}$
and the result on
$C_{\Phi (Q_1)}(A)$
follows.
Lemma 5.4. Let
$a\in Q_1\setminus \Phi (Q_1)$
, and set
$D:=C_{\Phi (Q_1)}(a)$
. Then
$|C_{Q_1}(D)|=q^7$
,
$C_{Q_1}(D)/\Phi (Q_1)$
is a natural
$\mathrm {SL}_2(q)$
-module for
$O^{p'}(\mathrm {Out}_{L_1}(Q_1))\cong \mathrm {SL}_2(q)$
,
$D=Z(C_{Q_1}(D))$
and if
$p=2$
, then
$\Phi (C_{Q_1}(D))=Z(Q_1)$
.
Proof. Take a and D as above. Then
$[a, D]=\{1\}$
and since D is normalized by
$L_1$
,
$[\langle a^{L_1}\rangle , D]=\{1\}$
. Since
$Q_1/\Phi (Q_1)$
is a direct sum of three natural
$\mathrm {SL}_2(q)$
-modules for
$L_1/Q_1$
, we have that
$\Phi (Q_1)\langle a^{L_1}\rangle $
has order
$q^7$
and centralizes D. If
$|C_{Q_1}(D)|>q^7$
, then for some b in
$C_{Q_1}(D)\setminus \langle a^{L_1}\rangle \Phi (Q_1)$
, we apply the same argument to enlarge
$C_{Q_1}(D)$
. It follows that
$|C_{Q_1}(D)|\geqslant q^9$
. But as argued in Lemma 5.3, we then have that
$|D\cap C_{\Phi (Q_1)}(c)|\geqslant q^3$
for some c with
$\langle c^{L_1}\rangle C_{Q_1}(D)=Q_1$
so that
$Z(Q_1)=D\cap C_{\Phi (Q_1)}(c)$
, a contradiction. Hence,
$|C_{Q_1}(D)|=q^7$
and
$C_{Q_1}(D)\trianglelefteq L_1$
. Indeed, by Lemma 2.30,
$C_{Q_1}(D)/\Phi (Q_1)$
is a natural
$\mathrm {SL}_2(q)$
-module for
$O^{p'}(\mathrm {Out}_L(Q_1))\cong \mathrm {SL}_2(q)$
. Since
$\Phi (Q_1)$
is self-centralizing in
$Q_1$
, we deduce that
$D\le Z(C_{Q_1}(D))\le \Phi (Q_1)$
so that
$D=Z(C_{Q_1}(D))$
.
By properties of
$Q_2$
, as in Lemma 2.36,
$C_{Q_1 \cap Q_2}(D)'=Z(S)$
and as
$\Phi (C_{Q_1}(D))\trianglelefteq L_1$
,
$Z(Q_1)\le \Phi (C_{Q_1}(D))$
. Assume that
$p=2$
. Since
$Q_2$
is irreducible for
$L_2/Q_2$
, we have that
$\Omega (Q_2)=Q_2$
and we deduce that there are involutions in
$C_{Q_1}(D)\setminus \Phi (Q_1)$
. Then for any maximal subgroup of A of
$\Phi (Q_1)$
containing
$Z(Q_1)$
, we have that
$C_{Q_1}(D)/A$
is either extraspecial or elementary abelian. If
$C_{Q_1}(D)/A$
is elementary abelian for all such A, then
$\Phi (C_{Q_1}(D))=Z(Q_1)$
, and so we assume that there is B with
$C_{Q_1}(D)/B$
extraspecial. Since
$L_1$
acts on
$C_{Q_1}(D)/B$
nontrivially and there are elements of order
$2$
in
$C_{Q_1}(D)\setminus \Phi (Q_1)$
, we have a contradiction by [Reference Delgado and Stellmacher8, (5.13)].
Lemma 5.5. Let
$x\in S\setminus Q_2$
. Then either:
-
1.
$p=2$
,
$|C_{Q_2/Z(S)}(x)|=q^4$
,
$|C_{\Phi (Q_1)/Z(S)}(x)|=q^3$
and writing D for the preimage in
$\Phi (Q_1)$
of
$C_{\Phi (Q_1)/Z(S)}(x)$
,
$C_{Q_2/Z(S)}(x)\le C_{Q_1\cap Q_2}(D)/Z(S)$
; or -
2. p is odd,
$|C_{Q_2/Z(S)}(x)|=q^3$
and
$C_{Q_2/Z(S)}(x)\le \Phi (Q_1)/Z(S)$
.
Proof. By [Reference Parker and Rowley21, Lemma 3.14 (iii)], we have that
$|C_{Q_2/Z(S)}(x)|=q^4$
when
$p=2$
and
$|C_{Q_2/Z(S)}(x)|=q^3$
when p is odd. Since
$S=Q_1Q_2$
,
$x=y_1y_2$
for some
$q_i\in Q_i$
, where
$i\in \{1,2\}$
. Furthermore, as
$[y_2, Q_2]\le Q_2'=Z(S)$
, we must have that
$C_{Q_2/Z(S)}(x)=C_{Q_2/Z(S)}(y_1)$
and by Lemma 5.3,
$|C_{\Phi (Q_1)/Z(S)}(y_1)|\geqslant q^3$
. Since
$C_S(\Phi (Q_1)/Z(S))=Q_2$
, it follows that
$|C_{\Phi (Q_1)/Z(S)}(y_1)|=q^3$
. Since
$\Phi (Q_1)\le Q_2$
, when p is odd we deduce that
$C_{Q_2/Z(S)}(x)=C_{\Phi (Q_1)/Z(S)}(x)\le \Phi (Q_1)/Z(S)$
and the result holds in this case. If
$p=2$
, then writing D for the preimage in
$\Phi (Q_1)$
of
$C_{\Phi (Q_1)/Z(S)}(x)$
, we have that
$[x, C_{Q_1}(D)]\le Z(Q_1)$
. In particular,
$[x, C_{Q_1\cap Q_2}(D)]\le Z(Q_1)$
and since
$|C_{Q_1\cap Q_2}(D)|=q^6$
and
$|Z(Q_1)|=q^2$
, we deduce that
$|C_{C_{Q_1\cap Q_2}(D)/Z(S)}(x)|=q^4$
and since
$|C_{Q_2/Z(S)}(x)|=q^4$
, we deduce that
$C_{Q_2/Z(S)}(x)\le C_{Q_1\cap Q_2}(D)/Z(S)$
. Indeed,
$C_{Q_1\cap Q_2}(D)/Z(S)=C_{Q_2/Z(S)}(x)(\Phi (Q_1)/Z(S))$
.
The above information will suffice for the determination of all potential essential subgroups of a saturated fusion system on S, which we achieve in Proposition 5.18. We now turn our focus to this endeavor, providing several structural features of a mooted essential subgroup of
$\mathcal {F}$
.
Proposition 5.6. Suppose that
$\mathcal {F}$
is a saturated fusion system supported on S and let
$E\in \mathcal {E}(\mathcal {F})$
. If
$E\le Q_2$
, then
$E=Q_2$
.
Proof. Suppose that
$E\le Q_2$
. Since
$\Omega (Z(S))=\Phi (Q_2)\le E$
, we have that
$E\trianglelefteq Q_2$
and that
$\Phi (E)\le \Omega (Z(S))$
. Assume first that E is elementary abelian. By Proposition 5.2,
$|E|\leqslant q^{5}$
and
$|N_{Q_2}(E)/E|\geqslant q^{4}$
. Since
$[Q_2, E]\le Q_2'= \Omega (Z(S))$
, Theorem 2.21 applied to E as an
$\mathrm {Out}_{\mathcal {F}}(E)$
-module gives a contradiction. Hence, E is not elementary abelian.
If
$\Phi (E)<\Omega (Z(S))$
, then for some maximal subgroup A of
$\Omega (Z(S))$
containing
$\Phi (E)$
, we have that
$Q_2/A$
is extraspecial and
$E/A$
is elementary abelian. Hence, by Lemma 2.36,
$|E/A|\leqslant p^{1+4n}$
so that
$|Q_2/E|\geqslant q^{4}$
. Since
$|[E/\Phi (E), Q_2]|\leqslant \Omega (Z(S))/\Phi (E)$
has order strictly less than q, Theorem 2.21 applied to
$E/\Phi (E)$
gives a contradiction.
Hence, we have that
$\Phi (E)=\Omega (Z(S))$
and
$Q_2$
centralizes the chain
$\{1\}\trianglelefteq \Phi (E)\trianglelefteq E$
so that by Lemma 2.13,
$E=Q_2$
.
Lemma 5.7. Suppose that
$\mathcal {F}$
is a saturated fusion system supported on S, and let
$E\in \mathcal {E}(\mathcal {F})$
. If
$Z(E)\not \le Q_2$
and
$Z(Q_1)\le E$
, then
$Z(Q_1)\le \Omega (Z(E))\le E\le Q_1$
,
$|E\cap \Phi (Q_1)|=q^4$
and
$\Omega (Z(E))(E\cap \Phi (Q_1))\in \mathcal {A}(E)\subseteq \mathcal {A}(S)$
.
Proof. Suppose that
$Z(E)\not \le Q_2$
and that
$Z(Q_1)\le E$
. Clearly, we have that
$Q_1\ne E\ne Q_2$
. Then
$[E, \Phi (Q_1)]\le [S, \Phi (Q_1)]=Z(Q_1)\le E$
so that
$\Phi (Q_1)\le N_S(E)$
. Now,
$Z(E)\cap Z(Q_1)\ge [Z(E), \Phi (Q_1)]Z(S)>Z(S)$
from which it follows that
$E\le Q_1$
and
$Z(Q_1)\le Z(E)$
. Since
$C_S(\Phi (Q_1))=\Phi (Q_1)$
and
$Z(E)\not \le Q_2$
, we deduce that
$\Phi (Q_1)\not \le E$
. Note that
$E\cap \Phi (Q_1)\le C_{\Phi (Q_1)}(Z(E))$
and since
$[E, \Phi (Q_1)]\le Z(Q_1)\le Z(E)$
,
$C_{\Phi (Q_1)}(Z(E))$
centralizes the chain
$\{1\}\trianglelefteq Z(E)\trianglelefteq E$
so that
$E\cap \Phi (Q_1)=C_{\Phi (Q_1)}(Z(E))$
. Thus,
$|E\cap \Phi (Q_1)|\in \{q^2, q^3, q^4\}$
by Lemma 5.3.
