2. Preliminaries: groups

We start with some elementary observations regarding the Thompson subgroup of a finite p-group and the related notion of failure to factorise modules. For a more in depth account of this phenomena, see [ Reference Kurzweil and Stellmacher27 , section 9·2].

See [ Reference Kurzweil and Stellmacher27 , 9·2·8] for parts (i) and (iii). Additionally, by part (d) of that result, we see that $\Omega_1(C_S(J(S)))\le \Omega_1(Z(J(S)))$ . Since $Z(J(S))\le C_S(J(S))$ , it is clear that $\Omega_1(C_S(J(S)))=\Omega_1(Z(J(S)))$ .

Let $a\in \bigcap_{A\in\mathcal{A}(S)} A$ . Then a has order p and $[a, A]=\{1\}$ for all $A\in\mathcal{A}(S)$ . By definition, $[a, J(S)]=\{1\}$ so that $a\in \Omega_1(C_S(J(S)))$ and $\bigcap_{A\in\mathcal{A}(S)} A\le \Omega_1(C_S(J(S)))$ . Now, for $x\in C_S(J(S))$ of order p, we have that $x\le C_S(J(S))\le C_S(A)$ for all $A\in\mathcal{A}(S)$ . Hence, $x\in \Omega_1(C_S(A))$ for all $A\in\mathcal{A}(S)$ . But now, $\langle x\rangle A$ is elementary abelian of order at least as large as A and by the definition of $\mathcal{A}(S)$ , we have that $x\in A$ . Therefore, $x\in \bigcap_{A\in\mathcal{A}(S)} A$ and $\Omega_1(C_S(J(S)))=\bigcap_{A\in\mathcal{A}(S)} A$ , completing the proof of (ii).

We will also make liberal use of several coprime action results, often without explicit reference.

In conclusion (iv) in the statement above, one can say a little more. The following is a classical result of Burnside, but the version we use follows from [ Reference Gorenstein20 , (I·5·1·4)].

Proof. See [ Reference Gorenstein20 , (I·5·3·3)].

Proof. See [ Reference Aschbacher4 , (24·2)].

Our final results in this section with regards to groups and modules concern the identification of some local actions within the groups $\textrm{Co}_1$ , $\textrm{Sp}_6(3)$ and M.

Proof. Let G be a minimal counterexample with respect to $|G|$ . By [ Reference Kurzweil and Stellmacher27 , 8·3·4(a)], $O_{3'}(G)=\langle C_{O_{3'}(G)}(a)\mid a\in A^\#\rangle$ and since $C_V(a)=C_V(A)$ for all $a\in A^\#$ , we have that $O_{3'}(G)$ normalises $C_V(A)$ . Set $T\;:\!=\;\langle A^{AO_{3'}(G)}\rangle$ so that $C_V(A)=C_V(T)\le C_V(O_{3'}(T))$ . By coprime action again, $V=[V, O_{3'}(T)]\times C_V(O_{3'}(T))$ . But now,

\[ C_{[V, O_{3'}(T)]}(A)\le [V, O_{3'}(T)]\cap C_V(A)=[V, O_{3'}(T)]\cap C_V(T)=\{1\}\]

and as A is a 3-group, we must have that $[V, O_{3'}(T)]=\{1\}$ . Since G acts faithfully on V, we infer that $O_{3'}(T)=\{1\}$ . Then, as $A\le T \le AO_{3'}(G)$ and $T\cap O_{3'}(G)\le O_{3'}(T)=\{1\}$ , we conclude that $A=T$ is normalised by $O_{3'}(G)$ . In particular, $[A, O_{3'}(G)]=\{1\}$ .

Since $O_3(G)=\{1\}$ and $F^*(G)$ is self-centralising in G, we have shown that G contains a component, K say, whose order is divisible by 3. Then $E\;:\!=\;\langle K^G\rangle$ is normalised by S and so we deduce that it contains Z(S). Note that since $m_3(S)=2$ and $O_3(E)\le O_3(G)=\{1\}$ , E contains at most two components of G whose orders are divisible by 3. Indeed, since S is a 3-group, we see that S normalises these components. If E contains exactly two components of G whose orders are divisible by 3, $K_1$ and $K_2$ say, then $K_i\cap S\trianglelefteq S$ for $i\in\{1,2\}$ so that $Z(S)\le K_1\cap K_2\le Z(E)$ . Hence, $Z(S)\le O_3(Z(E))\le O_3(G)$ , a contradiction.

Thus, $E=K$ is quasisimple. Now, $K=\langle Z(S)^K\rangle=\langle Z(S)^G\rangle$ and so K is a component of $H\;:\!=\;\langle Z(S)^G\rangle S$ so that $H=O^{3'}(H)$ is almost-quasisimple. Note that $O_3(H)$ is trivial for otherwise $Z(S)\le O_3(H)\cap K\le O_3(K)\le O_3(G)$ and since $O_3(G)=\{1\}$ , this is a contradiction. Hence, by minimality, either $H\cong \textrm{PSL}_3(3)$ or $2.\textrm{M}_{12}$ ; or $G=H$ . In the former case, we deduce that $H=K\trianglelefteq G$ and since $S\le H$ and $G=O^{3'}(G)$ , we have that $G=H$ .

Hence, a minimal counterexample of this lemma is almost quasisimple. Now, $|A|^2=3^4\gt3^3=|V/C_V(A)|$ so that V is a 2F-module for G with offender A in the language of [ Reference Guralnick, Lawther and Malle24 ]. By [ Reference Guralnick, Lawther and Malle24 , Table 1], G is isomorphic to either a group of Lie type in characteristic 3 or $2.\textrm{M}_{12}$ . The groups of Lie type in characteristic 3 with Sylow 3-subgroup isomorphic to S are well known (see [ Reference Gorenstein, Lyons and Solomon21 , (3·3)]), and so we have that $G\cong 2.\textrm{M}_{12}$ , $\textrm{PSL}_3(3)$ or $\textrm{SU}_3(3)$ . Now, $\textrm{SU}_3(3)$ has only one non-trivial module of dimension 6 over $\textrm{GF}(3)$ , the natural module. But for this module, we have that $|C_V(B)|=3^2$ for any subgroup B of the Sylow 3-subgroup which has order 9.

In the following proposition, MAGMA is used to verify certain calculations. The actual code itself may be found in [ Reference van Beek44 , appendix A].

Proof. Since Q is extraspecial, $O_5(G)=\{1\}$ and $G=O^{5'}(G)$ , applying [ 47 ] we have that G is isomorphic to a subgroup of $\textrm{Sp}_6(5)$ and $Q/Z(Q)$ may be identified with the natural module for $\textrm{Sp}_6(5)$ in this action. We appeal to [ Reference Bray, Holt and Roney-Dougal9 , Table 8·28, Table 8·29] for the list of maximal subgroups of $\textrm{Sp}_6(5)$ . These are

\[2.\textrm{J}_2,\, \textrm{Sp}_2(5)\circ \textrm{GO}_3(5),\, \textrm{GU}_3(5).2, \textrm{Sp}_2(5^3).3,\, \textrm{Sp}_2(5)^3:\textrm{Sym}(3),\, \textrm{Sp}_2(5)\times \textrm{Sp}_4(5),\]

\[5^6:\textrm{GL}_3(5),\, 5^{3+4}:(\textrm{GL}_2(5)\times \textrm{Sp}_2(5))\,\,\text{and}\,\,5^{1+4}_+:(C_4\times \textrm{Sp}_4(5)).\]

Aiming for a contradiction, assume throughout that $G\not\cong 2.\textrm{J}_2$ .

We compute that the maximal subgroups in which a Sylow 5-subgroup fixes a subspace of dimension 1 are $2.\textrm{J}_2$ , $\textrm{Sp}_2(5)\circ \textrm{GO}_3(5)$ , $5^6:\textrm{GL}_3(5)$ , $5^{3+4}:(\textrm{GL}_2(5)\times \textrm{Sp}_2(5))$ and $5^{1+4}_+:(C_4\times \textrm{Sp}_4(5))$ . We refer to these subgroups as $M_1,\ldots, M_5$ respectively. In $M_2$ , one can compute that there is a 5-element which fixes a subspace of dimension 3 and as a Sylow 5-subgroup of $M_2$ has order 25, G cannot be isomorphic to a subgroup of $M_2$ . If G is isomorphic to a subgroup of $M_3$ , then as $O_5(G)=\{1\}$ , G projects as a subgroup of $\textrm{GL}_3(5)$ . But every subgroup of $\textrm{GL}_3(5)$ which has a Sylow 5-subgroup of order 25 has a normal 5-subgroup, a contradiction.

Similarly, if G is isomorphic to a subgroup of $M_4$ , then G is isomorphic to a subgroup of $\textrm{GL}_2(5)\times \textrm{Sp}_2(5)$ . Indeed, since $G=\langle S^G\rangle$ , $|S|=25$ and $O_5(G)=\{1\}$ , it follows that $G\cong \textrm{SL}_2(5)\times \textrm{Sp}_2(5)$ . Hence, $GO_5(M_4)=O^{5'}(M_4)$ . Let $L\le G$ be such that $L\trianglelefteq G$ and $L\cong \textrm{SL}_2(5)\cong \textrm{Sp}_2(5)$ . Then L contains a Sylow 2-subgroup T of $LO_5(M_4)\trianglelefteq G$ . By a calculation (see [ Reference van Beek44 , appendix A]), we have that $C_{GO_5(M_4)}(T)\cong 2\times \textrm{SL}_2(5)$ . Since $C_G(T)\cong 2\times \textrm{SL}_2(5)$ , we have that $C_{GO_5(M_4)}(T)=C_G(T)$ . However, for $R\in\textrm{Syl}_5(C_{GO_5(M_4)}(T))$ , we have that $|C_V(R)|=5^5$ , a clear contradiction.

If G is isomorphic to a subgroup of $M_5$ then G is isomorphic to a subgroup of $C_4\times \textrm{Sp}_4(5)$ . Since $G=\langle S^G\rangle$ , we see that G is isomorphic to a subgroup of $\textrm{Sp}_4(5)$ . Using MAGMA (see [ Reference van Beek44 , appendix A]), since $O_5(G)=\{1\}$ , $|S|=25$ and $G=\langle S^G\rangle$ , we calculate that $G\cong \textrm{SL}_2(25)$ or $\textrm{SL}_2(5)\times \textrm{SL}_2(5)$ . Moreover, the center of $\textrm{Sp}_4(5)$ is equal to the center of a Sylow 2-subgroup of $\textrm{Sp}_4(5)$ and it follows from computations that G centralises the center of a Sylow 2-subgroup of $L_5\;:\!=\;O^{5'}(M_5)$ , which we denote by T. Then $G=G'\le C_{L_5}(T)'\cong \textrm{Sp}_4(5)$ and so G is contained in a specified complement to $O_5(M_5)$ in $L_5$ . But then we calculate for all such candidates for G that $|C_V(S)|=5^4$ , a contradiction.

Hence, G is isomorphic to a proper subgroup of $M_1\cong 2.\textrm{J}_2$ . But, appealing to [ Reference Conway, Curtis, Norton, Parker and Wilson13 ] for a list of maximal subgroups of $\textrm{J}_2$ , the only maximal subgroups of $2.\textrm{J}_2$ which have a Sylow 5-subgroup of order 25 also have a normal 5-subgroup, a contradiction.

3. Preliminaries: fusion systems

We now let S be a finite p-group and $\mathcal{F}$ a saturated fusion system on S, referring to [ Reference Aschbacher, Kessar and Oliver5, Reference Craven14 ] for standard terminology and results regarding fusion systems. We use the remainder of this section to reaffirm some important concepts regarding fusion systems which pertain to this work, and mention some vital results from other sources in the literature.

We begin with the notion of isomorphism for fusion systems.

Definition 3·1. Let $\mathcal{F}$ be a saturated fusion system on a p-group S and let $\alpha\;:\; S\to T$ be a group isomorphism. Define $\mathcal{F}^\alpha$ to be the fusion system on T with

\[\textrm{Hom}_{\mathcal{F}^\alpha}(P, Q)=\{\alpha^{-1}\gamma \alpha \mid \gamma\in\textrm{Hom}_{\mathcal{F}}(P\alpha^{-1}, Q\alpha^{-1}\}\]

for $P,Q\le T$ .

We then say that a fusion system $\mathcal{E}$ over a p-group T is isomorphic to $\mathcal{F}$ , written $\mathcal{E}\cong \mathcal{F}$ , if there a group isomorphism $\alpha: S\to T$ with $\mathcal{E}=\mathcal{F}^\alpha$ .

Remark. We note that our definition of isomorphism coincides with morphisms defined in [ Reference Aschbacher, Kessar and Oliver5 , definition II·2·2] which are surjective and have trivial kernel.

Importantly, for G a finite group, $S\in\textrm{Syl}_p(G)$ and K a normal $p'$ -subgroup of G, writing $\overline{G}\;:\!=\;G/K$ , we have that $\mathcal{F}_S(G)\cong \mathcal{F}_{\overline{S}}(\overline{G})$ . This is often viewed as one of the main attractions for working with fusion systems in place of finite groups.

We recall that $\mathcal{F}$ is realizable if there is a finite group G and $S\in\textrm{Syl}_p(G)$ such that $\mathcal{F}=\mathcal{F}_S(G)$ , and $\mathcal{F}$ is exotic otherwise. By the above observation, if we aim to show that $\mathcal{F}$ is realised by a finite group G, then we may as well assume that $O_{p'}(G)=\{1\}$ .

Notation. Let $\mathcal{F}$ be a fusion system and let $\mathcal{F}_1, \mathcal{F}_2$ be fusion subsystems of $\mathcal{F}$ . That is, $\mathcal{F}_i$ is a subcategory of $\mathcal{F}$ which is itself a fusion system. Write $\langle \mathcal{F}_1, \mathcal{F}_2\rangle_S$ for the smallest subsystem of $\mathcal{F}$ supported on S which contains both $\mathcal{F}_1$ and $\mathcal{F}_2$ .

At various points, we may also write $\langle \mathcal{M}_1, \mathcal{M}_2,\ldots\rangle_S$ where $\mathcal{M}_i$ is some set of morphisms contained in $\mathcal{F}$ and by this we mean the smallest subsystem of $\mathcal{F}$ supported on S which contains $\mathcal{M}_i$ for all i. We also mix the two conventions e.g. $\langle \mathcal{F}_1, \mathcal{M}_1, \mathcal{M}_2\rangle_S$ is the smallest subsystem of $\mathcal{F}$ supported on S containing $\mathcal{F}_1$ , $\mathcal{M}_1$ and $\mathcal{M}_2$ .

We emphasise that saturation is not imposed here. So even if $\mathcal{F}$ , $\mathcal{F}_1$ and $\mathcal{F}_2$ are saturated, then $\langle \mathcal{F}_1, \mathcal{F}_2\rangle_S$ need not be saturated.

We denote the set of $\mathcal{F}$ -centric subgroups of $\mathcal{F}$ by $\mathcal{F}^c$ and the fully $\mathcal{F}$ -normalised, $\mathcal{F}$ -centric-radical subgroups of $\mathcal{F}$ by $\mathcal{F}^{frc}$ , referring to [ Reference Aschbacher, Kessar and Oliver5 , definition I·2·4, definition I·3·1] for the appropriate definitions. We present the following result as a lemma, but in truth it may be considered as part of the definition of saturation of a fusion system.

Proof. Since P is fully $\mathcal{F}$ -normalised and $\mathcal{F}$ is saturated, P is receptive, as defined in [ Reference Aschbacher, Kessar and Oliver5 , definition I·2·2]. Hence, for $\alpha\in N_{\textrm{Aut}_{\mathcal{F}}(P)}(\textrm{Aut}_R(P))$ and $N_\alpha\;:\!=\;\{g\in N_S(P) |{}^\alpha c_g\in \textrm{Aut}_S(P)\}$ , we have $P\lt R\le N_\alpha\le N_S(P)$ such that there is $\widehat{\alpha}\in\textrm{Hom}_{\mathcal{F}}(N_\alpha, S)$ with $R\widehat{\alpha}=R$ and $\widehat{\alpha}|_P=\alpha$ . Indeed, $\widehat{\alpha}$ restricts to $\overline{\alpha}\in \textrm{Aut}_{\mathcal{F}}(R)$ , as desired.

Often, the morphisms we choose to lift in Lemma 3·2 can chosen to lift all the way to certain essential subgroups of $\mathcal{F}$ .

Proof. See [ Reference Aschbacher, Kessar and Oliver5 , proposition I·3·3(a)].

In later sections, our treatment of saturated fusion systems will focus specifically on the actions associated to essential subgroups, and the morphisms lifted to them. The reasoning behind this is that a saturated fusion system is completely determined by this information. This observation is contained in the following theorem.

Theorem 3·5 (Alperin – Goldschmidt Fusion Theorem). Let $\mathcal{F}$ be a saturated fusion system over a p-group S and let $\mathcal{E}^0(\mathcal{F})$ be a set of representatives of the $\mathcal{F}$ -conjugacy classes of $\mathcal{E}(\mathcal{F})$ . Then

\[\mathcal{F}=\langle \textrm{Aut}_{\mathcal{F}}(Q), \textrm{Aut}_{\mathcal{F}}(S) \mid Q\in\mathcal{E}^0(\mathcal{F})\rangle_S.\]

Proof. See [ Reference Aschbacher, Kessar and Oliver5 , theorem I·3·5] and [ Reference Craven14 , proposition 7·25].

We will make frequent use of the following lemma which comes as a result of Lemma 3·2 and the Alperin–Goldschmidt theorem.

Proof. Let $Q\lt R\le N_S(Q)$ with Q $\mathcal{F}$ -centric and $\alpha\in N_{\textrm{Aut}_{\mathcal{F}}(Q)}(\textrm{Aut}_R(Q))$ . In particular, Q is fully $\mathcal{F}$ -centralised.

Suppose that there is $E\in\mathcal{E}(\mathcal{F})$ such that E is $\textrm{Aut}_{\mathcal{F}}(S)$ -invariant and for all $P\in Q^{\mathcal{F}}$ , E is the unique essential subgroup of $\mathcal{F}$ which properly contains P. By Lemma 3·2, there is $\widetilde{\alpha}\in \textrm{Aut}_{\mathcal{F}}(R)$ with $\widetilde{\alpha}|_Q=\alpha$ . By the Alperin–Goldschmidt theorem, we may write $\widetilde{\alpha}=(\alpha_1\circ \ldots \circ \alpha_n)|_R$ where $\alpha_i\in \textrm{Aut}_{\mathcal{F}}(F)$ for $F\in \mathcal{E}(\mathcal{F})\cup \{S\}$ . Now, Q is properly contained in exactly one essential subgroup (namely E), and as $R\gt Q$ we must have that $\alpha_1\in \textrm{Aut}_{\mathcal{F}}(S)$ or $\alpha_1\in \textrm{Aut}_{\mathcal{F}}(E)$ . Notice that if $R\le E$ , then as E is $\textrm{Aut}_{\mathcal{F}}(S)$ -invariant, we may replace $\alpha_1$ by $\alpha_1|_{E}\in\textrm{Aut}_{\mathcal{F}}(E)$ and $Q\alpha_1\lt R\alpha_1\le E$ . If $R\not\le E$ , then $\alpha_1\in\textrm{Aut}_{\mathcal{F}}(S)$ and $R\alpha_1\not\le E$ .

Now, $Q\alpha_1\lt R\alpha_1$ and $Q\alpha_1$ is properly contained in exactly one essential subgroup, and so $\alpha_2\in \textrm{Aut}_{\mathcal{F}}(S)$ or $\alpha_2\in \textrm{Aut}_{\mathcal{F}}(E)$ . Again, if $R\alpha_1\le E$ (so that $R\le E$ ) then as E is $\textrm{Aut}_{\mathcal{F}}(S)$ -invariant, we may arrange that $\alpha_2\in\textrm{Aut}_{\mathcal{F}}(E)$ . Otherwise, $\alpha_2\in\textrm{Aut}_{\mathcal{F}}(S)$ . Continuing in this fashion, we see that either $R\le E$ and we may take $\widehat{\alpha}=\alpha_1\circ \ldots \circ \alpha_n\in \textrm{Aut}_{\mathcal{F}}(E)$ ; or $R\not\le E$ and $\widehat{\alpha}=\alpha_1\circ \ldots \circ \alpha_n\in \textrm{Aut}_{\mathcal{F}}(S)$ . In the latter case, since $Q\le E$ , we have that $\widehat{\alpha}|_E$ is still a lift of $\alpha$ , and so the first statement of (i) holds. In particular, in either case we see that $\alpha$ normalises $\textrm{Aut}_{E}(Q)$ .