If
$|E\cap \Phi (Q_1)|\leqslant q^3$
, then
$|N_S(E)/E|\geqslant |\Phi (Q_1)E/E|\geqslant q^2$
. Since
$|[\Phi (Q_1), E]|\leqslant |Z(Q_1)|=q^2$
, an application of Theorem 2.21 yields that
$|E\cap \Phi (Q_1)|=q^3$
,
$N_S(E)=\Phi (Q_1)E$
and
$\Phi (E)\cap Z(Q_1)=\{1\}$
. Note that
$[C_E(O^{p'}(\mathrm {Out}_{\mathcal {F}}(E))), \Phi (Q_1)]=\{1\}$
so that
$C_E(O^{p'}(\mathrm {Out}_{\mathcal {F}}(E)))\le C_S(\Phi (Q_1))=\Phi (Q_1)$
. Let
$e\in [E, O^{p'}(\mathrm {Out}_{\mathcal {F}}(E))]\setminus (E\cap \Phi (Q_1))$
. Indeed, we can choose e to lie in the preimage of the unique noncentral chief factor for
$\mathrm {Out}_{\mathcal {F}}(E)$
in E with the property
$[e, x]\ne 1$
for any
$x\in \Phi (Q_1)\setminus (E\cap \Phi (Q_1))$
. But by Lemma 5.3,
$|C_{\Phi (Q_1)}(e)|=q^4$
, a contradiction since
$|E\cap \Phi (Q_1)|=q^3$
.
Hence, we have that
$|E\cap \Phi (Q_1)|=q^4$
. Note that
$[\Phi (Q_1), E]\le Z(Q_1)\le \Omega (Z(E))$
, and so
$\Omega (Z(E))$
contains all noncentral chief factors for
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(E))$
in E. Now,
$A:=\Omega (Z(E))(E\cap \Phi (Q_1))$
is elementary abelian and so has order at most
$q^5$
. But
$\Phi (Q_1)$
centralizes
$\Omega (Z(E))\cap \Phi (Q_1)$
and an application of Theorem 2.21 yields that
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(E))\cong \mathrm {SL}_2(q)$
and
$|\Omega (Z(E))/(\Omega (Z(E))\cap \Phi (Q_1))|=q$
so that
$|A|=q^5$
,
$m_p(E)=m_p(S)=5n$
and
$A\in \mathcal {A}(E)\cap \mathcal {A}(S)$
.
With the aim of demonstrating that
$E\in \{Q_1, Q_2\}$
, we suppose that E is an essential subgroup of
$\mathcal {F}$
chosen maximally such that
$Q_1\ne E\ne Q_2$
. By Proposition 5.6,
$E\not \le Q_2$
and so either E is maximally essential or
$E\le Q_1$
and
$Q_1\in \mathcal {E}(\mathcal {F})$
.
Lemma 5.8. Suppose that
$\mathcal {F}$
is a saturated fusion system supported on S. If
$Z(E)\not \le Q_2$
, then p is odd,
$E=\Omega (Z(E))$
,
$E\cap Z(Q_1)=Z(S)$
,
$S=EQ_1$
and
$N_S(E)=EZ(Q_1)$
.
Proof. We suppose first that
$Z(Q_1)\le E$
and let
$A=\Omega (Z(E))(E\cap \Phi (Q_1))$
so that
$A\in \mathcal {A}(S)$
by Lemma 5.7. In addition,
$Z(Q_1)\le \Omega (Z(E))\le E\le Q_1$
and
$|E\cap \Phi (Q_1)|=q^4$
. Assume that p is odd. Note that
$E\cap Q_2/Z(S)\le C_{Q_2/Z(S)}(\Omega (Z(E)))\le \Phi (Q_1)/Z(S)$
by Lemma 5.5. Since
$|\Omega (Z(E))\Phi (Q_1)/\Phi (Q_1)|=q$
, we conclude that
$C_{Q_1}(E\cap \Phi (Q_1))=\Omega (Z(E))C_{Q_1\cap Q_2}(E\cap \Phi (Q_1))$
.
If
$Q_1\in \mathcal {E}(\mathcal {F})$
, then
$p'$
-elements of
$O^{p'}(\mathrm {Aut}_{\mathcal {F}}(Q_1))$
necessarily act faithfully on
$\Phi (Q_1)$
(this follows from the three subgroups lemma) and since
$[S, \Phi (Q_1)]\le Z(Q_1)$
, we deduce that
$O^{p'}(\mathrm {Aut}_{\mathcal {F}}(Q_1))$
acts faithfully on
$Z(Q_1)$
and Theorem 2.21 implies that
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(Q_1))\cong \mathrm {SL}_2(q)$
. Then
$E\cap \Phi (Q_1)$
is normalized by
$O^{p'}(\mathrm {Aut}_{\mathcal {F}}(Q_1))$
and so too is
$C_{Q_1}(E\cap \Phi (Q_1))/\Phi (Q_1)$
. Indeed,
$C_{Q_1}(E\cap \Phi (Q_1))/\Phi (Q_1)$
is a natural module for
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(Q_1))$
by Theorem 2.21 and since
$O^{p'}(\mathrm {Aut}_{\mathcal {F}}(Q_1))$
is transitive on subgroups of order q in this action by Lemma 2.20, we infer that there is
$\alpha \in O^{p'}(\mathrm {Aut}_{\mathcal {F}}(Q_1))$
with
$\Omega (Z(E))\alpha \le Q_2$
but
$\Omega (Z(E))\not \le \Phi (Q_1)$
. Then
$[E\alpha , \Omega (Z(E))\alpha ]=[E, \Omega (Z(E))]\alpha =\{1\}$
and applying Lemma 5.5, we conclude that
$E\alpha \le Q_2$
. But then
$Q_2\le N_S(E\alpha )$
and since E is fully normalized, we have that
$N_S(E)=E\Phi (Q_1)$
has index at most
$q^3$
in S and is contained in
$Q_1$
. But
$N_S(E)\cap Q_2=\Phi (Q_1)(E\cap Q_2)\le \Phi (Q_1)$
and we have a contradiction.
Hence, we continue assuming now that
$Q_1\not \in \mathcal {E}(\mathcal {F})$
so that E is maximally essential. For
$\beta \in O^{p'}(\mathrm {Aut}_{\mathcal {F}}(E))$
of order
$q-1$
and normalizing
$\mathrm {Aut}_S(E)$
,
$\beta $
lifts to
$\widehat {\beta }$
in
$\mathrm {Aut}_{\mathcal {F}}(S)$
. Hence, upon restriction,
$\beta $
normalizes
$Z(S)$
. Since
$E\le Q_1$
, we have that
$\Phi (E)\le \Phi (Q_1)$
and as
$N_S(E)=E\Phi (Q_1)$
, we deduce that
$O^p(O^{p'}(\mathrm {Aut}_{\mathcal {F}}(E)))$
centralizes
$\Phi (E)$
. Moreover,
$[\Phi (Q_1), E]\le Z(Q_1)\le \Omega (Z(E))$
and writing
$P_E$
for the preimage in E of
$[E/\Phi (E), O^p(O^{p'}(\mathrm {Aut}_{\mathcal {F}}(E)))]$
and
$C_E$
for the preimage in E of
$C_{E/\Phi (E)}(O^p(O^{p'}(\mathrm {Aut}_{\mathcal {F}}(E))))$
, we have that
$P_E\le \Omega (Z(E))\Phi (E)$
and
$P_E\cap C_E\le \Phi (E)$
. Indeed, since
$[E, \Phi (Q_1)]\le Z(Q_1)$
and as
$\beta $
also normalizes
$Z(Q_1)\cap C_E$
, we infer that
$[\Phi (Q_1), C_E]\le Z(Q_1)\cap C_E\le Z(S)$
and
$C_E\le Q_2$
. Then
$C_E\le E\cap Q_2\le \Phi (Q_1)$
so that
$E=P_EC_E=\Omega (Z(E))(E\cap \Phi (Q_1))$
is elementary abelian and
$E=\Omega (Z(E))$
. Thus,
$E=P_E\times C_E$
,
$|P_E|=q^2$
and
$|C_E|=q^3$
. Again, since
$\beta $
normalizes
$Z(S)$
, we calculate that either
$Z(S)\le [E, O^{p'}(\mathrm {Aut}_{\mathcal {F}}(E))]$
or
$Z(S)\le C_E$
. Since
$E\not \le Q_2$
,
$[E, \Phi (Q_1)]\ne Z(S)$
and we have that
$Z(S)\le C_E$
. Now,
$Z(S)\le C_E\le \Phi (Q_1)$
so that
$C_E\trianglelefteq Q_2$
. Moreover,
$N_S(E)=E\Phi (Q_1)$
,
$N_S(E)/C_E\cong T\in \mathrm {Syl}_p(\mathrm {PSL}_3(q))$
and
$Z(N_S(E)/C_E)=Z(Q_1)C_E/C_E$
. Now,
$C_{Q_1\cap Q_2}(N_S(E)/Z(Q_1)C_E)$
normalizes E and so
$C_{Q_1\cap Q_2}(N_S(E)/Z(Q_1)C_E)\le \Phi (Q_1)$
. Then
$Q_1\cap Q_2/\Phi (Q_1)$
has order
$q^3$
and embeds in the automorphism group of
$N_S(E)/C_E$
. By [Reference Parker and Rowley20, Proposition 5.3], we have a contradiction.
Assume now that
$p=2$
. Since
$Z(E)\not \le Q_2$
, we have by Lemma 5.5 that
$E\cap Q_2\le C_{Q_1\cap Q_2}(E\cap \Phi (Q_1))$
. Indeed,
$C_{Q_1}(E\cap \Phi (Q_1))=\Omega (Z(E))C_{Q_1\cap Q_2}(E\cap \Phi (Q_1))$
. Let B be an
$L_1$
-conjugate of A contained in
$Q_1\cap Q_2$
. Since
$A\cap \Phi (Q_1)$
is normalized by
$L_1$
, we deduce that
$B\cap \Phi (Q_1)=A\cap \Phi (Q_1)$
has order
$q^4$
. Now,
$[A, B]\le Z(Q_1)$
by Lemma 5.4 so that
$[A, B, E]=\{1\}$
. Moreover,
$[A, E, B]=[E\cap \Phi (Q_1), E, B]\le [Z(Q_1), B]=\{1\}$
and the three subgroups lemma gives that
$[B, E, A]=\{1\}$
. Since
$[B, E]\le \Phi (Q_1)$
, we have that
$[B, E]\le C_{\Phi (Q_1)}(\Omega (Z(E)))=E\cap \Phi (Q_1)$
so that
$B\le N_S(E)$
. But now,
$B\Phi (Q_1)\le N_S(E)$
and since
$|N_S(E)/E|=q$
and
$|\Phi (Q_1)/E\cap \Phi (Q_1)|=q$
, we must have that
$|E\cap B\Phi (Q_1)|=q^5$
and
$E\cap B\Phi (Q_1)\le Q_2$
. But
$E\cap B\Phi (Q_1)$
centralizes A so that
$\Omega (E\cap B\Phi (Q_1))\le C_{B\Phi (Q_1)}(A)=E\cap \Phi (Q_1)$
. Then
$\{1\}\ne \mho ^1(E\cap B\Phi (Q_1))\le Z(S)\cap \mho ^1(E)$
. If
$E\not \le C_{Q_1}(E\cap \Phi (Q_1))$
, then
$|\Phi (E)\cap Z(Q_1)|\geqslant |[E, E\cap \Phi (Q_1)]\mho ^1(E)\cap Z(Q_1)|>q$
and as
$|\Phi (Q_1)E/E|=q$
, Theorem 2.21 provides a contradiction. Hence,
$E\le C_{Q_1}(E\cap \Phi (Q_1))$
so that
$A=\Omega (Z(E))$
. From this, we deduce that
$E=A$
. But
$A\cap B\Phi (Q_1)=E\cap \Phi (Q_1)$
has order
$q^4$
, a contradiction.
Therefore, if
$Z(E)\not \le Q_2$
, then
$Z(Q_1)\not \le E$
. If there is
$x\in (Z(Q_1)\cap E)\setminus Z(S)$
, then
$Z(E)\le Q_1$
so that
$[Z(E), Z(Q_1)]=\{1\}$
and
$Z(Q_1)$
centralizes the chain
$\{1\}\trianglelefteq Z(E)\trianglelefteq E$
, a contradiction. Thus,
$Z(Q_1)\cap E=Z(S)$
and since E is S-centric, we deduce that
$E\not \le Q_1$
. Furthermore,
$[E, Z(Q_1)]=Z(S)\le \Omega (Z(E))$
, and so we must have that
$\Omega (Z(E))\not \le Q_1$
. But now,
$|Z(Q_1)E/E|=q$
and
$C_{\Omega (Z(E))}(Z(Q_1))=\Omega (Z(E))\cap Q_1$
has index
$|\Omega (Z(E))Q_1/Q_1|$
in
$\Omega (Z(E))$
. By Theorem 2.21, we have that
$|\Omega (Z(E))Q_1/Q_1|=q$
and
$S=\Omega (Z(E))Q_1$
. Then
$E=\Omega (Z(E))(E\cap Q_1)$
. Furthermore, since
$\Omega (Z(E))\not \le Q_1$
,
$E\cap Q_1\le E\cap Q_1\cap Q_2$
so that
$\Phi (E)=\Phi (\Omega (Z(E))(E\cap Q_1\cap Q_2))=\Phi (E\cap Q_1\cap Q_2)\le Z(S)$
. Since
$[Z(Q_1), \Omega (Z(E))]=Z(S)$
, we have by Theorem 2.21 that
$\Phi (E)=\{1\}$
and E is elementary abelian. Again, since
$E\not \le Q_1$
,
$E\cap Q_1\le E\cap Q_1\cap Q_2$
. If
$p=2$
, then
$E\le (E\cap Q_1)(E\cap Q_2)\le Q_2$
, a contradiction by Proposition 5.6. Hence, p is odd.
Thus, whenever
$Z(E)\not \le Q_2$
, we see that
$Q_1\not \ge E\not \le Q_2$
and E is maximally essential in
$\mathcal {F}$
.
Proposition 5.9. Suppose that
$\mathcal {F}$
is a saturated fusion system supported on S. Then
$Z(E)\le Q_2$
.
Proof. Aiming for a contradiction, we assume that
$Z(E)\not \le Q_2$
so that by Lemma 5.8,
$E=\Omega (Z(E))\not \le Q_2$
,
$S=EQ_1$
,
$N_S(E)=EZ(Q_1)$
and p is odd. Since
$E\not \le Q_2$
, we have that
$EQ_2\cap Q_1\not \le Q_1\cap Q_2$
. However,
$[EQ_2\cap Q_1, E\cap \Phi (Q_1)]\le Z(S)$
and we deduce that
$EQ_2\le Q_2C_{Q_1}(E\cap \Phi (Q_1))$
. Then
$[Z(C_{Q_1}(E\cap \Phi (Q_1))), EQ_2]=Z(S)\le E$
so that
$Z(Q_1)(E\cap \Phi (Q_1))=Z(C_{Q_1}(E\cap \Phi (Q_1)))$
and
$|(E\cap \Phi (Q_1))Z(Q_1)|\in \{q^2, q^3, q^4\}$
. But now,
$\Phi (Q_1)\le N_S(N_S(E))$
. In particular,
$\Phi (Q_1)$
normalizes
$EZ(Q_1)/C_E(O^{p'}(\mathrm {Aut}_{\mathcal {F}}(E)))\cong T\in \mathrm {Syl}_p(\mathrm {PSL}_3(q))$
. Moreover, if
$x\in \Phi (Q_1)$
has
$[x, EZ(Q_1)]\le Z(S)C_E(O^{p'}(\mathrm {Aut}_{\mathcal {F}}(E)))$
, then
$x\in N_S(E)\cap \Phi (Q_1)=Z(Q_1)(E\cap \Phi (Q_1))$
. Applying [Reference Parker and Rowley20, Proposition 5.3],
$|\Phi (Q_1)/Z(Q_1)(E\cap \Phi (Q_1)|<q^2$
and we deduce that
$|Z(Q_1)(E\cap \Phi (Q_1))|=q^4$
and
$|N_S(N_S(E))/N_S(E)|<q^2$
.
Let t be a nontrivial element in
$Z(O^{p'}(\mathrm {Aut}_{\mathcal {F}}(E)))$
. Since we have chosen E to be maximally essential, applying the Alperin–Goldschmidt theorem and using that E is receptive, t lifts to
$\widehat {t}\in \mathrm {Aut}_{\mathcal {F}}(S)$
and normalizes
$Q_2$
. Note that
$\widehat {t}$
centralizes
$Z(Q_1)(E\cap Q_1)/Z(S)$
. Write
$C_t$
to be the preimage in
$Q_2$
of
$C_{Q_2/Z(S)}(\widehat {t})$
. Then
$[C_t, \widehat {t}, C_t]=\{1\}$
and since
$Z(S)$
is inverted by
$\widehat {t}$
, we deduce that
$C_t$
is abelian. Indeed, we must have that
$|[Q_1\cap Q_2/\Phi (Q_1), \widehat {t}]|\geqslant q^2$
. Moreover, we have that
$\Phi (Q_1)/Z(Q_1)(E\cap \Phi (Q_1))$
is inverted by
$\widehat {t}$
, for otherwise
$[E, t, C_{\Phi (Q_1)}(\widehat {t})]\le Z(S)$
and it follows that
$E\le Q_2$
. Therefore,
$E[Q_1\cap Q_2, \widehat {t}]/(E\cap \Phi (Q_1))Z(Q_1)$
is inverted by
$\widehat {t}$
is so is abelian. But then
$\Phi (Q_1)[Q_1\cap Q_2, \widehat {t}]\le N_S(N_S(E))$
. But then
$|N_S(N_S(E))/N_S(E)|\geqslant q^3$
, a contradiction. Hence,
$Z(E)\le Q_2$
, as desired.
Proposition 5.10. Suppose that
$\mathcal {F}$
is a saturated fusion system supported on S. Then
$Z(E)\le \Phi (Q_1)$
,
$Z(Q_1)\le E$
and
$\Phi (Q_1)\le N_S(E)$
.
Proof. Since
$Z(E)\le Q_2$
by Proposition 5.9, it follows immediately from Lemma 5.5 that
$Z(E)\le \Phi (Q_1)$
when p is odd. When
$p=2$
, we have that
$Z(E)\le Q_1\cap Q_2$
. In either case, since
$Z(Q_1)$
centralizes the chain
$\{1\}\trianglelefteq Z(E)\trianglelefteq E$
, we deduce that
$Z(Q_1)\le E$
. Then
$[\Phi (Q_1), E]\le [\Phi (Q_1), S]=Z(Q_1)\le E$
and
$\Phi (Q_1)\le N_S(E)$
. It remains to prove that
$Z(E)\le \Phi (Q_1)$
whenever
$p=2$
. Aiming for a contradiction, assume otherwise, noting that
$Z(E)\le Q_1\cap Q_2$
. We have that
$|E\cap \Phi (Q_1)|\leqslant q^4$
since
$Z(E)\not \le \Phi (Q_1)$
and as
$E\not \le Q_2$
, Lemma 5.5 yields that
$Z(E)\cap \Phi (Q_1)$
has index at most q in
$Z(E)$
.
Suppose first that
$Z(E)$
contains no noncentral chief factors for
$O^{2'}(\mathrm {Aut}_{\mathcal {F}}(E))$
so that
$Z(S)$
is normalized by
$O^{2'}(\mathrm {Aut}_{\mathcal {F}}(E))$
. Note that
$E\not \le Q_1$
for otherwise
$Z(Q_1)\le Z(E)$
and since
$\Phi (Q_1)$
centralizes
$E/Z(Q_1)$
, we would have a contradiction. Set
$Z_E$
to be the preimage in E of
$Z(E/Z(S))$
so that
$Z_E$
is normalized by
$O^{2'}(\mathrm {Aut}_{\mathcal {F}}(E))$
and contains
$Z(Q_1)$
. Now, since
$L_1/Q_1$
acts on
$Q_1/\Phi (Q_1)$
as a direct sum of three natural
$\mathrm {SL}_2(q)$
-modules, we either have that
$E\cap Q_1\le Q_1\cap Q_2$
so that
$\Omega (E)\le Q_2$
; or
$Z_E\le Q_1\cap Q_2$
. In either case, since
$[\Phi (Q_1), E]\le Z(Q_1)\le Z_E\cap \Omega (E)$
and
$[\Phi (Q_1), Q_2]\le Z(S)\le Z(E)$
,
$\Phi (Q_1)$
centralizes a chain and we have a contradiction.