Assume now that $Q\not\in\mathcal{E}(\mathcal{F})$ . Then by [ Reference Aschbacher, Kessar and Oliver5 , proposition I·3·3], $\textrm{Aut}_{\mathcal{F}}(Q)$ is generated by maps $\alpha\in N_{\textrm{Aut}_{\mathcal{F}}(Q)}(\textrm{Aut}_R(Q))$ for some $Q\lt R\le N_S(Q)$ . But all such maps normalise $\textrm{Aut}_{E}(Q)$ and as $Q\lt E$ and Q is $\mathcal{F}$ -centric, $\{1\}\lt\textrm{Out}_{E}(Q)\trianglelefteq \textrm{Out}_{\mathcal{F}}(Q)$ and Q is not $\mathcal{F}$ -radical. This completes the proof of (i).

For the proof of (ii), we follow the same proof scheme as for (i). However, this time we note that as each $\mathcal{F}$ -conjugate of Q is not properly contained in any essential subgroup of $\mathcal{F}$ , the Alperin–Goldschmidt theorem promises that $\widetilde{\alpha}$ is a composition of restrictions of maps in $\textrm{Aut}_{\mathcal{F}}(S)$ and so we deduce that $\widehat{\alpha}\in \textrm{Aut}_{\mathcal{F}}(S)$ . In particular, $\widehat{\alpha}$ normalises any $\textrm{Aut}_{\mathcal{F}}(S)$ -invariant subgroup of S containing Q. Applying [ Reference Aschbacher, Kessar and Oliver5 , proposition I·3·3], $\textrm{Aut}_{\mathcal{F}}(Q)$ is generated by maps $\alpha\in N_{\textrm{Aut}_{\mathcal{F}}(Q)}(\textrm{Aut}_R(Q))$ for some $Q\lt R\le N_S(Q)$ and all such maps lift to maps which normalise any $\textrm{Aut}_{\mathcal{F}}(S)$ -invariant subgroup B of S containing Q. Since these maps also normalise Q, they must normalise $N_B(Q)$ and we deduce that $\textrm{Aut}_B(Q)\trianglelefteq \textrm{Aut}_{\mathcal{F}}(Q)$ . As Q is $\mathcal{F}$ -centric, and taking $B=S$ , we have that $\{1\}\lt\textrm{Out}_{S}(Q)\trianglelefteq \textrm{Out}_{\mathcal{F}}(Q)$ and Q is not $\mathcal{F}$ -radical. This completes the proof of (ii), and so completes the proof of the lemma.

Throughout the later portions of this work, we will often employ computational methods to determine a list of potential essential subgroups of a fusion system $\mathcal{F}$ supported on a given p-group S via the fusion systems package in MAGMA [ Reference Parker and Semeraro36, Reference Parker and Semeraro37 ].

Roughly speaking, the algorithm first determines a list a subgroups of S which are self-centralising in S, a prerequisite to being essential. Since the groups with a strongly p-embedded subgroup are “known”, the isomorphism type of $N_S(E)/E$ for a potential essential subgroup E should have a prescribed form too. Then further checks are carried out which verify that certain internal conditions in E hold which necessarily hold if E is essential in some saturated fusion system supported on S. These checks and more are described in [ Reference Parker and Semeraro36 ].

The following result is a useful tool for identifying automisers of essential subgroups.

Proof. Since $E=C_S(W)$ , we infer that $\textrm{Inn}(E)=C_{\textrm{Aut}_S(E)}(W)$ so that $C_G(W)$ is a $p'$ -group. In particular, $G/C_G(W)$ has a strongly p-embedded subgroup and so too does $L/C_L(W)\cong LC_G(W)/C_G(W)=O^{p'}(G/C_G(W))$ by [ Reference Henke26 , remark 3·5]. Then W is an FF-module for $L/C_L(W)$ and we apply [ Reference Henke26 , theorem 5·6] to obtain the result.

The next two results of this section are pivotal in creating exotic fusion systems from p-fusion categories while maintaining saturation. The first of these techniques we refer to as “pruning.”

Proof. We apply [ Reference Parker and Semeraro36 , lemma 6·4], taking $K=H_{\mathcal{F}}(P)$ where $H_{\mathcal{F}}(P)$ denotes the subgroup of $\textrm{Aut}_{\mathcal{F}}(P)$ which is generated by $\mathcal{F}$ -automorphisms of P which is extend to $\mathcal{F}$ -isomorphisms between strictly larger subgroups of S. By that result, the fusion system $\langle \mathcal{G}, K\rangle_S$ is saturated. However, for $P\lt R\le S$ we have that $\textrm{Hom}_{\mathcal{F}}(R, S)=\textrm{Hom}_{\mathcal{G}}(R, S)$ and we conclude that $K\subseteq \mathcal{G}$ so that $\mathcal{G}$ is saturated.

Also included in [ Reference Parker and Semeraro36 , lemma 6·4] is the statement that $P\not\in \mathcal{E}(\mathcal{G})$ . Since $\textrm{Aut}_{\mathcal{G}}(E)=\textrm{Aut}_{\mathcal{F}}(E)$ for all $E\in \mathcal{E}(\mathcal{F})\setminus \{P^{\mathcal{F}}\}$ , we ascertain that $\mathcal{E}(\mathcal{G})=\mathcal{E}(\mathcal{F})\setminus \{P^{\mathcal{F}}\}$ .

Proof. See [ Reference Broto, Levi and Oliver10 , proposition 5·1] or [ Reference Semeraro42 , theorem C].

We recall the notion of normaliser fusion systems from [ Reference Aschbacher, Kessar and Oliver5 , section I·6], noting that for P a fully $\mathcal{F}$ -normalised subgroup, $N_{\mathcal{F}}(P)$ is a saturated fusion subsystem of $\mathcal{F}$ . We say P is normal in $\mathcal{F}$ if $\mathcal{F}=N_{\mathcal{F}}(P)$ and we denote by $O_p(\mathcal{F})$ the unique largest normal subgroup of $\mathcal{F}$ . The following proposition connects normal subgroups of $\mathcal{F}$ , strongly closed subgroups of $\mathcal{F}$ in the sense of [ Reference Aschbacher, Kessar and Oliver5 , definition I·4·1], and the essential subgroups of $\mathcal{F}$ .

Moreover, if Q is an abelian subgroup of S, then $Q\trianglelefteq \mathcal{F}$ if and only if Q is strongly closed in $\mathcal{F}$ .

Proof. See [ Reference Aschbacher, Kessar and Oliver5 , proposition I·4·5] and [ Reference Aschbacher, Kessar and Oliver5 , corollary I·4·7].

Fundamental to our analysis of fusion systems is the application of a plethora of known results from finite group theory. Particularly, given a fully normalised subgroup Q, we wish to understand the actions induced by $N_{\mathcal{F}}(Q)$ and to do this, we wish to work in a finite group which models the behaviour of this normaliser subsystem.

Proof. See [ Reference Aschbacher, Kessar and Oliver5 , theorem I·4·9].

As with finite groups, we desire a more global sense of normality in fusion systems, not just restricted to p-subgroups. That is, we are interested in subsystems of a fusion system $\mathcal{F}$ which are normal. We use the notion of normality provided in [ Reference Aschbacher, Kessar and Oliver5 , definition I·6·1], noting that this condition is stronger than some of other definitions in the literature.

By [ Reference Craven15 , theorem A], a proper, non-trivial normal subsystem of $\mathcal{F}$ with respect to one of the accepted definitions of normality gives rise to a proper, non-trivial normal subsystem of $\mathcal{F}$ with respect to the other accepted definitions. Thus, we can unambiguously declare $\mathcal{F}$ to be simple if it has no proper, non-trivial normal subsystems and so, for our purposes, the distinction between the definitions of normality is unimportant.

Of particular importance in our case is the normal subsystem $O^{p'}(\mathcal{F})$ of $\mathcal{F}$ , and more generally, the saturated subsystems of $\mathcal{F}$ of index prime to p, as in [ Reference Aschbacher, Kessar and Oliver5 , definition I·7·3]. The following result characterises some of the most important properties of these subsystems.

Lemma 3·12. Fix a saturated fusion system $\mathcal{F}$ over a p-group S, and set $\mathcal{E}_0\;:\!=\;\langle O^{p'}(\textrm{Aut}_{\mathcal{F}}(P)) \mid P\le S\rangle_S$ , as a (not necessarily saturated) fusion system on S. Define

\[\textrm{Aut}_{\mathcal{F}}^0(S)\;:\!=\;\langle \alpha\in\textrm{Aut}_{\mathcal{F}}(S)\mid \alpha|_P\in\textrm{Hom}_{\mathcal{E}_0}(P, S)\text{, some}\,\,P\in\mathcal{F}^c\rangle\]

and let $\mathcal{E}$ be a saturated fusion system on S of index prime to p in $\mathcal{F}$ . Then

1. (i) $\textrm{Aut}_{\mathcal{F}}^0(S)\le \textrm{Aut}_{\mathcal{E}}(S)\le \textrm{Aut}_{\mathcal{F}}(S)$ and each group L with $\textrm{Aut}_{\mathcal{F}}^0(S)\le L\le \textrm{Aut}_{\mathcal{F}}(S)$ gives rise to a unique saturated fusion subsystem of index prime to p in $\mathcal{F}$ ;

2. (ii) $\mathcal{E}\trianglelefteq \mathcal{F}$ if and only if $\textrm{Aut}_{\mathcal{E}}(S)\trianglelefteq \textrm{Aut}_{\mathcal{F}}(S)$ ; and

3. (iii) there is a unique minimal saturated subsystem $O^{p'}(\mathcal{F})\trianglelefteq \mathcal{F}$ of index prime to p, and $\textrm{Aut}_{O^{p'}(\mathcal{F})}(S)=\textrm{Aut}_{\mathcal{F}}^0(S)$ .

In particular, $\textrm{Aut}_{\mathcal{F}}^0(S)=\textrm{Aut}_{\mathcal{F}}(S)$ implies that $\mathcal{F}=O^{p'}(\mathcal{F})$ , and $O^{p'}(O^{p'}(\mathcal{F}))=O^{p'}(\mathcal{F})$ .

Proof. See [ Reference Aschbacher, Kessar and Oliver5 , theorem I·7·7].

If $\mathcal{E}$ is a saturated subsystem of index prime to p in $\mathcal{F}$ with $[\textrm{Aut}_{\mathcal{F}}(S) : \textrm{Aut}_{\mathcal{E}}(S)]=r$ , then we say that $\mathcal{E}$ has index r in $\mathcal{F}$ .

We provide a short lemma characterising essential subgroups in saturated subsystems of index prime to p.