Hence,
$Z(E)$
contains a noncentral chief factor for
$O^{2'}(\mathrm {Aut}_{\mathcal {F}}(E))$
and applying Theorem 2.21 we infer that
$|E\cap \Phi (Q_1)|=q^4$
,
$|Z(E)/Z(E)\cap \Phi (Q_1)|=q$
and
$N_S(E)=E\Phi (Q_1)$
. In particular,
$A:=Z(E)(E\cap \Phi (Q_1))$
is elementary abelian of order
$q^5$
. If
$E\not \le Q_1$
, then
$Z(S)=[E, Z(Q_1)]\le \Phi (E)$
. Note that if
$Z(Q_1)\le \Phi (E)$
, then
$\Phi (Q_1)$
centralizes
$E/\Phi (E)$
, and so we must have that
$[E, E\cap \Phi (Q_1)]\le Z(S)$
. Now, for
$e\in \Omega (E)\cap Q_1$
an involution, we have that
$[e, E\cap \Phi (Q_1)]\le Z(S)$
, and so either
$e\in E\cap Q_2$
or
$[e, E\cap \Phi (Q_1)]=\{1\}$
. In the latter case,
$[e, A]=\{1\}$
so that
$e\in A\le Q_2$
. Hence,
$Z(Q_1)\le \Omega (E)\le Q_2$
and as
$Z(S)\le \Phi (E)\le \Phi (E)\Omega (E)\le Q_2$
,
$\Phi (Q_1)$
centralizes the chain
$\Phi (E)\trianglelefteq \Phi (E)\Omega (E)\trianglelefteq E$
, a contradiction.
Finally, we deduce that
$Z(E)$
contains a noncentral chief factor for
$O^{2'}(\mathrm {Aut}_{\mathcal {F}}(E))$
and
$E\le Q_1$
. In particular,
$Z(Q_1)\le Z(E)\le Q_1\cap Q_2$
and so
$Z(S)\cap \Phi (E)=\{1\}$
by Theorem 2.21. Thus,
$[E\cap Q_2, E\cap Q_2]=\{1\}$
and since
$A\le E\cap Q_2$
is maximal abelian, we infer that
$A=E\cap Q_2$
. Moreover, any
$e\in E\setminus A$
is not contained in
$Q_2$
and centralizes
$Z(E)$
and Lemma 5.5 implies that
$E\le C_{Q_1}(E\cap \Phi (Q_1))$
. Hence,
$A=\Omega (E)=Z(E)$
. But
$[Q_2, A]\le Z(S)\le A$
and we calculate that
$E=C_S(A)$
so that E is normalized by
$Q_2$
. Since
$N_S(E)=E\Phi (Q_1)$
, this is a contradiction.
Proposition 5.11. Suppose that
$\mathcal {F}$
is a saturated fusion system supported on S. Then
$\Phi (Q_1)\le E$
,
$Q_1\cap Q_2\le N_S(E)$
and
$m_p(E)=m_p(S)$
.
Proof. Aiming for a contradiction, assume throughout that
$\Phi (Q_1)\not \le E$
. Since
$Z(S)\le Z(E)$
,
$[\Phi (Q_1), E/Z(E)]\le Z(Q_1)Z(E)/Z(E)$
and
$[\Phi (Q_1), Z(E)]=\{1\}$
, we must have by Theorem 2.21 that
$|E\cap \Phi (Q_1)|\geqslant q^4$
and
$Z(E)\cap Z(Q_1)=Z(S)$
. Another application of Theorem 2.21 yields that
$\Phi (E)\cap Z(Q_1)=Z(S)$
. In particular,
$[E,E\cap \Phi (Q_1)]=Z(S)$
and we have that
$|E\cap \Phi (Q_1)|=q^4$
.
Since
$[\Phi (Q_1), E/Z(E)]=Z(Q_1)Z(E)/Z(E)$
, Theorem 2.21 implies that
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(E))\cong \mathrm {SL}_2(q)$
,
$N_S(E)=E\Phi (Q_1)$
and there is a unique noncentral chief factor for
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(E))$
inside E. Indeed,
$O^{p'}(\mathrm {Aut}_{\mathcal {F}}(E))$
acts trivially on
$Z(S)$
and normalizes D, the preimage in E of
$Z(E/Z(S))$
.
Assume that
$\Omega (D)\not \le Q_2$
. In particular,
$[D, \Phi (Q_1)]\not \le Z(S)$
, and so we must have that
$[\Omega (D), \Phi (Q_1)]Z(S)=Z(Q_1)$
. Hence,
$[\Phi (Q_1), E]\le Z(Q_1)\le \Omega (D)$
and
$\Omega (D)/Z(E)$
contains the unique noncentral chief factor for
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(E))$
inside E. If
$p=2$
, then
$\Omega (D)=(\Omega (D)\cap Q_1)(\Omega (D)\cap Q_2)$
by Proposition 5.2 and
$(\Omega (D)\cap Q_1)\not \le Q_2$
. Note that if
$\Omega (D)\cap Q_2\not \le Q_1\cap Q_2$
, then
$\Phi (Q_1)\not \ge [\Omega (D)\cap Q_1, \Omega (D)\cap Q_2]\le Z(S)$
, a contradiction. Hence,
$\Omega (D)\cap Q_2\le Q_1$
and
$\Omega (D)\le Q_1$
. But then
$Z(Q_1)\le \Omega (Z(\Omega (D)))$
so that
$[\Phi (Q_1), E]=Z(Q_1)\le \Omega (Z(\Omega (D)))$
and
$\Omega (Z(\Omega (D)))$
contains the unique noncentral chief factor for
$O^{2'}(\mathrm {Out}_{\mathcal {F}}(E))$
inside E. But then for r an odd order element of
$O^{2'}(\mathrm {Aut}_{\mathcal {F}}(E))$
,
$[r, E, D]=\{1\}$
and
$[E, D, r]=\{1\}$
and by the three subgroups lemma,
$[r, D]\le Z(E)$
so that
$[D, r]=[D, r, r]\le [Z(E), r]=\{1\}$
, a contradiction since D contains a noncentral chief factor.
If p is odd, then
$C_{(E\cap Q_2)/Z(S)}(D)\le C_{Q_2/Z(S)}(D)\cap E/Z(S)\le (E\cap \Phi (Q_1))/Z(S)$
by Lemma 5.5. Hence,
$E\cap Q_2\le \Phi (Q_1)$
. Since
$[E, Z(Q_1)]\ne \{1\}$
,
$E\not \le Q_1$
and since
$[E, E\cap Q_1]\le \Phi (E)\le E\cap Q_2\le \Phi (Q_1)$
, we deduce that
$E\cap Q_1\le E\cap Q_1\cap Q_2\le \Phi (Q_1)$
. But now, since
$|N_S(E)/E|=q$
and D contains the unique noncentral chief factor within E, we deduce that
$S=DQ_1$
and
$E=D(E\cap \Phi (Q_1))$
is nonabelian. Hence,
$E'=Z(S)$
. Note that
$C_{E}(O^p(\mathrm {Aut}_{\mathcal {F}}(E)))\le \Phi (Q_1)$
from which it follows that
$C_{E}(O^p(\mathrm {Aut}_{\mathcal {F}}(E)))=Z(E)$
is of order
$q^3$
. Furthermore, we have that
$\Phi (Q_1)E/Z(E)\cong T\in \mathrm {Syl}_p(\mathrm {PSL}_3(q))$
and
$Z(\Phi (Q_1)E/Z(E))=Z(Q_1)Z(E)/Z(E)$
. Now,
$Q_1\cap Q_2$
normalizes
$Z(E)$
and
$\Phi (Q_1)E$
and so acts on
$\Phi (Q_1)E/Z(E)$
. Moreover, for
$x\in Q_1\cap Q_2$
, if
$[x, \Phi (Q_1)E]\le Z(E)Z(Q_1)$
, then
$x\in N_{Q_2}(E)=\Phi (Q_1)$
. Applying [Reference Parker and Rowley20, Proposition 5.3],
$|(Q_1\cap Q_2)E/\Phi (E)E|<q^2$
, a contradiction.
Therefore,
$\Omega (D)\le Q_2$
and
$[\Omega (D), \Phi (Q_1)]=Z(S)\le Z(E) $
. Then, as
$[E, \Phi (Q_1)]\le Z(Q_1)\le \Omega (D)$
,
$\Phi (Q_1)$
centralizes the chain
$\{1\}\trianglelefteq Z(E)\trianglelefteq \Omega (D)\trianglelefteq E$
, a contradiction. Hence,
$\Phi (Q_1)\le E$
. Since
$\Phi (Q_1)\in \mathcal {A}(S)$
, we have that
$m_p(S)=m_p(E)=5n$
and
$\Phi (Q_1)\in \mathcal {A}(E)$
. Since
$[E, Q_1\cap Q_2]\le [S, Q_1\cap Q_2]=\Phi (Q_1)\le E$
, we have that
$Q_1\cap Q_2\le N_S(E)$
.
Since
$\Phi (Q_1)\le E$
and
$\Phi (Q_1)\in \mathcal {A}_{\trianglelefteq }(S)$
,
$\Phi (Q_1)\in \mathcal {A}_{\trianglelefteq }(E)\ne \emptyset $
, and so
$J_{\trianglelefteq }(E)$
is defined. We will use this group and its properties frequently in the following lemmas and propositions.
Proposition 5.12. Suppose that
$\mathcal {F}$
is a saturated fusion system supported on S and let
$E\in \mathcal {E}(\mathcal {F})$
. Then either
$J_{\trianglelefteq }(E)\le Q_1$
or
$J_{\trianglelefteq }(E)\le Q_2$
.
Proof. Aiming for a contradiction, we suppose throughout that there are
$A,C\in \mathcal {A}_{\trianglelefteq }(E)$
with
$A\not \le Q_1$
and
$C\not \le Q_2$
. Since
$\Phi (Q_1)$
is a normal elementary abelian subgroup of E and
$m_p(S)=m_p(E)=5n$
, we have that
$|A|=|C|=q^5$
. Moreover, since
$A\not \le Q_1$
,
$A\cap Z(Q_1)=C_{Z(Q_1)}(A)=Z(S)$
. Then
$B:=(A\cap Q_1)Z(Q_1)$
is elementary abelian of order
$|A\cap Q_1|q$
and we conclude that
$|A\cap Q_1|=q^4$
and
$S=AQ_1$
. Since
$A\not \le Q_1$
and A centralizes
$A\cap Q_1$
, we have that
$A\cap Q_1\le Q_1\cap Q_2$
and
$B\le Q_1\cap Q_2$
. If
$p=2$
, then by Proposition 5.2, we have that
$A\le Q_2$
. If p is odd, then
$[A, B]=[A, Z(Q_1)]=Z(S)$
and by Lemma 5.5, since
$|B/Z(S)|=q^4$
, we have again that
$A\le Q_2$
. In particular,
$Q_2=A(Q_1\cap Q_2)$
.
Since
$Q_2=C_S(\Phi (E)/Z(S))$
, we have that
$Z(Q_1)=[C, \Phi (E)]Z(S)\le C$
so that
$C\le Q_1$
. Let
$c\in C\setminus (C\cap Q_2)$
. Then by Lemma 5.3, for
$F_c:=C_{\Phi (Q_1)}(c)$
,
$|F_c|=q^4$
,
$|C_{Q_1}(F_c)|=q^7$
and
$|C_{Q_1}(F_c)\cap Q_2|=q^6$
. Note that
$[c, A]\Phi (Q_1)=C_{Q_1}(F_c)\cap Q_2$
and as A normalizes C, we deduce that
$|(C\cap C_{Q_1}(F_c))\Phi (Q_1)|>q^6$
. But then,
$(C\cap C_{Q_1}(F_c))F_c$
is elementary abelian of order strictly larger than
$q^5$
, a contradiction. Hence, either
$J_{\trianglelefteq }(E)\le Q_1$
or
$J_{\trianglelefteq }(E)\le Q_2$
.
Proposition 5.13. Suppose that
$\mathcal {F}$
is a saturated fusion system supported on S and let
$E\in \mathcal {E}(\mathcal {F})$
. If
$Q_2\le E$
, then
$E=Q_2$
.
Proof. Since
$Q_2\le E$
, we ascertain immediately that
$E\trianglelefteq S$
,
$Z(E)=Z(S)$
and
$Q_2$
is the preimage in E of
$J(E/Z(E))$
. Thus,
$Q_2$
is characteristic in E. Aiming for a contradiction, we assume throughout that
$Q_2<E$
, and so there is
$x\in (E\cap Q_1)\setminus (Q_1\cap Q_2)$
. In particular,
$Z(S)[Q_2, x]\le \Phi (E)$
. We remark that
$\Phi (E)<Q_1\cap Q_2$
, else S centralizes
$E/\Phi (E)$
.
Suppose first that p is odd. By Lemma 5.5, we have that
$|C_{Q_2/Z(S)}(x)|=q^3$
, and so we deduce that
$|Z(S)[Q_2, x]|=q^6$
. Indeed, it follows that
$|Z(S)[Q_1\cap Q_2, x]|=q^5$
and since
$x\in Q_1$
,
$[x, Q_1\cap Q_2]\le \Phi (Q_1)$
and we deduce that
$\Phi (Q_1)=Z(S)[Q_1\cap Q_2, x]\le \Phi (E)$
. Since
$[S, E]\le Q_1\cap Q_2\le Q_2$
and
$[S, Q_1\cap Q_2]\le \Phi (Q_1)$
, we deduce that
$O^{p'}(\mathrm {Aut}_{\mathcal {F}}(E))$
centralizes
$E/Q_2$
. Moreover, Theorem 2.21 reveals that
$|\Phi (E)|=q^7$
,
$|S/E|=q$
and
$Q_2/\Phi (E)$
is a natural module for
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(E))\cong \mathrm {SL}_2(q)$
. Now,
$E/\Phi (E)$
splits by coprime action, and we can arrange that x lies in the preimage in E of
$C_{E/\Phi (E)}(O^{p'}(\mathrm {Out}_{\mathcal {F}}(E)))$
. Hence,
$E_x:=\langle x\rangle \Phi (E)$
is normalized by
$O^{p'}(\mathrm {Aut}_{\mathcal {F}}(E))$
. Note that the preimage of
$Z(E_x/Z(S))$
is equal to
$C_{\Phi (Q_1)}(x)$
of order
$q^4$
and normalized by
$O^{p'}(\mathrm {Aut}_{\mathcal {F}}(E))$
. Note we can also choose y in the preimage in E of
$C_{E/\Phi (E)}(O^{p'}(\mathrm {Out}_{\mathcal {F}}(E)))$
with
$y\in (E\cap Q_1)\setminus (Q_1\cap Q_2)$
and
$C_{\Phi (Q_1)}(x)\ne C_{\Phi (Q_1}(y)$
, and we may form
$E_y$
normalized by
$O^{p'}(\mathrm {Aut}_{\mathcal {F}}(E))$
. Then
$Z_2(E)=C_{\Phi (Q_1)}(x)\cap C_{\Phi (Q_1)}(y)$
and
$\Phi (Q_1)=C_{\Phi (Q_1)}(x)C_{\Phi (Q_1)}(y)$
is normalized by
$O^{p'}(\mathrm {Aut}_{\mathcal {F}}(E))$
. But then
$Z(Q_1)=[E, \Phi (Q_1)]$
is normalized by
$O^{p'}(\mathrm {Aut}_{\mathcal {F}}(E))$
and as
$Q_1$
centralizes the chain
$\{1\}\trianglelefteq Z(Q_1)\trianglelefteq \Phi (Q_1)$
, we have that all
$p'$
-elements in
$O^{p'}(\mathrm {Aut}_{\mathcal {F}}(E))$
centralize
$\Phi (Q_1)$
. Since
$\Phi (Q_1)$
is self-centralizing in E, the three subgroups lemma implies that all
$p'$
-elements in
$O^{p'}(\mathrm {Aut}_{\mathcal {F}}(E))$
centralize E, a contradiction.
Hence, we have that
$p=2$
. We employ the fusion systems package in MAGMA [Reference Parker and Semeraro23] when
$q=2$
to deduce that
$E=Q_2$
. Hence, we may assume that
$q>2$
. As in the odd prime case, we recognize that
$O^{2'}(\mathrm {Aut}_{\mathcal {F}}(E))$
centralizes
$E/Q_2$
. Let
$x\in (E\cap Q_1)\setminus Q_2$
, and write
$C_x$
for the preimage in
$Q_2$
of
$C_{Q_2/Z(S)}(x)$
. Then, applying Lemma 5.5,
$C_x=[x, Q_2]Z(S)$
is of order
$q^5$
and is not contained in
$\Phi (Q_1)$
. Indeed,
$C_x\cap \Phi (Q_1)=C_{\Phi (Q_1)}(x)$
has order
$q^4$
. We suppose first that
$C_x$
is the preimage in
$Q_2$
of
$C_{Q_2/Z(S)}(E)$
so that
$|S/E|\geqslant q^2$
. Note that as
$E_x:=\langle x\rangle Q_2$
is normalized by
$O^{p'}(\mathrm {Aut}_{\mathcal {F}}(Q_2))$
, so too is
$C_x=[E_x, Q_2]$
. Indeed,
$|Q_2/C_x|=|C_x/Z(S)|=q^4$
and at least one of these quotients contains a noncentral chief factor for
$O^{2'}(\mathrm {Out}_{\mathcal {F}}(E))$
. Note that
$C_x=[x, Q_2]Z(S)\le \Phi (E)$
, and so
$Q_2/C_x$
contains a noncentral chief factor. Indeed, applying Proposition 2.33 and Lemma 2.30 we have that
$C_x=\Phi (E)$
and
$Q_2/\Phi (E)$
is irreducible for
$O^{2'}(\mathrm {Out}_{\mathcal {F}}(E))\cong \mathrm {SL}_2(q^2)$
. We assume now that there are
$x,y\in E\setminus Q_2$
with
$C_x\cap \Phi (Q_1)\ne C_y\cap \Phi (Q_1)$
. Indeed, applying Lemma 5.5 we deduce that
$|C_x\cap C_y|=q^3$
and
$|C_xC_y|=q^7$
. Since
$C_xC_y\le \Phi (E)$
, we deduce that
$C_xC_y=\Phi (E)$
for otherwise, using the structure of
$Q_1/\Phi (Q_1)$
as a
$L_1/Q_1$
-module, we would have that
$\Phi (E)=Q_1\cap Q_2$
and S would centralize
$E/\Phi (E)$
. Hence,
$|S/E|\geqslant q$
and
$O^{2'}(\mathrm {Out}_{\mathcal {F}}(E))$
acts nontrivially on
$Q_2/C_xC_y$
. Applying Proposition 2.33 and Lemma 2.30, we have that
$Q_2/C_xC_y$
is a natural module for
$O^{2'}(\mathrm {Out}_{\mathcal {F}}(E))\cong \mathrm {SL}_2(q)$
.
Therefore, we have that
$p=2$
,
$q>2$
and
$O^{2'}(\mathrm {Out}_{\mathcal {F}}(E))\cong \mathrm {SL}_2(q^a)$
for
$a\in \{1,2\}$
. Set
$K_E$
the group of order
$q^a-1$
obtained by lifting a Hall
$2'$
-subgroup of
$N_{O^{2'}(\mathrm {Aut}_{\mathcal {F}}(E))}(\mathrm {Aut}_S(E))$
to
$\mathrm {Aut}(S)$
, set
$K_2$
a Hall
$2'$
-subgroup of
$\mathrm {Aut}_{L_2}(S)$
, and set
$H:=\langle K_E, K_2\rangle \le \mathrm {Aut}(S)$
. Let
$\widehat {K_E}$
be the subgroup of
$\mathrm {Aut}(S/Q_2)$
induced by
$K_E$
, and define
$\widehat {K_2}$
and
$\widehat {H}$
similarly. In particular,
$K_E\cong \widehat {K_E}$
,
$K_2\cong \widehat {K_2}$
and
$\overline {H}:=\langle \widehat {K_2}, \widehat {K_E}\rangle \le \widehat {H}$
. Since
$\widehat {K_2}$
is transitive on involutions in
$S/Q_2$
, and
$\widehat {K_E}$
fixes a subgroup of
$S/Q_2$
of order
$q^{3-a}$
, we deduce that
$\widehat {K_2}\cap \widehat {K_E}=\{1\}$
. Indeed,
$|\overline {H}|\geqslant |\widehat {K_E}||\widehat {K_2}|=(q^3-1)(q^a-1)$
. Now,
$\widehat {K_2}$
is a Singer cycle of
$\mathrm {Aut}(S/Q_2)\cong \mathrm {GL}_{3n}(2)$
and [Reference Kantor15] yields that
$\mathrm {GL}_{3n/s}(2^s)\trianglelefteq \overline {H}\le \varGamma L_{3n/s}(2^s)$
for some s which divides
$3n$
. If
$s=3n$
, then
$\overline {H}$
lies in the normalizer of a Singer cycle. However, the normalizer of a Singer cycle in
$\mathrm {GL}_{3n}(2)$
has order
$3n(q^3-1)$
by [Reference Huppert14, 7.3 Satz] which is only divisible by
$\widehat {H}$
if
$a=1$
and
$n=2$
. In this case, we compute directly in
$\mathrm {GL}_6(2)$
that
$\widehat {K_2}\trianglelefteq \overline {H}$
and that no element of order
$3$
in
$\overline {H}$
fixes a subgroup of
$S/Q_2$
of order
$2^4$
, a contradiction since
$\widehat {K_E}\le \overline {H}$
.