Proof. By [ Reference Aschbacher, Kessar and Oliver5 , lemma I·7·6(a)], the centric subgroups of $\mathcal{F}$ and $\mathcal{B}$ coincide.

It is clear that any fully $\mathcal{F}$ -normalised subgroup of S is also fully $\mathcal{B}$ -normalised. Suppose that P is a fully $\mathcal{B}$ -normalised subgroup of S which is not fully $\mathcal{F}$ -normalised and choose $Q\le S$ a fully $\mathcal{F}$ -normalised $\mathcal{F}$ -conjugate of P. Choose $\phi\in\textrm{Hom}_{\mathcal{F}}(P, Q)$ . By [ Reference Aschbacher, Kessar and Oliver5 , lemma I·7·6(a)] the “Frattini condition” holds and so there is $\alpha\in\textrm{Aut}_{\mathcal{F}}(S)$ and $\phi_0\in\textrm{Hom}_{\mathcal{B}}(P, Q\alpha^{-1})$ such that $\phi=\phi_0\circ \alpha$ . Since $\alpha\in \textrm{Aut}_{\mathcal{F}}(S)$ we have that $|N_S(Q)|=|N_S(Q\alpha^{-1})|$ and since P is fully $\mathcal{B}$ -normalised $|N_S(P)|\geq |N_S(Q\alpha^{-1})|$ , yielding a contradiction. Hence, any fully $\mathcal{B}$ -normalised subgroup is also fully $\mathcal{F}$ -normalised.

Finally, since $O^{p'}(\textrm{Aut}_{\mathcal{F}}(P))\le \textrm{Aut}_{\mathcal{B}}(P)\le \textrm{Aut}_{\mathcal{F}}(P)$ for all $P\le S$ , $\textrm{Out}_{\mathcal{F}}(P)$ has a strongly p-embedded subgroup if and only if $O^{p'}(\textrm{Out}_{\mathcal{F}}(P))$ does, and we conclude that $\mathcal{E}(\mathcal{F})=\mathcal{E}(\mathcal{B})$ .

We close this section with a result concerning strongly closed subgroups of fusion systems, and how they might be used to verify the exoticity of certain saturated fusion systems. In the analogous definition for finite groups, a result of Foote, building on work of Goldschmidt, promises that when $p=2$ , the only simple groups G which contain a proper non-trivial strongly closed subgroup $T\lt S\in\textrm{Syl}_2(G)$ are $\textrm{PSU}_3(2^n)$ and $\textrm{Sz}(2^n)$ . Work of Flores and Foote [ Reference Flores and Foote18 ] complements the result in the odd prime case, using the classification of finite simple groups. From their results, we deduce the following consequence ready for use in fusion systems.

Proof. Assume that $\mathcal{F}$ and A satisfy the hypotheses of the lemma, and suppose that there is a finite group G with $\mathcal{F}=\mathcal{F}_S(G)$ . We may as well choose G such that $O_{p'}(G)=\{1\}$ . Then A is a proper non-trivial strongly closed subgroup of G. Following [ Reference Flores and Foote18 ], let $\mathcal{O}_A(G)$ be the largest normal subgroup N of G such that $A\cap N\in\textrm{Syl}_p(N)$ . Then $\mathcal{O}_A(G)\cap A$ is a strongly closed subgroup of G. By the minimality of A, and using that $O_{p'}(G)=\{1\}$ , we deduce that either $A\in\textrm{Syl}_p(\mathcal{O}_A(G))$ or $\mathcal{O}_A(G)=\{1\}$ . In the former case, we have that $\mathcal{F}_A(\mathcal{O}_A(G))\trianglelefteq \mathcal{F}_S(G)=\mathcal{F}$ , a contradiction. Hence, $\mathcal{O}_A(G)=\{1\}$ . Applying [ Reference Flores and Foote18 , theorem 1·1] when $p=2$ and [ Reference Flores and Foote18 , theorem 1·3] when p is odd, we conclude that A is elementary abelian. But then by Proposition 3·10 we have that $A\trianglelefteq \mathcal{F}$ so that $\mathcal{F}_A(A)$ is a normal subsystem of $\mathcal{F}$ supported on A, another contradiction. Hence, no such G exists and $\mathcal{F}$ is exotic.

This result provides an alternate check on exoticity distinct from the techniques currently used in the literature, albeit still relying on the classification of finite simple groups.

4. Fusion Systems on a Sylow 3-subgroup of $\textrm{Co}_1$

In this section, we classify all saturated fusion systems supported on a 3-group S which is isomorphic to a Sylow 3-subgroup of the sporadic simple group $\textrm{Co}_1$ , validating Theorem A. Utilising the Atlas [ Reference Conway, Curtis, Norton, Parker and Wilson13 ], we extract the following 3-local maximal subgroups from $G\;:\!=\;\textrm{Co}_1$ :

\[M_1\cong 3^6\;:\;2.\textrm{M}_{12}\]

\[M_2\cong 3^{1+4}_+:\textrm{Sp}_4(3).2\]

\[M_3\cong 3^{3+4}\;:\;2.(\textrm{Sym}(4)\times \textrm{Sym}(4))\]

and remark that for a given $S\in\textrm{Syl}_3(G)$ , $M_i$ can be chosen such that $S\in\textrm{Syl}_3(M_i)$ . We record that $|S|=3^9$ and $J(S)=O_3(M_1)$ (where J(S) is as defined in Definition 2·1). We denote $\textbf{J}\;:\!=\;O_3(M_1)$ , $\textbf{Q}\;:\!=\;O_3(M_2)$ and $\textbf{R}\;:\!=\;O_3(M_3)$ .

In addition, S is isomorphic to a Sylow 3-subgroup of $\textrm{Sp}_6(3)$ and in this isomorphism we recognise the subgroups $E_1, E_2, E_3\le S$ whose images correspond to the unipotent radicals of the minimal parabolic subgroups of $\textrm{Sp}_6(3)$ . Indeed, $E_1, E_2, E_3$ are also essential subgroups of $\mathcal{F}_S(\textrm{Co}_1)$ such that

\[N_G(E_1)=M_1\cap M_2\cong 3^{1+4}_+.3^3:(2\times \textrm{GL}_2(3))\]

\[N_G(E_2)=M_1\cap M_3\cong 3^6.3^2:(2\times \textrm{GL}_2(3))\]

\[N_G(E_3)=M_2\cap M_3\cong 3^{3+4}.3:(2\times \textrm{GL}_2(3))\]

In an abuse of notation, we suppress the isomorphism between S and a Sylow 3-subgroup of $\textrm{Sp}_6(3)$ and let $E_1, E_2, E_3$ be subgroups of $\textrm{Co}_1$ or of $\textrm{Sp}_6(3)$ where appropriate.

We also note the following characterizations of $E_1, E_2$ and $E_3$ from their embeddings in S.

• (i) $E_1$ is the unique subgroup of S of order $3^8$ such that $\Phi(E_1)=\Phi(Y)$ , where Y is the preimage in S of $Z(S/\textbf{J})$ and has order $3^7$ .

• (ii) $E_2=C_S(Z_2(S))$ .

• (iii) $E_3$ is the unique subgroup X of S of order $3^8$ which is not equal to $E_1$ but satisfies $\mho^1(X)=Z(S)$ .

In particular, $E_1$ , $E_2$ and $E_3$ are characteristic subgroups of S, and so too is $\textbf{R}=E_2\cap E_3$ . In what follows, we take several liberties with the determination of various characteristic subgroups of the $E_i$ , but all of these properties are easily verified by computer (e.g. using MAGMA and taking S to be a Sylow 3-subgroup of $\textrm{Sp}_6(3)$ ).

Proof. This is a consequence of the Borel–Tits theorem [ Reference Gorenstein, Lyons and Solomon21 , corollary 3·1·6].

We record one final subgroup of G. Let $X\trianglelefteq M_1$ with $M_1/X\cong \textrm{M}_{12}$ and consider the maximal subgroup $H\cong \textrm{Alt}(4)\times \textrm{Sym}(3)$ of $M_1/X$ . Define $E_4$ to be the largest normal 3-subgroup of the preimage of H in $M_1$ so that

\[N_G(E_4)=N_{M_1}(E_4)\cong 3^6.3:(\textrm{SL}_2(3)\times 2).\]

Then $E_4$ is an essential subgroup of $\mathcal{F}_S(\textrm{Co}_1)$ , $E_4$ is not contained in any other essential subgroup of $\mathcal{F}_S(\textrm{Co}_1)$ and $[N_G(S): N_{N_G(S)}(E_4)]=6$ .

We note that non-trivial elements of $E_1/\textbf{J}$ and $E_2/\textbf{J}$ comprise of elements of type 3A in $N_G(\textbf{J})/\textbf{J}\cong 2.\textrm{M}_{12}$ and non-trivial elements of $E_4/\textbf{J}$ correspond to elements of type 3B in ATLAS terminology [ Reference Conway, Curtis, Norton, Parker and Wilson13 ]. In particular, for $x\in R\in E_4^{\mathcal{F}}$ with $x\not\in \textbf{J}$ , x acts on $\textbf{J}$ unlike any element of $E_1$ or $E_2$ .

Proof. See [ Reference Sawabe41 ].

We now move onto to the classification of all saturated fusion systems on S. Throughout we suppose that $\mathcal{F}$ is a saturated fusion system on a 3-group S such that S is isomorphic to a Sylow 3-subgroup of $\textrm{Co}_1$ .

We utilise the fusion systems package in MAGMA [ Reference Parker and Semeraro36, Reference Parker and Semeraro37 ] to verify the following proposition. The code and outputs are included in [ Reference van Beek44 , appendix A].

For the duration of this section, we will frequently use that $\textbf{J}=J(S)=J(E_i)$ is a characteristic subgroup of $E_i$ for $i\in\{1,2,4\}$ . This follows from Proposition 2·2 (iii).

Proof. We note first that $E_4$ is contained in no other essential subgroup of $\mathcal{F}$ and so by the Alperin–Goldschmidt theorem, $\{E_4^{\mathcal{F}}\}=\{E_4^{\textrm{Aut}_{\mathcal{F}}(S)}\}$ . In particular, if any $\mathcal{F}$ -conjugate of $E_4$ is essential, then every $\mathcal{F}$ -conjugate of $E_4$ is. Since $\textbf{J}$ is invariant under $\textrm{Aut}_{\mathcal{F}}(S)$ we may as well assume, aiming for a contradiction, that $E_4\in\mathcal{E}(\mathcal{F})$ .