Hence,
$s<3n$
and
$\overline {H}$
is not solvable. We note that H is trivial on
$Z(S)$
. Since
$C_H(Z(Q_1))\trianglelefteq H$
and
$|Z(Q_1)/Z(S)|=q$
, we deduce that
$C_H(Z(Q_1))$
has a composition factor isomorphic to
$\mathrm {PSL}_{3n/s}(2^s)$
. Moreover, the three subgroups lemma yields that
$C_H(Z(Q_1))$
centralizes
$S/Q_1$
so odd order elements of
$C_H(Z(Q_1))$
act faithfully on
$Q_1/Q_1\cap Q_2$
. By the three subgroups lemma, we have that
$C_{\mathrm {Aut}(Q_1)}(Z(Q_1))\cap C_{\mathrm {Aut}(Q_1)}(\Phi (Q_1)/Z(Q_1))$
centralizes
$Q_1/\Phi (Q_1)$
from which we infer that
$L_1/Q_1$
embeds faithfully in
$C_{\mathrm {Aut}(Q_1)}(\Phi (Q_1)/Z(Q_1))$
. In particular, the subgroups generated by induced actions of
$L_1$
and
$C_H(Z(Q_1))$
on
$Q_1/\Phi (Q_1)$
commute and intersect trivially in
$\mathrm {Aut}(Q_1/\Phi (Q_1))$
. Set T to be the subgroup of
$\mathrm {Aut}(Q_1/Q_1\cap Q_2)$
induced by a Hall
$2'$
-subgroup of
$N_{L_1}(S)$
so that T has order
$q-1$
, and write C the subgroup of
$\mathrm {Aut}(Q_1/Q_1\cap Q_2)$
induced by
$C_H(Z(Q_1))$
. Hence, we have a subgroup of shape
$C\times T$
contained in
$\mathrm {Aut}(Q_1/Q_1\cap Q_2)\cong \mathrm {GL}_{3n}(2)$
, and C has a normal subgroup isomorphic to
$\mathrm {SL}_{3n/s}(2^s)$
which we denote
$C^*$
. Indeed,
$C^*$
is also normalized by the subgroup of
$\mathrm {Aut}(Q_1/Q_1\cap Q_2)$
induced by H, which we denote
$\widetilde H$
. Since
$K_2$
restricts faithfully to
$Q_1/Q_1\cap Q_2$
, we have that
$X:=\langle \widetilde H, T\rangle $
contains a Singer cycle of
$\mathrm {Aut}(Q_1/Q_1\cap Q_2)$
. Applying [Reference Kantor15] again, and using that
$C^*\trianglelefteq X$
, we must have that
$\mathrm {GL}_{3n/s}(2^s)\trianglelefteq X\trianglelefteq \varGamma L_{3n/s}(2^s)$
and as T centralizes
$C^*$
and intersects
$C^*$
trivially, we have a contradiction. This completes the proof.
Proposition 5.14. Suppose that
$\mathcal {F}$
is a saturated fusion system supported on S, and let
$E\in \mathcal {E}(\mathcal {F})$
. Then either
$E=Q_2$
or
$\Phi (Q_1)\le J_{\trianglelefteq }(E)\le Q_1$
.
Proof. We will suppose that
$E\ne Q_2$
with the aim of proving that
$\Phi (Q_1)\le J_{\trianglelefteq }(E)\le Q_1$
. By Proposition 5.13, we may assume that
$E\not \le Q_2$
. By Proposition 5.11, we have that
$\Phi (Q_1)\le E$
, and so it will be enough to show that
$J_{\trianglelefteq }(E)\le Q_1$
.
Aiming for a contradiction, we assume that
$J_{\trianglelefteq }(E)\not \le Q_1$
. Applying Proposition 5.12, we see that
$J_{\trianglelefteq }(E)\le Q_2$
and there is
$A\in \mathcal {A}_{\trianglelefteq }(E)$
with
$A\not \le Q_1$
. Indeed, it follows that
$Q_2=A(Q_1\cap Q_2)$
. Note, if
$Q_1\cap Q_2\not \le E$
, then
$[E, Q_1\cap Q_2]\le \Phi (Q_1)\le J_{\trianglelefteq }(E)$
. But now,
$Z(S)=[A, \Phi (E)]\le [J_{\trianglelefteq }(E), \Phi (Q_1)]\le \Phi (J_{\trianglelefteq }(E))\le Z(S)$
. Hence,
$Q_1\cap Q_2$
centralizes the chain
$\{1\}\trianglelefteq \Phi (J_{\trianglelefteq }(E))\trianglelefteq J_{\trianglelefteq }(E)\trianglelefteq E$
, a contradiction. Thus,
$Q_2=A(Q_1\cap Q_2)\le E$
and by Proposition 5.13,
$E=Q_2$
, a contradiction.
Lemma 5.15. If
$\Phi (Q_1)\le J_{\trianglelefteq }(E)\not \le Q_1\cap Q_2$
, then
$E\le Q_1$
and
$E\cap Q_2>\Phi (Q_1)$
.
Proof. By Proposition 5.14,
$Z(Q_1)\le Z(J_{\trianglelefteq }(E))\le \Phi (Q_1)\le J_{\trianglelefteq }(E)\le E\cap Q_1$
. Aiming for a contradiction, we assume that that there is
$A\in \mathcal {A}_{\trianglelefteq }(E)$
with
$A\not \le Q_2$
. If
$E\not \le Q_1$
, then for
$a\in A\setminus (A\cap Q_2)$
, we have that for
$D_a:=C_{\Phi (Q_1)}(a)$
,
$|D_a|=q^4$
and
$L:=C_{Q_1\cap Q_2}(D_a)=[E, a]\Phi (Q_1)$
. Indeed, we have that
$|(L\cap A)D_a|\geqslant q^5$
so that
$\langle a\rangle (L\cap A)D_a$
is elementary abelian of order strictly larger than
$q^5$
, a contradiction. Hence,
$E\le Q_1$
.
Since
$\Phi (Q_1)<E< Q_1$
we have that
$Z(Q_1)\le Z(E)<\Phi (Q_1)$
. Aiming for a contradiction, suppose that
$E\cap Q_2=\Phi (Q_1)$
. Then
$Q_1\cap Q_2$
normalizes E,
$|E(Q_1\cap Q_2)/E|=q^3$
and
$\Phi (Q_1)$
has index at most
$q^3$
in E. Since
$Z(E)\le \Phi (Q_1)$
and
$[Q_1\cap Q_2, \Phi (Q_1)]=Z(S)$
, an application of Theorem 2.21 to
$E/Z(E)$
and
$Z(E)$
yields that
$Z(E)=Z(Q_1)\ge \Phi (E)$
,
$|E/\Phi (Q_1)|=q^3$
,
$N_S(E)=Q_1=E(Q_1\cap Q_2)$
and
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(E))\cong \mathrm {SL}_2(q^3)$
. Since E is maximally chosen, the only other essential of
$\mathcal {F}$
that could contain E is
$Q_1$
.
If
$Q_1$
is essential in
$\mathcal {F}$
, then as
$[S, \Phi (Q_1)]\le Z(Q_1)$
, we deduce that
$\mathrm {Aut}_{\mathcal {F}}(Q_1)$
acts trivially on
$\Phi (Q_1)/Z(Q_1)$
and so normalizes the subspaces described in Lemma 5.3. In particular, for
$D\le \Phi (Q_1)$
of order
$q^4$
with
$C_{Q_1}(D)$
of order
$q^7$
, we have that
$\mathrm {Aut}_{\mathcal {F}}(Q_1)$
normalizes
$C_{Q_1}(D)$
. By Theorem 2.21,
$\mathrm {Out}_{\mathcal {F}}(Q_1)$
induces a natural
$\mathrm {SL}_2(q)$
-action on
$C_{Q_1}(D)/\Phi (Q_1)$
. Choose
$\alpha \in \mathrm {Aut}_{\mathcal {F}}(Q_1)$
with
$(E\cap C_{Q_1}(D))\alpha =C_{Q_1\cap Q_2}(D)$
. Since
$Q_1=N_S(E)$
, E is fully
$\mathcal {F}$
-normalized and
$E\le Q_1$
, it follows that
$E\alpha $
is also essential in
$\mathcal {F}$
. Moreover,
$Z(E)\alpha =Z(E\alpha )=Z(Q_1)$
and since E is fully normalized,
$E\alpha \ne Q_1\cap Q_2$
. But now, for all
$e\in E\alpha \setminus C_{Q_1\cap Q_2}(D)$
,
$[e, E\alpha \cap Q_2]\le [E\alpha , E\alpha ]=[E, E]\alpha =Z(Q_1)$
. Then
$|C_{E\alpha \cap Q_2/Z(S)}(e)|\geqslant q^4$
and Lemma 5.5 provides a contradiction. Hence,
$Q_1$
is not essential in
$\mathcal {F}$
and E is maximally essential.
Let r be an element of order
$q-1$
in
$O^{p'}(\mathrm {Aut}_{\mathcal {F}}(E))$
which normalizes
$\mathrm {Aut}_S(E)=\mathrm {Aut}_{Q_1}(E)$
. Using the receptiveness of E and the Alperin–Goldschmidt theorem, r lifts an automorphism
$\widehat {r}\in \mathrm {Aut}_{\mathcal {F}}(S)$
such that
$\widehat {r}|_E=r$
. Since r centralizes
$Z(Q_1)$
, we have that
$[\widehat {r}, S]\le C_S(Z(Q_1))=Q_1$
by the three subgroups lemma. Since
$\widehat {r}|_E=r$
and
$Q_1=N_S(E)$
, we have that
$[\widehat {r}, Q_1]\le E$
. Hence,
$[\widehat {r}, S]\le E$
. Since
$Q_2$
is characteristic in S,
$\widehat {r}$
normalizes
$Q_2$
so that
$[\widehat {r}, Q_2]\le [\widehat {r}, S]\cap Q_2=E\cap Q_2=\Phi (Q_1)$
. But then
$[\widehat {r}, Q_2, \Phi (Q_1)]=\{1\}$
,
$[Q_2, \Phi (Q_1), \widehat {r}]=\{1\}$
so that
$[\widehat {r}, \Phi (Q_1), Q_2]=\{1\}$
by the three subgroups lemma. But
$C_{\Phi (Q_1)}(Q_2)=Z(S)$
and
$\Phi (Q_1)=[\Phi (Q_1), r]Z(Q_1)$
, and we have a contradiction.