Since $\textbf{J}=J(E_4)$ , we have that $N_{\mathcal{F}}(E_4)\le N_{\mathcal{F}}(\textbf{J})$ and so $N_{\mathcal{F}}(E_4)=N_{N_{\mathcal{F}}(\textbf{J})}(E_4)$ . In particular, if $E_4\in\mathcal{E}(\mathcal{F})$ then $E_4\in\mathcal{E}(N_{\mathcal{F}}(\textbf{J}))$ . By the uniqueness of models provided by Theorem 3·11, for H a model of $N_{\mathcal{F}}(\textbf{J})$ , we have that $N_{\mathcal{F}}(E_4)=\mathcal{F}_{N_S(E_4)}(N_H(E_4))$ so that $\textrm{Out}_{\mathcal{F}}(E_4)=N_H(E_4)/E_4$ . Since $O^{3'}(H/\textbf{J})\cong \textrm{PSL}_3(3)$ , we have that $N_H(E_4)\le N_H(S)$ and so $N_H(E_4)/E_4$ does not have a strongly 3-embedded subgroup, a contradiction.

Proof. Suppose that $\mathcal{F}$ is a saturated fusion system on S with $E_4\in\mathcal{E}(\mathcal{F})$ . Then, as $\textbf{J}=J(E_4)$ , $N_{\mathcal{F}}(E_4)\le N_{\mathcal{F}}(\textbf{J})$ and so $E_4$ is also essential in $N_{\mathcal{F}}(\textbf{J})$ . Since $E_4\not\trianglelefteq S$ and $|E_4/\textbf{J}|=3$ , Proposition 3·10 implies that $\textbf{J}=O_3(N_{\mathcal{F}}(\textbf{J}))$ . By Theorem 3·11 there is a finite group H with $S\in\textrm{Syl}_3(H)$ , $N_{\mathcal{F}}(\textbf{J})=\mathcal{F}_S(H)$ and $F^*(H)=\textbf{J}$ . Then, $O^{3'}(H)/\textbf{J}$ is determined by Lemma 2·8. Using that $E_4\in\mathcal{E}(\mathcal{F})$ and applying Lemma 4·4, we conclude that $O^{3'}(H)/\textbf{J}\cong 2.\textrm{M}_{12}$ .

Suppose now that $O^{3'}(\textrm{Out}_{\mathcal{F}}(\textbf{J}))\cong 2.\textrm{M}_{12}$ and again set H to be a model for $N_{\mathcal{F}}(\textbf{J})$ so that $O^{3'}(H)/\textbf{J}\cong 2.\textrm{M}_{12}$ . We examine the maximal subgroups of $2.\textrm{M}_{12}$ as can be found in the Atlas [ Reference Conway, Curtis, Norton, Parker and Wilson13 ], and identify them with their preimage in $O^{3'}(H)$ . Then there are three classes of maximal 3-local subgroups $H_1$ , $H_2$ , $H_3$ , and we may arrange in each case that $S\cap H_i\in\textrm{Syl}_3(H_i)$ . These groups have the same shape as $N_G(E_1), N_G(E_2)$ and $N_G(E_4)$ respectively. Indeed, in each case, $|N_S(O_3(H_i))/O_3(H_i)|=3$ and so we deduce that $H_i/O_3(H_i)$ contains a strongly 3-embedded subgroup for $i\in\{1,2,3\}$ . Since $\textbf{J}=J(O_3(H_i))$ , we have that $O^{3'}(\textrm{Out}_{\mathcal{F}}(O_3(H_i)))=O^{3'}(\textrm{Out}_{H}(O_3(H_i)))=O^{3'}(\textrm{Out}_{H_i}(O_3(H_i)))$ contains a strongly 3-embedded subgroup for all $i\in\{1,2,3\}$ . Moreover, each $O_3(H_i)$ is fully $\mathcal{F}$ -normalised and as $\textbf{J}$ is $\mathcal{F}$ -centric, so too is $O_3(H_i)$ for $i\in\{1,2,3\}$ . Applying Proposition 4·3, we have that $O_3(H_1), O_3(H_2), O_3(H_3)$ are equal to $E_1, E_2$ and $E_4\alpha$ for some $\alpha\in\textrm{Aut}_{\mathcal{F}}(S)$ . Hence, $E_1, E_2, E_4\in\mathcal{E}(\mathcal{F})$ , as required.

As a consequence of the above lemma, we have the following observation. Let $R\le S$ be such that $|R/\textbf{J}|=3$ and $R\not\trianglelefteq S$ . The for a saturated fusion system $\mathcal{F}$ on S, if R is $\mathcal{F}$ -essential then $R\in E_4^{\mathcal{F}}$ and $R/\textbf{J}$ corresponds to a subgroup of order 3 generated by an element of type 3B in $O^{3'}(\textrm{Out}_{\mathcal{F}}(\textbf{J}))\cong 2.\textrm{M}_{12}$ . Moreover, under these conditions and for $x\in S\setminus \textbf{J}$ such that $x\textbf{J}$ is an element of type 3B in $O^{3'}(\textrm{Out}_{\mathcal{F}}(\textbf{J}))\cong 2.\textrm{M}_{12}$ , we have that $\langle x\rangle \textbf{J}$ is an essential subgroup of $\mathcal{F}$ which is $\mathcal{F}$ -conjugate to $E_4$ .

Proof. Assume that $E_1\in\mathcal{E}(\mathcal{F})$ . We calculate that $Z(S)=Z(E_1)$ is of order 3, and $\Phi(E_1)=\textbf{J}\cap \textbf{Q}$ is elementary abelian of order $3^3$ with $C_S(\Phi(E_1))=\textbf{J}$ . Let $K\;:\!=\;C_{\textrm{Aut}_{\mathcal{F}}(E_1)}(\Phi(E_1))$ so that $\textrm{Aut}_{\textbf{J}}(E_1)\in\textrm{Syl}_3(K)$ and K normalises $\textrm{Inn}(E_1)$ . In particular, $[K, \textrm{Inn}(E_1)]\le K\cap \textrm{Inn}(E_1)=\textrm{Aut}_{\textbf{J}}(E_1)$ and K centralises the quotient $E_1/\textbf{J}$ . Now, as $\textbf{J}$ is elementary abelian, $K/C_K(\textbf{J})$ is a $3'$ -group and centralises $Z(E_1)=C_{\textbf{J}}(\textrm{Inn}(E_1))\le \Phi(E_1)$ . Applying the A $\times$ B-lemma, with $K|_{\textbf{J}}$ , $\textrm{Inn}(E_1)|_{\textbf{J}}$ and $\textbf{J}$ in the roles of A, B and V we deduce that K centralises $\textbf{J}$ , and so K centralises the chain $\{1\}\trianglelefteq \textbf{J}\trianglelefteq E_1$ . By Lemma 2·6, K is a 3-group. Thus, $K=\textrm{Aut}_{\textbf{J}}(E_1)$ and so we infer that $\textrm{Aut}_{\mathcal{F}}(E_1)/K$ acts faithfully on $\Phi(E_1)$ . Since $\textrm{Aut}_S(E_1)$ centralises $Z(E_1)=Z(S)$ and $\textrm{Inn}(E_1)=C_{\textrm{Aut}_S(E_1)}(\Phi(E_1)/Z(S))$ , we conclude that $O^{3'}(\textrm{Out}_{\mathcal{F}}(E_1))$ acts faithfully on $\Phi(E_1)/Z(S)$ of order $3^2$ . Then as $O_3(O^{3'}(\textrm{Out}_{\mathcal{F}}(E_1)))=\{1\}$ , considering subgroups of $\textrm{SL}_2(3)$ yields that $O^{3'}(\textrm{Out}_{\mathcal{F}}(E_1))\cong \textrm{SL}_2(3)$ and $\Phi(E_1)/Z(S)$ is its natural module.

We note that for $r\in O^{3'}(\textrm{Out}_{\mathcal{F}}(E_1))$ , if r centralises $E_1/\textbf{J}$ , then as $[E_1, \Phi(E_1)]=Z(S)$ , we have by the three subgroups lemma that $[r, \Phi(E_1), E_1]=\{1\}$ so that r centralises $\Phi(E_1)/Z(S)$ , a contradiction. Hence, $O^{3'}(\textrm{Out}_{\mathcal{F}}(E_1))\cong \textrm{SL}_2(3)$ and $E_1/\textbf{J}$ is its natural module. Set $V\;:\!=\;\textbf{J}/\Phi(E_1)$ of order $3^3$ . Then for $T=Z(O^{3'}(\textrm{Out}_{\mathcal{F}}(E_1)))$ we have by coprime action that $V=[V,T]\times C_V(T)$ . However, $\textrm{Out}_S(E_1)$ acts indecomposably on V and we conclude that $V=[V, T]$ or $V=C_V(T)$ is an irreducible 3-dimensional $\textrm{SL}_2(3)$ -module. Thus, $V=C_V(T)$ is a natural $\Omega_3(3)$ -module for $O^{3'}(\textrm{Out}_{\mathcal{F}}(E_1))/Z(O^{3'}(\textrm{Out}_{\mathcal{F}}(E_1)))\cong \textrm{PSL}_2(3)$ .

Proof. Assume that $E_2\in \mathcal{E}(\mathcal{F})$ . One can calculate that that $\Phi(E_2)=[E_2, \textbf{J}]$ is of order $3^5$ and is contained in $\textbf{J}$ , and $|Z(E_2)|=3^3$ and $|Z_2(S)|=3^2$ . By Proposition 2·2 (iii), $\textbf{J}=J(E_2)$ and $\textbf{J}/\Phi(E_2)$ is of order 3 and centralised by S. Hence, $O^{3'}(\textrm{Out}_{\mathcal{F}}(E_2))$ acts trivially on $\textbf{J}/E_2$ and so must act faithfully on $E_2/\textbf{J}$ of order $3^2$ by Lemma 2·5 and coprime action. Since $O_3(O^{3'}(\textrm{Out}_{\mathcal{F}}(E_2)))=\{1\}$ , we deduce that $O^{3'}(\textrm{Out}_{\mathcal{F}}(E_2))\cong\textrm{SL}_2(3)$ and $E_2/\textbf{J}$ is its natural module. Letting $r\in O^{3'}(\textrm{Out}_{\mathcal{F}}(E_2))$ , if r centralised $\Phi(E_2)/Z(E_2)$ then by coprime action, $[r, \textbf{J}, E_2]=\{1\}$ . Moreover, since $[E_2, \textbf{J}, r]\le Z(E_2)$ we conclude by the three subgroups lemma that $[E_2, r, \textbf{J}]\le Z(E_2)$ . But $\textbf{J}$ is abelian so that $[E_2, r, \textbf{J}]=[[E_2, r]\textbf{J}, \textbf{J}]=[E_2, \textbf{J}]=\Phi(E_2)$ , a contradiction. Hence, $\Phi(E_2)/Z(E_2)$ is a 2-dimensional faithful module for $O^{3'}(\textrm{Out}_{\mathcal{F}}(E_2))$ and so is a natural $\textrm{SL}_2(3)$ -module.

Proof. Assume that $E_3\in \mathcal{E}(\mathcal{F})$ . One may calculate that $Z_2(S)=Z_2(E_3)$ and so $C_{E_3}(Z_2(E_3))=E_2\cap E_3=\textbf{R}\trianglelefteq \textrm{Out}_{\mathcal{F}}(E_3)$ . Since S centralises $E_3/\textbf{R}$ , we must have that $O^{3'}(\textrm{Out}_{\mathcal{F}}(E_3))$ centralises $E_3/\textbf{R}$ . Moreover, one can calculate that $\Phi(E_3)$ is of order $3^5$ and so by Lemma 2·5 and coprime action, $O^{3'}(\textrm{Out}_{\mathcal{F}}(E_3))$ acts faithfully on $\textbf{R}/\Phi(E_3)$ which has order $3^2$ . As in Lemma 4·6, we conclude that $O^{3'}(\textrm{Out}_{\mathcal{F}}(E_3))\cong\textrm{SL}_2(3)$ and $\textbf{R}/\Phi(E_3)$ is a natural module.

Let $r\in O^{3'}(\textrm{Out}_{\mathcal{F}}(E_3))$ of $3'$ -order. We note that $Z(S)\lt Z_2(E_3)=Z_2(S)\lt Z(\textbf{R})=Z(E_2)$ and $Z(E_2)$ has order $3^3$ . It follows that $O^3(O^{3'}(\textrm{Out}_{\mathcal{F}}(E_3)))$ acts trivially on $Z(\textbf{R})$ . Assume that r centralises $\Phi(E_3)/Z(\textbf{R})$ . Then by coprime action r centralises $\Phi(E_3)$ . One can calculate that $C_{E_3}(\Phi(E_3)))=Z_2(E_3)\le \Phi(E_3)$ so that $[r, E_3]=\{1\}$ by coprime action and the three subgroups lemma, a contradiction. Hence, $\Phi(E_3)/Z(\textbf{R})$ is a 2-dimensional faithful module for $O^{3'}(\textrm{Out}_{\mathcal{F}}(E_3))$ and so is a natural $\textrm{SL}_2(3)$ -module.

1. (i) $\mathcal{F}=N_{\mathcal{F}}(S)$ ; or

2. (ii) $\mathcal{F}=N_{\mathcal{F}}(E_i)$ where $O^{3'}(\textrm{Out}_{\mathcal{F}}(E_i))\cong \textrm{SL}_2(3)$ for some $i\in\{1,2,3\}$ .

Proof. If $\mathcal{E}(\mathcal{F})=\emptyset$ , then outcome (i) is satisfied by the Alperin–Goldschmidt theorem. Thus, we may assume that $E_i$ is the unique essential subgroup of $\mathcal{F}$ . Indeed, we must have that $E_i$ is invariant under $\textrm{Aut}_{\mathcal{F}}(S)$ and so $E_i\trianglelefteq \mathcal{F}$ . Then Lemma 4·6, Lemma 4·7 and Lemma 4·8 complete the proof in case (ii).

Proof. Assume that $E_1\in\mathcal{E}(\mathcal{F})$ . By Lemma 4·6, $O^{3'}(\textrm{Out}_{\mathcal{F}}(E_1))\cong \textrm{SL}_2(3)$ normalises $\textbf{J}$ and $\Phi(E_1)$ , and $V\;:\!=\;\textbf{J}/\Phi(E_1)$ is a irreducible 3-dimensional module for $O^{3'}(\textrm{Out}_{\mathcal{F}}(E_1))/T\cong \textrm{PSL}_2(3)$ where $T=C_{O^{3'}(\textrm{Out}_{\mathcal{F}}(E_1))}(V)$ . For $U\;:\!=\;E_1/\Phi(E_1)$ we have that $U/V$ has order 9 and as T acts non-trivially on U by coprime action, we deduce that $C_U(T)=V$ and $U=[U, T]\times V$ where [U,T] is a natural $\textrm{SL}_2(3)$ -module.

For X the preimage in $E_1$ of [U,T], we have that $X\trianglelefteq S$ and $X\cap \textbf{J}=\Phi(E_1)$ . Moreover, since $X/\Phi(E_1)$ is an irreducible module for $\textrm{Out}_{\mathcal{F}}(E_1)$ , we deduce that $|\Omega_1(X)|\ne 3^4$ . With this information, we calculate that there are 3 subgroups of $E_1$ satisfying these properties including X. Furthermore, since $E_1, \Phi(E_1)$ and $\textbf{J}$ are all characteristic subgroups of S, we have that $X\alpha$ also satisfies these properties for all $\alpha\in\textrm{Aut}(S)$ , and we calculate that under the action of $\textrm{Aut}(S)$ , all 3 subgroups of $E_1$ are conjugate (see [ Reference van Beek44 , appendix A] for the explicit code for these calculations). Finally, since $\textbf{Q}$ satisfies these properties, we conclude that there is $\alpha\in\textrm{Aut}(S)$ such that $X=\textbf{Q}\alpha$ . By the module decomposition of U above, X is the unique such $\textrm{Aut}(S)$ -conjugate of $\textbf{Q}$ which is $\textrm{Aut}_{\mathcal{F}}(E_1)$ -invariant.

By definition, $\mathcal{F}^{\alpha^{-1}}$ is a saturated fusion system on S which is isomorphic to $\mathcal{F}$ , for $\alpha\in\textrm{Aut}(S)$ . Furthermore, it follows from the above lemma that there is $\alpha\in\textrm{Aut}(S)$ such that $\textbf{Q}$ is the unique subgroup of S in its $\textrm{Aut}(S)$ -conjugacy class which is both $\textrm{Aut}_{\mathcal{F}^{\alpha^{-1}}}(E_1)$ -invariant and $\textrm{Aut}_{\mathcal{F}^{\alpha^{-1}}}(S)$ -invariant. Since we are only interested in investigating the possibilities of $\mathcal{F}$ up to isomorphism, we may as well assume for the remainder of this section that $\textbf{Q}$ is $\textrm{Aut}_{\mathcal{F}}(E_1)$ -invariant whenever $E_1\in\mathcal{E}(\mathcal{F})$ . Indeed, $\textbf{Q}$ is the preimage in $E_1$ of $[E_1/\Phi(E_1), Z(O^{3'}(\textrm{Out}_{\mathcal{F}}(E_1)))]$ .

1. (i) $E_4^{\mathcal{F}}\cap \mathcal{E}(\mathcal{F})=\emptyset$ , $\mathcal{E}(N_{\mathcal{F}}(\textbf{J}))=\{E_1, E_2\}$ and $O^{3'}(\textrm{Out}_{\mathcal{F}}(\textbf{J}))\cong \textrm{PSL}_3(3)$ ; or

2. (ii) $E_4\in\mathcal{E}(\mathcal{F})$ , $\mathcal{E}(N_{\mathcal{F}}(\textbf{J}))=\{E_1, E_2, E_4^{\mathcal{F}}\}$ and $O^{3'}(\textrm{Out}_{\mathcal{F}}(\textbf{J}))\cong 2.\textrm{M}_{12}$ .

Moreover, in each case, if $E_3\not\in\mathcal{E}(\mathcal{F})$ then $\mathcal{F}=N_{\mathcal{F}}(\textbf{J})$ .

Proof. By Proposition 4·3, $\mathcal{E}(N_{\mathcal{F}}(\textbf{J}))\subseteq \{E_1, E_2, E_3, E_4^{\mathcal{F}}\}$ . We note that as $\textbf{J}=J(E_1)=J(E_2)=J(E_4)$ , $\textrm{Out}_{\mathcal{F}}(E_i)=\textrm{Out}_{N_{\mathcal{F}}(\textbf{J})}(E_i)$ for $i\in\{1,2,4\}$ . Furthermore, since $E_i$ is self-centralising in S and fully normalised in $\mathcal{F}$ , we see that $E_i\in\mathcal{E}(N_{\mathcal{F}}(\textbf{J}))$ if and only if $E_i\in\mathcal{E}(\mathcal{F})$ for $i\in\{1,2,4\}$ . Since $\textbf{J}\not\le E_3$ , we necessarily have that $E_3\not\in\mathcal{E}(N_{\mathcal{F}}(\textbf{J}))$ by Proposition 3·10.

Suppose that $\{E_1, E_2\}\subseteq \mathcal{E}(\mathcal{F})$ . Let X be the largest subgroup normalised by $\textrm{Aut}_{\mathcal{F}}(E_1)$ and $\textrm{Aut}_{\mathcal{F}}(E_2)$ . Since $\textbf{J}=J(E_1)=J(E_2)$ , we have that $\textbf{J}\le X\le E_1\cap E_2$ . Furthermore, by Lemma 4·6, $E_1/\textbf{J}$ is irreducible under $\textrm{Aut}_{\mathcal{F}}(E_1)$ and we deduce that $X=\textbf{J}$ and $\textbf{J}=O_3(N_{\mathcal{F}}(\textbf{J}))$ . Indeed, $\textrm{Out}_{\mathcal{F}}(\textrm{J})$ satisfies the hypothesis of Lemma 2·8 and we deduce that $O^{3'}(\textrm{Out}_{\mathcal{F}}(\textbf{J}))\cong \textrm{PSL}_3(3)$ or $2.\textrm{M}_{12}$ . In the former case, we have by Lemma 4·4 that $E_4^{\mathcal{F}}\cap \mathcal{E}(\mathcal{F})=\emptyset$ , and so (i) holds. In the latter case, we have by Lemma 4·5 that $E_4\in\mathcal{E}(\mathcal{F})$ , and so (ii) holds. Finally, since $\textbf{J}=J(S)$ and $\textbf{J}$ is invariant under $\textrm{Aut}_{\mathcal{F}}(S)$ , Proposition 3·10 and Proposition 4·3 imply that if $E_3\not\in\mathcal{E}(\mathcal{F})$ then $\mathcal{F}=N_{\mathcal{F}}(\textbf{J})$ .