Lemma 5.16. If
$\Phi (Q_1)\le J_{\trianglelefteq }(E)\not \le Q_1\cap Q_2$
, then
$p=2$
,
$Z(E)=Z(Q_1)$
,
$Q_1=N_S(E)$
and
$O^{2'}(\mathrm {Aut}_{\mathcal {F}}(E))$
acts trivially on
$Z(Q_1)$
.
Proof. We have that
$E\cap Q_2>\Phi (Q_1)$
by Lemma 5.15 so that
$Z(Q_1)=[E, \Phi (Q_1)]\le \Phi (E)\le \Phi (Q_1)$
. By Lemma 5.3, we deduce that
$|Z(E)|\in \{q^2, q^3, q^4\}$
. Note that
$C_{Q_1}(E/Z(E))$
centralizes
$E/Z(E)\Phi (E)$
and
$Z(E)\Phi (E)/\Phi (E)$
. We deduce by Lemma 2.11 that
$C_{Q_1}(E/Z(E))\le E$
and
$E/Z(E)$
contains a noncentral chief factor for
$O^{p'}(\mathrm {Aut}_{\mathcal {F}}(E))$
. Applying Theorem 2.21, since
$[Q_1, Z(E)\Phi (E)]\le Z(Q_1)\le \Phi (E)$
, we have that
$|Q_1/E|\leqslant |[E, Q_1]Z(E)/Z(E)|$
and we deduce that
$|Z(E)|\in \{q^2, q^3\}$
. Moreover, if
$|Z(E)|=q^3$
, then Theorem 2.21 implies that
$E=C_{Q_1}(Z(E))$
and
$Q_1=N_S(E)$
, a contradiction since in this instance,
$E\trianglelefteq S$
. Hence,
$Z(E)=Z(Q_1)$
.
If p is odd, then for
$a\in E\setminus E\cap Q_2$
, we have that
$|[a, Q_2/Z(Q_1)|=q^4$
. We deduce that
$|C_{Q_2/Z(Q_1)}(a)|=q^3$
and as
$\Phi (Q_1)/Z(Q_1)\le C_{Q_2/Z(Q_1)}(a)$
, we have
$\Phi (Q_1)/Z(Q_1)=C_{Q_2/Z(Q_1)}(a)$
. It follows that
$Z(E/Z(Q_1))=\Phi (Q_1)/Z(Q_1)$
and that
$Q_1$
centralizes the chain
$\{1\}\trianglelefteq Z(Q_1)\trianglelefteq \Phi (Q_1)\trianglelefteq E$
, a contradiction since
$E<Q_1$
. Hence,
$p=2$
. Clearly,
$N_S(E)\in \{S, Q_1\}$
. Assume that
$E\trianglelefteq S$
. Since
$Z(Q_1)=Z(E)$
and
$Q_1/\Phi (Q_1)$
is a direct sum of natural
$\mathrm {SL}_2(q)$
-modules for
$L_1/Q_1$
, we infer that
$C_{\Phi (Q_1)}(Q_1\cap Q_2\cap E)=Z(Q_1)$
. Indeed, a calculation as in the odd case yields that either
$Z(E/Z(Q_1))=\Phi (Q_1)/Z(Q_1)$
and we obtain a contradiction as before, or
$E\le C_{Q_1}(C_{\Phi (Q_1)}(a))(Q_1\cap Q_2)$
and the preimage of
$Z(E/Z(Q_1))$
is equal to
$C_{Q_1\cap Q_2}(C_{\Phi (Q_1)}(a))$
. But now,
$[Q_1, E]\le C_{Q_1\cap Q_2}(C_{\Phi (Q_1)}(a))$
,
$[Q_1, \Phi (Q_1)]=Z(Q_1)=Z(E)$
,
$\Phi (Q_1)$
has index q in
$C_{Q_1\cap Q_2}(C_{\Phi (Q_1)}(a))$
and
$|Q_1/E|\geqslant q^2$
. Hence, Theorem 2.21 provides a contradiction. Therefore,
$p=2$
and
$Q_1=N_S(E)$
. Indeed, since
$Q_1$
centralizes
$Z(Q_1)$
, we deduce that
$O^{2'}(\mathrm {Aut}_{\mathcal {F}}(Q_1))$
acts trivially on
$Z(Q_1)$
.
Lemma 5.17. We have that
$\Phi (Q_1)\le J_{\trianglelefteq }(E)\le Q_1\cap Q_2$
.
Proof. Note first that
$(E\cap Q_2)/Z(S)$
is elementary abelian and
$E\le Q_1$
by Lemma 5.16. Assume that there is
$F\trianglelefteq E$
with
$F\not \le Q_2$
,
$F/Z(S)$
elementary abelian and
$|F|\geqslant |E\cap Q_2|$
. Let
$f\in F\setminus Q_2$
. Then
$(\Phi (Q_1)\cap F)/Z(S)\le C_{\Phi (Q_1)/Z(S)}(f)=C_{\Phi (Q_1)}(f)/Z(S)$
by Lemma 5.3. Moreover, by Lemma 5.5, we deduce that
$|F/F\cap Q_2|\leqslant q^5/|F\cap \Phi (Q_1)|$
. Hence,
$$ \begin{align*} |F|\leqslant q^5|F\cap Q_2|/|\Phi(Q_1)\cap F|&=|\Phi(Q_1)||F\cap Q_2|/|\Phi(Q_1)\cap F|\\ &=|\Phi(Q_1)(F\cap Q_2)|\leqslant |E\cap Q_2|\leqslant |F| \end{align*} $$
and we infer that every inequality is an equality. Therefore,
$E\cap Q_2=(F\cap Q_2)\Phi (Q_1)$
. Using that
$E\cap Q_2>\Phi (Q_1)$
, for
$e\in (F\cap Q_2)\setminus \Phi (Q_1)$
, we have that
$F/Z(S)\le C_{Q_1/Z(S)}(e)$
. By Lemma 5.5, we have that
$C_{Q_1/Z(S)}(e)=C_{Q_1}(C_{\Phi (Q_1)}(e))(Q_1\cap Q_2)/Z(S)$
. But
$(F\cap Q_2)/Z(S)\le C_{(Q_1\cap Q_2)/Z(S)}(f)$
and another application of Lemma 5.5 implies that
$F\cap Q_2\le C_{Q_1\cap Q_2}(C_{\Phi (Q_1)}(e))$
and we deduce that
$F\le C_{Q_1}(C_{\Phi (Q_1)}(e))$
. Then
$F\cap \Phi (Q_1)=C_{\Phi (Q_1)}(f)=C_{\Phi (Q_1)}(e)=Z(F(E\cap Q_2))$
is of order
$q^4$
. Indeed,
$F\cap \Phi (Q_1)=Z(E\cap Q_2)$
does not depend on F.
Since
$Z(S)$
is normalized by
$O^{2'}(\mathrm {Aut}_{\mathcal {F}}(E))$
so too is
$Z(E\cap Q_2)$
and hence, so too is
$C_E(Z(E\cap Q_2))$
. Now,
$[Q_1\cap Q_2, E]\le \Phi (Q_1)\le C_E(Z(E\cap Q_2))$
and
$[Q_1\cap Q_2, Z(E\cap Q_2)]\le Z(S)$
, and so each odd order element of
$\langle \mathrm {Aut}_{Q_1\cap Q_2}(E)^{\mathrm {Aut}_{\mathcal {F}}(E)}\rangle $
acts faithfully on
$C_E(Z(E\cap Q_2))/Z(E\cap Q_2)$
. But
$|(Q_1\cap Q_2)E/E|\geqslant q^2$
and
$E\cap Q_2$
has index q in
$C_E(Z(E\cap Q_2))$
and is centralized modulo
$Z(E\cap Q_2)$
by
$Q_1\cap Q_2$
. An application of Theorem 2.21 provides a contradiction.
Hence,
$E\cap Q_2/Z(S)=J_{\trianglelefteq }(E/Z(S))$
and since
$Z(S)$
is
$O^{2'}(\mathrm {Aut}_{\mathcal {F}}(E))$
-invariant so too is
$E\cap Q_2$
. Then
$Q_1\cap Q_2$
centralizes the chain
$\{1\}\trianglelefteq Z(S)\trianglelefteq E\cap Q_2\trianglelefteq E$
so that
$Q_1\cap Q_2\le E$
by Lemma 2.13 and
$Z(Q_1)=Z(E)$
. Note that for
$a\in Q_1\setminus (Q_1\cap Q_2)$
since
$|C_{Q_1\cap Q_2/Z(S)}(a)|\leqslant q^4$
, we deduce that
$|[a, Q_1\cap Q_2]Z(S)|\geqslant q^4$
so that
$|\Phi (E)|\geqslant q^4$
. Letting
$b\in Q_1\setminus (Q_1\cap Q_2)$
with
$C_{Q_2/Z(S)}(a)\ne C_{Q_2/Z(S)}(b)$
, we have that
$[a, Q_1\cap Q_2][b, Q_1\cap Q_2]Z(S)=\Phi (Q_1)$
so that
$\Phi (E)=\Phi (Q_1)$
. In this scenario,
$Q_1$
centralizes the chain
$\{1\}\trianglelefteq Z(Q_1)\trianglelefteq \Phi (Q_1)\trianglelefteq E$
, a contradiction. Letting
$A\in \mathcal {A}_{\trianglelefteq }(E)$
with
$A\not \le Q_1\cap Q_2$
, it follows that for
$a\in A\setminus (A\cap Q_2)$
,
$E\le C_{Q_1}(C_{\Phi (Q_1)}(a))(Q_1\cap Q_2)$
and
$|Q_1/E|\geqslant q^2$
. But now,
$[Q_1, E]=\Phi (Q_1)$
and since
$|Q_1/E|=q^2>q=|\Phi (Q_1)/\Phi (E)|$
, an application of Theorem 2.21 gives a contradiction. Thus, we have shown that
$\Phi (Q_1)\le J_{\trianglelefteq }(E)\le Q_1\cap Q_2$
.
As promised, we finally demonstrate that
$\mathcal {E}(\mathcal {F})\subseteq \{Q_1, Q_2\}$
.
Proposition 5.18. Suppose that
$\mathcal {F}$
is a saturated fusion system supported on S, and let
$E\in \mathcal {E}(\mathcal {F})$
. Then
$E\in \{Q_1, Q_2\}$
.