Proof. Suppose that $\{E_1, E_3\}\subseteq \mathcal{E}(\mathcal{F})$ and let X be the largest subgroup of S normalised by both $\textrm{Aut}_{\mathcal{F}}(E_1)$ and $\textrm{Aut}_{\mathcal{F}}(E_3)$ . Then $X\le E_1\cap E_3$ so that $\textbf{J}\not\le X$ . Since $\textrm{Aut}_{\mathcal{F}}(E_1)$ acts irreducibly on $\textbf{J}/\Phi(E_1)$ , by the choice of $\textbf{Q}$ following Lemma 4·10 we have that $X\le \textbf{Q}$ . We note that $Z(S)=Z(E_1)=Z(E_3)$ so that $Z(S)\le X$ .

Assume first that $X=Z(S)$ , let $G_i$ be a model for $N_{\mathcal{F}}(E_i)$ , where $i\in\{1,3\}$ , and $G_{13}$ be a model for $N_{\mathcal{F}}(S)$ . Since $E_1$ and $E_3$ are $\textrm{Aut}_{\mathcal{F}}(S)$ -invariant, we can arrange that there are injective maps $\alpha_i:G_{13}\to G_i$ for $i\in\{1,3\}$ . Furthermore, since $Z(S)\trianglelefteq G_1, G_3$ , we may form injective maps $\alpha_i^*:G_{13}/Z(S)\to G_i/Z(S)$ so that the tuple $(G_1/Z(S), G_3/Z(S), G_{13}/Z(S), \alpha_1^*, \alpha_3^*)$ satisfies the hypothesis of [ Reference Delgado and Stellmacher16 , theorem A]. Since $|S/Z(S)|=3^8$ and $|Z(S/Z(S))|=3$ , comparing with the outcomes provided by [ Reference Delgado and Stellmacher16 , theorem A], we have a contradiction.

Thus, $Z(S)\lt X$ and we deduce that $Z(S)\lt X\cap Z_2(E_1)\le \Phi(E_1)$ . By Lemma 4·6, $\textrm{Aut}_{\mathcal{F}}(E_1)$ is irreducible on $\Phi(E_1)/Z(S)$ and so we have that $\Phi(E_1)\le X$ . If $X=\Phi(E_1)$ then $|X|=3^3$ and $X\cap Z_2(S)\gt Z(S)$ . Hence, $|X\Phi(E_3)/\Phi(E_3)|\leq 3$ and as $\textrm{Aut}_{\mathcal{F}}(E_3)$ acts irreducibly on $\textbf{R}/\Phi(E_3)$ by Lemma 4·8 we deduce that $X\le \Phi(E_3)$ . Similarly, $|XZ_2(S)/Z_2(S)|\leq 3$ and as $\textrm{Aut}_{\mathcal{F}}(E_3)$ acts irreducibly on $\Phi(E_3)/Z_2(S)$ by Lemma 4·8 we deduce that $X=Z_2(S)$ , a contradiction since $X/Z(S)$ is a natural $\textrm{SL}_2(3)$ -module for $O^{3'}(\textrm{Out}_{\mathcal{F}}(E_1))$ . Hence, $\Phi(E_1)\lt X$ . Finally, since $X\le \textbf{Q}$ and $\textrm{Aut}_{\mathcal{F}}(E_1)$ acts irreducibly on $\textbf{Q}/\Phi(E_1)$ by Lemma 4·6, we have that $X=\textbf{Q}$ .

We have that $O^{3'}(\textrm{Out}_{\mathcal{F}}(\textbf{Q}))$ acts faithfully on $\textbf{Q}$ . By [ 47 ], we deduce that $O^{3'}(\textrm{Out}_{\mathcal{F}}(\textbf{Q}))$ is isomorphic to a subgroup of $O^{3'}(\textrm{Out}(\textbf{Q}))\cong\textrm{Sp}_4(3)$ . Hence, $\textrm{Out}_S(\textbf{Q})\in\textrm{Syl}_3(\textrm{Out}(\textbf{Q}))$ and $O^{3'}(\textrm{Out}_{\mathcal{F}}(\textbf{Q}))$ is an overgroup of $\textrm{Out}_S(\textbf{Q})$ with no non-trivial normal 3-subgroups. By [ Reference Bray, Holt and Roney-Dougal9 , Table 8·12], any maximal subgroup of $\textrm{Sp}_4(3)$ which contains a Sylow 3-subgroup is a parabolic subgroup so has a normal 3-subgroups. Hence, $O^{3'}(\textrm{Out}_{\mathcal{F}}(\textbf{Q}))$ is contained in no maximal subgroups so that $O^{3'}(\textrm{Out}_{\mathcal{F}}(\textbf{Q}))\cong \textrm{Sp}_4(3)$ .

We note that the maximal abelian subgroups of $\textbf{Q}$ have order $3^3$ and so $\textbf{Q}\textbf{J}=E_1$ . In particular, $E_2\not\ge \textbf{Q}\not\le E_4$ and neither $E_2$ nor $E_4$ are essential in $N_{\mathcal{F}}\textbf{(Q)}$ by Proposition 3·10. Since $E_1, E_3$ are $\mathcal{F}$ -centric, normal in S and satisfy $\textrm{Out}_{\mathcal{F}}(E_i)=\textrm{Out}_{N_{\mathcal{F}}(\textbf{Q})}(E_i)$ , we deduce that $E_1, E_3\in\mathcal{E}(\mathcal{F})$ if and only if $E_1, E_3\in\mathcal{E}(N_{\mathcal{F}}(\textbf{Q}))$ . By Lemma 4·5, if $E_4^{\mathcal{F}}\cap \mathcal{E}(\mathcal{F})\ne \emptyset$ , then $E_2\in\mathcal{E}(\mathcal{F})$ and so by Proposition 4·3, if $E_2\not\in\mathcal{E}(\mathcal{F})$ then $\mathcal{E}(\mathcal{F})=\{E_1, E_3\}$ . In particular, since we have arranged that $\textbf{Q}$ is $\textrm{Aut}_{\mathcal{F}}(S)$ -invariant by Lemma 4·10, applying Proposition 3·10 we see that $\mathcal{F}=N_{\mathcal{F}}(\textbf{Q})$ , completing the proof.

Proof. Suppose that $\{E_2, E_3\}\subseteq \mathcal{E}(\mathcal{F})$ . By Lemma 4·8, we have that $\textbf{R}=E_2\cap E_3$ is characteristic in $E_3$ . Recall from Lemma 4·7 that for $V\;:\!=\;E_2/\Phi(E_2)$ and $L\;:\!=\;O^{3'}(\textrm{Out}_{\mathcal{F}}(E_2))\cong \textrm{SL}_2(3)$ , $V=[V,L]\times C_V(L)$ where [V,L] has order $3^2$ and $C_V(L)=\textbf{J}/\Phi(E_2)$ .

We claim that $\textbf{R}$ is the preimage of [V,L] in $E_2$ and so is normalised by $\textrm{Aut}_{\mathcal{F}}(E_2)$ . First, observe that $[E_2, E_3]\Phi(E_2)/\Phi(E_2)$ has order 3 and is contained in $([V,L]\cap \textbf{R})/\Phi(E_2)$ . Since $E_3$ is $\textrm{Aut}_{\mathcal{F}}(S)$ -invariant, we deduce that either $\textbf{R}$ is the preimage of [V,L], or L centralises $\textbf{R}/[E_2, E_3]\Phi(E_2)$ . In the latter case, we deduce that $C_V(L)\le \textbf{R}/\Phi(E_2)$ so that $\textbf{J}\le \textbf{R}$ , a contradiction. Hence, $\textbf{R}$ is the preimage in $E_2$ of [V,L] and so is normalised by $\textrm{Aut}_{\mathcal{F}}(E_2)$ .

Since $\Phi(\textbf{R})=Z(\textbf{R})$ , $|\textbf{R}/\Phi(\textbf{R})|=3^4$ and applying Lemma 2·5, we deduce that $O^{3'}(\textrm{Out}_{\mathcal{F}}(\textbf{R}))$ is isomorphic to a subgroup $\textrm{SL}_4(3)$ . Set $\overline{\textbf{R}}=\textbf{R}/\Phi(\textbf{R})$ . We note that $|C_{\overline{\textbf{R}}}(\textrm{Out}_S(\textbf{R}))|=3$ and that $\overline{\textbf{R}}=\langle C_{\overline{\textbf{R}}}(\textrm{Out}_S(\textbf{R}))^{\textrm{Out}_{\mathcal{F}}(\textbf{R})}\rangle$ by the actions of $N_{\textrm{Out}_{\mathcal{F}}(\textbf{R})}(\textrm{Out}_{E_i}(\textbf{R}))\cong \textrm{Aut}_{\mathcal{F}}(E_i)/\textrm{Aut}_{\textbf{R}}(E_i)$ for $i\in\{2,3\}$ . In particular, $\textrm{Out}_{\mathcal{F}}(\textbf{R})$ stabilises no subspaces of $\overline{R}$ and $\overline{R}$ is indecomposable under $\textrm{Out}_{\mathcal{F}}(\textbf{R})$ . Moreover, $|N_{O^{3'}(\textrm{Out}_{\mathcal{F}}(\textbf{R}))}(\textrm{Out}_{E_i}(\textbf{R}))|$ is divisible by 8 and we deduce that $O^{3'}(\textrm{Out}_{\mathcal{F}}(\textbf{R}))\not\cong \textrm{(P)SL}_2(9)$ . Comparing with [ Reference Bray, Holt and Roney-Dougal9 , Table 8·8], we deduce that $O^{3'}(\textrm{Out}_{\mathcal{F}}(\textbf{R}))$ is isomorphic to a subgroup of $\textrm{SO}_4^+(3)$ or $\textrm{Sp}_4(3)$ . In the latter case, we check against the tables of maximal subgroups of $\textrm{Sp}_4(3)$ [ Reference Bray, Holt and Roney-Dougal9 , Table 8·12] and find no suitable candidates which contain $O^{3'}(\textrm{Out}_{\mathcal{F}}(\textbf{R}))$ . In the former case, since $|\textrm{SO}_4^+(3)|_3=3^2$ and comparing orders we deduce that $O^{3'}(\textrm{Out}_{\mathcal{F}}(\textbf{R}))\cong O^{3'}(\textrm{SO}_4^+(3))=\Omega_4^+(3)$ , as desired.