Proof. We suppose, ultimately for a contradiction, that E is an essential subgroup chosen maximally such that
$Q_1\ne E\ne Q_2$
. By Lemma 5.17,
$\Phi (Q_1)\le J_{\trianglelefteq }(E)\le Q_1\cap Q_2$
. Assume first that
$\Phi (Q_1)=J_{\trianglelefteq }(E)$
and
$E\cap Q_2>\Phi (Q_1)$
. Since
$E\not \le Q_2$
, it follows that
$[E, \Phi (Q_1)]=Z(Q_1)$
. If
$Q_1\not \le E$
, then
$\langle \mathrm {Aut}_{Q_1}(E)^{\mathrm {Aut}_{\mathcal {F}}(E)}\rangle $
contains elements of order coprime to p and we may choose such an element r. Since
$Q_1$
centralizes the chain
$\{1\}\trianglelefteq Z(Q_1)\trianglelefteq \Phi (Q_1)$
, we deduce that r centralizes
$\Phi (Q_1)$
. But then, by the three subgroups lemma,
$[r, E]\le C_E(\Phi (Q_1))\le C_S(\Phi (Q_1))=\Phi (Q_1)$
so that r acts trivially on E, a contradiction. Hence,
$Q_1<E$
and
$Z(E)=Z(S)$
. But then
$\Phi (S)=Q_1\cap Q_2=[E, Q_1]\le \Phi (E)$
and S centralizes
$E/\Phi (E)$
, a contradiction. Hence, if
$\Phi (Q_1)=J_{\trianglelefteq }(E)$
, then
$E\cap Q_2=\Phi (Q_1)$
. But now,
$[E, Q_1\cap Q_2]\le \Phi (Q_1)$
and
$[Q_1\cap Q_2, \Phi (Q_1)]=Z(S)$
and since
$|Z(S)|=q<|(Q_1\cap Q_2)E/E|=q^3$
, applying Theorem 2.21 yields a contradiction.
Therefore, we have that
$\Phi (Q_1)<J_{\trianglelefteq }(E)\le Q_1\cap Q_2$
. Then
$\Phi (J_{\trianglelefteq }(E))=Z(S)$
is characteristic in E. Since
$[E, Q_1\cap Q_2]\le \Phi (Q_1)\le J_{\trianglelefteq }(E)$
, and
$[Q_1\cap Q_2, J_{\trianglelefteq }(E)]\le \Phi (Q_1\cap Q_2)=Z(S)=\Phi (J_{\trianglelefteq }(E))$
,
$Q_1\cap Q_2$
centralizes the chain
$\{1\}\trianglelefteq \Phi (J_{\trianglelefteq }(E))\trianglelefteq J_{\trianglelefteq }(E)\trianglelefteq E$
so that
$Q_1\cap Q_2\le E$
and
$E\trianglelefteq S$
. Now, if
$Q_2\not \le E$
, then
$\langle \mathrm {Aut}_{Q_2}(E)^{\mathrm {Aut}_{\mathcal {F}}(E)}\rangle $
contains elements of order coprime to p. Choose such an element r. Since
$[Q_2, J_{\trianglelefteq }(E)]\le \Phi (Q_2)=Z(S)=\Phi (J_{\trianglelefteq }(E))$
, it follows that
$[r, J_{\trianglelefteq }(E)]=\{1\}$
. But then, by the three subgroups lemma,
$[r, E]\le C_{E}(J_{\trianglelefteq }(E))\le J_{\trianglelefteq }(E)$
and r acts trivially on E, a contradiction. Hence,
$Q_2\le E$
and Proposition 5.13 provides a contradiction.
Finally, we classify all saturated fusion systems supported on a Sylow p-subgroup of
${}^3\mathrm {D}_4(p^n)$
. As remarked in the Introduction and after Theorem 2.37, we need to employ a
$\mathcal {K}$
-group hypothesis when p is an odd prime to deduce
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(Q_2))\cong \mathrm {SL}_2(q^3)$
acting on
$Q_2/Z(S)$
as a triality module.
Theorem 5.19. Let
$\mathcal {F}$
be a saturated fusion system on a Sylow p-subgroup of
${}^3\mathrm {D}_4(p^n)$
, and assume that
$\mathrm {Aut}_{\mathcal {F}}(Q_2)$
is a
$\mathcal {K}$
-group if p is odd. Then either:
-
1.
$\mathcal {F}=\mathcal {F}_S(S:\mathrm {Out}_{\mathcal {F}}(S))$
; -
2.
$\mathcal {F}=\mathcal {F}_S(Q_1: \mathrm {Out}_{\mathcal {F}}(Q_1))$
, where
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(Q_1))\cong \mathrm {SL}_2(p^n)$
; -
3.
$\mathcal {F}=\mathcal {F}_S(Q_2: \mathrm {Out}_{\mathcal {F}}(Q_2))$
, where
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(Q_2))\cong \mathrm {SL}_2(p^{3n})$
; -
4.
$\mathcal {F}=\mathcal {F}_S(G)$
, where
$F^*(G)=O^{p'}(G)={}^3\mathrm {D}_4(p^n)$
.
Proof. If
$\mathcal {E}(\mathcal {F})=\emptyset $
, then outcome 1 holds. Assume that
$Q_1$
is essential. Since
$\Phi (Q_1)$
is self-centralizing, it follows from the three subgroups lemma that
$p'$
-elements of
$\mathrm {Aut}_{\mathcal {F}}(Q_1)$
act faithfully
$\Phi (Q_1)$
. Since S centralizes
$\Phi (Q_1)/Z(Q_1)$
, we deduce that
$p'$
-elements of
$O^{p'}(\mathrm {Aut}_{\mathcal {F}}(Q_1))$
act faithfully on
$Z(Q_1)$
and since
$C_S(Z(Q_1))=Q_1$
, Theorem 2.21 yields that
$Z(Q_1)$
is a natural module for
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(Q_1))\cong \mathrm {SL}_2(q)$
. If
$Q_1$
is the only essential, then outcome 2 holds.
Assume that
$Q_2$
is essential. Since
$\mathrm {Out}_{\mathcal {F}}(Q_2)$
is a
$\mathcal {K}$
-group when p is odd and
$\mathrm {Out}_S(Q_2)$
is elementary abelian of order
$q^3$
, we have by Lemma 2.32, Proposition 2.33 and [Reference Parker and Stroth24, (2.5), (3.3)] (which in turn uses [Reference Gorenstein, Lyons and Solomon9, Theorem 7.6.1]) that
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(Q_2))\cong \mathrm {(P)}SL_2(q^3)$
. Applying Lemma 2.32 and Lemma 2.30, there is a unique nontrivial composition factor in
$Q_2/Z(S)$
for
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(Q_2))$
. Set
$U:=\langle Z(Q_1)/Z(S)^{\mathrm {Aut}_{\mathcal {F}}(Q_2)}\rangle $
. Then
$C_U(\mathrm {Out}_S(Q_2))=Z(Q_1)/Z(S)$
has order q and comparing with Lemma 2.30, either
$U=Z(Q_1)/Z(S)$
and
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(Q_2))$
is trivial on
$Z(Q_1)/Z(S)$
, or
$U=Q_2/Z(S)$
is a triality module for
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(Q_2))\cong \mathrm {SL}_2(q^3)$
. With the aim of forcing a contradiction, we assume that
$U=Z(Q_1)/Z(S)$
. Then for r of
$p'$
-order in
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(Q_2))$
, we have by coprime action that
$[r, Z(Q_1)]=\{1\}$
and the three subgroups lemma yields that
$[r, Q_2]\le Q_1\cap Q_2$
. Since
$Q_1\cap Q_2/Z(Q_1)$
is of order
$q^6$
, applying Lemma 2.30 we have that
$Q_1\cap Q_2/Z(Q_1)$
is the unique nontrivial composition factor in
$Q_2/Z(S)$
for
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(Q_2))$
. Since
$\mathrm {Out}_S(Q_2)$
is quadratic on
$Q_1\cap Q_2/Z(Q_1)$
, we deduce that
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(Q_2))\cong \mathrm {SL}_2(q^3)$
and
$Q_1\cap Q_2/Z(Q_1)$
has the structure of a natural module. If
$p=2$
, then for all
$x\in \mathrm {Out}_S(Q_2)$
we have that
$|C_{Q_1\cap Q_2/Z(S)}(x)|=q^4$
and by properties of natural
$\mathrm {SL}_2(q)$
-modules, we conclude that
$C_{Q_1\cap Q_2/Z(S)}(x)=\Phi (Q_1)/Z(S)$
, a contradiction since
$[\Phi (Q_1), \mathrm {Out}_S(Q_2)]=Z(Q_1)$
. If p is odd, then
$Q_1\cap Q_2/Z(S)$
splits into a direct sum of a natural module and
$Z(Q_1)/Z(S)=C_{Q_1\cap Q_2/Z(S)}(O^{p'}(\mathrm {Out}_{\mathcal {F}}(Q_2)))$
, and so we deduce again that
$C_{Q_1\cap Q_2/Z(S)}(\mathrm {Out}_S(Q_2))=\Phi (Q_1)/Z(S)$
, again a contradiction. Hence,
$Q_2/Z(S)$
is a triality module for
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(Q_2))\cong \mathrm {SL}_2(q^3)$
. If
$Q_1$
is not essential, then outcome 3 holds.
Hence, we are left with the case where
$\mathcal {E}(\mathcal {F})=\{Q_1, Q_2\}$
. If
$O_p(\mathcal {F})\ne \{1\}$
, then
$O_p(\mathcal {F})\cap Z(S)\ne \{1\}$
and since
$Z(Q_1)$
is a natural
$\mathrm {SL}_2(q)$
-module for
$O^{p'}(\mathrm {Out}_{\mathcal {F}}(Q_1))$
, we have that
$Z(Q_1)\le O_p(\mathcal {F})$
. Then, as
$\mathrm {Out}_{\mathcal {F}}(Q_2)$
is irreducible on
$Q_2/Z(S)$
, we deduce that
$Q_2\le O_p(\mathcal {F})\not \le Q_1$
, a contradiction by [Reference Aschbacher, Kessar and Oliver2, Proposition I.4.5]. Hence,
$O_p(\mathcal {F})=\{1\}$
and
$\mathcal {F}$
satisfies outcome (4) by Corollary 2.39.
Funding statement
The author is supported by the Heilbronn Institute for Mathematical Research. Part of this work was carried out at the Isaac Newton Institute for Mathematical Sciences during the programme GRA2, supported by the EPSRC (EP/K032208/1).
Competing interests
The authors have no competing interest to declare.
Ethical standards
The research meets all ethical guidelines, including adherence to the legal requirements of the study country.
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