Since $\textbf{R}\trianglelefteq S$ is of order $3^7$ , contained in $E_2$ and does not contain $\textbf{J}$ (for otherwise $\textbf{J}\le E_3$ ), we see that $E_1\not\ge \textbf{R}\not\le E_4$ and neither $E_1$ nor $E_4$ are essential in $N_{\mathcal{F}}\textbf{(R)}$ by Proposition 3·10. Since $E_2, E_3$ are $\mathcal{F}$ -centric, normal in S and satisfy $\textrm{Out}_{\mathcal{F}}(E_i)=\textrm{Out}_{N_{\mathcal{F}}(\textbf{R})}(E_i)$ , we deduce that $E_2, E_3\in\mathcal{E}(\mathcal{F})$ if and only if $E_2, E_3\in\mathcal{E}(N_{\mathcal{F}}(\textbf{R}))$ . By Lemma 4·5, if $E_4^{\mathcal{F}}\cap \mathcal{E}(\mathcal{F})\ne \emptyset$ , then $E_1\in\mathcal{E}(\mathcal{F})$ and so by Proposition 4·3, if $E_1\not\in\mathcal{E}(\mathcal{F})$ then $\mathcal{E}(\mathcal{F})=\{E_2, E_3\}$ . Since $E_2$ and $E_3$ are characteristic subgroups of S, so too is $\textbf{R}$ . Hence, $\textbf{R}$ is $\textrm{Aut}_{\mathcal{F}}(S)$ -invariant and so if $E_1\not\in\mathcal{E}(\mathcal{F})$ , then applying Proposition 3·10, we have that $\mathcal{F}=N_{\mathcal{F}}(\textbf{R})$ , completing the proof.

Hence, as consequence of Proposition 4·3, Lemma 4·5 and Proposition 4·9-Proposition 4·13, we have proved the following result.

We now complete the classification of all saturated fusion systems supported on S. As evidenced in Proposition 4·12 and Proposition 4·13, the structure of $O^{3'}(\textrm{Out}_{\mathcal{F}}(\textbf{Q}))$ and $O^{3'}(\textrm{Out}_{\mathcal{F}}(\textbf{R}))$ is fairly rigid and the flexibility we exploit is in the possible choices of actions for $\textrm{Out}_{\mathcal{F}}(\textbf{J})$ .

The identification of the fusion systems of $\textrm{Sp}_6(3)$ and $\textrm{Aut}(\textrm{Sp}_6(3))$ is proved using a result of Onofrei [ Reference Onofrei34 ] which identifies a parabolic system in $\mathcal{F}$ . Further restrictions then identify $\textrm{Sp}_6(3)$ from an associated chamber system. We remark that in the case of parabolic systems in groups, the definition is meant to abstractly capture a set of minimal parabolics containing a “Borel”, in analogy with groups of Lie type in defining characteristic. We cannot hope to capture the rich theory of parabolic systems in groups (and fusion systems) here, but we refer to [ Reference Meixner, Kantor, Liebler, Payne and Shult29 ] for a survey of this area in the group theory case, and refer to [ Reference Onofrei34 ] for the fusion system parallel.

Proof. Let $\mathcal{F}_{ij}\;:\!=\;\langle N_{\mathcal{F}}(E_i), N_{\mathcal{F}}(E_j)\rangle_S$ for $i,j\in\{1,2,3\}$ , noting that $N_{\mathcal{F}}(S)\le N_{\mathcal{F}}(E_i)$ for all $i\in\{1,2,3\}$ . Then $\textbf{J}\trianglelefteq \mathcal{F}_{12}$ and as $E_4\not\in\mathcal{E}(\mathcal{F})$ , Proposition 4·11 along with the Alperin–Goldschmidt theorem imply that $\mathcal{F}_{12}=N_{\mathcal{F}}(\textbf{J})$ . Applying Proposition 4·12 we have that $\mathcal{F}_{13}=N_{\mathcal{F}}(\textbf{Q})$ , and Proposition 4·13 yields that $\mathcal{F}_{23}=N_{\mathcal{F}}(\textbf{R})$ .

Let $\alpha\in \textrm{Hom}_{N_{\mathcal{F}}(E_i)\cap N_{\mathcal{F}}(E_j)}(P, Q)$ for $P,Q\le S$ , $i\ne j$ and $i,j\in \{1,2,3\}$ . Since $E_i\trianglelefteq N_{\mathcal{F}}(E_i)\cap N_{\mathcal{F}}(E_j)$ , there is $\widehat{\alpha}\in\textrm{Hom}_{N_{\mathcal{F}}(E_i)\cap N_{\mathcal{F}}(E_j)}(PE_i, QE_i)$ with $\widehat{\alpha}|_P=\alpha$ . But $E_j\trianglelefteq N_{\mathcal{F}}(E_i)\cap N_{\mathcal{F}}(E_j)$ and so there is $\widetilde \alpha\in \textrm{Hom}_{N_{\mathcal{F}}(E_i)\cap N_{\mathcal{F}}(E_j)}(PE_iE_j, QE_iE_j)$ with $\widetilde\alpha|_{PE_i}=\widehat{\alpha}$ . Since $E_iE_j=S$ , we have shown that for all $\alpha\in\textrm{Hom}_{N_{\mathcal{F}}(E_i)\cap N_{\mathcal{F}}(E_j)}(P, Q)$ , there is $\widetilde \alpha\in \textrm{Aut}_{N_{\mathcal{F}}(E_i)\cap N_{\mathcal{F}}(E_j)}(S)$ with $\widetilde \alpha|_P=\alpha$ . Hence, $S\trianglelefteq N_{\mathcal{F}}(E_i)\cap N_{\mathcal{F}}(E_j)$ so that $N_{\mathcal{F}}(S)=N_{\mathcal{F}}(E_i)\cap N_{\mathcal{F}}(E_j)$ whenever $i\ne j$ . Hence, $\{\mathcal{F}_i: i\in\{1,2,3\}\}$ is a family of parabolic subsystems in the sense of [ Reference Onofrei34 , definition 5·1].

In fact, following [ Reference Onofrei34 , definition 7·4], $\mathcal{F}$ has a family of parabolic subsystems of type $\mathfrak{M}$ , where $\mathfrak{M}$ is the diagram associated to $\mathcal{F}$ described in that definition. By Proposition 4·11, Proposition 4·12 and Proposition 4·13, $\mathfrak{M}$ is exactly the Dynkin diagram corresponding to the group $\textrm{Sp}_6(3)$ and so is a spherical diagram. Then [ Reference Onofrei34 , proposition 7·5 (ii)] implies that $\mathcal{F}$ is the fusion system of a finite simple group G of Lie type in characteristic p extended by diagonal and field automorphisms. Then $N_{\mathcal{F}}(\textbf{Q})=\mathcal{F}_S(N_G(\textbf{Q}))$ and as $O^{3'}(\textrm{Out}_{\mathcal{F}}(\textbf{Q}))\cong \textrm{Sp}_4(3)$ acts irreducibly on $\textbf{Q}/Z(S)$ , we conclude that $N_G(\textbf{Q})=O^3(N_G(\textbf{Q}))$ so that $G=O^3(G)$ . Comparing with the structure of the Sylow 3-subgroups of the finite simple groups of Lie type (as can be found in [ Reference Gorenstein, Lyons and Solomon21 , section 3·3]), we deduce that $\mathcal{F}=\mathcal{F}_S(G)$ where $\textrm{Inn}(\textrm{Sp}_6(3))\le G\le \textrm{Aut}(\textrm{Sp}_6(3)))$ .

Proof. We observe first that $\mathcal{G}\;:\!=\;\mathcal{F}_S(\textrm{Co}_1)$ satisfies the hypothesis of the proposition and that $\mathcal{G}=\langle \textrm{Aut}_{\mathcal{G}}(E_1), \textrm{Aut}_{\mathcal{G}}(E_2), \textrm{Aut}_{\mathcal{G}}(E_3), \textrm{Aut}_{\mathcal{G}}(E_4), \textrm{Aut}_{\mathcal{G}}(S)\rangle$ by the Alperin–Goldschmidt theorem. By Lemma 4·10, there is $\alpha\in \textrm{Aut}_{\mathcal{F}}(S)$ such that $\textbf{Q}\alpha$ is $\textrm{Aut}_{\mathcal{F}}(S)$ -invariant and $\textrm{Aut}_{\mathcal{F}}(E_1)$ -invariant. Since we are only interested in determining $\mathcal{F}$ up to isomorphism, we may replace $\mathcal{F}$ by $\mathcal{F}^\alpha$ and assume that $\textbf{Q}$ is $\textrm{Aut}_{\mathcal{F}}(S)$ -invariant and $\textrm{Aut}_{\mathcal{F}}(E_1)$ -invariant. We have that $\textbf{Q}$ is $\textrm{Aut}_{\mathcal{G}}(E_1)$ -invariant and $\textrm{Aut}_{\mathcal{G}}(E_1)$ -invariant by construction.

Since $\textbf{J}$ is characteristic in $E_1, E_2 $ and $E_4$ , $\textbf{Q}\trianglelefteq N_{\mathcal{F}}(E_3)$ and $\textbf{Q}\trianglelefteq N_{\mathcal{G}}(E_3)$ , we see that $\mathcal{G}=\langle N_{\mathcal{G}}(\textbf{J}), N_{\mathcal{G}}(\textbf{Q})\rangle_S$ and $\mathcal{F}=\langle N_{\mathcal{F}}(\textbf{J}), N_{\mathcal{F}}(\textbf{Q})\rangle_S$ . Hence, upon showing that $N_{\mathcal{G}}(\textbf{J})=N_{\mathcal{F}}(\textbf{J})$ and $N_{\mathcal{G}}(\textbf{Q})=N_{\mathcal{F}}(\textbf{Q})$ , we will have shown that $\mathcal{F}=\mathcal{G}$ and the proof will be complete.

Applying Proposition 4·11 and Proposition 4·12, since $E_4\in\mathcal{E}(\mathcal{F})$ , we have that $O^{3'}(\textrm{Aut}_{\mathcal{F}}(\textbf{J}))\cong 2.\textrm{M}_{12}$ and $O^{3'}(\textrm{Out}_{\mathcal{F}}(\textbf{Q}))\cong \textrm{Sp}_4(3)$ . We may lift the $3'$ -order morphisms in $N_{O^{3'}(\textrm{Aut}_{\mathcal{F}}(\textbf{J}))}(\textrm{Aut}_S(\textbf{J}))$ to morphisms in $\textrm{Aut}_{\mathcal{F}}(S)$ by Lemma 3·2, which then restrict faithfully to morphisms of $\textrm{Aut}_{\mathcal{F}}(\textbf{Q})$ by Lemma 4·10. Similarly, any morphism in $N_{\textrm{Aut}_{\mathcal{F}}(\textbf{Q})}(\textrm{Aut}_S(\textbf{Q}))$ lift to morphisms in $\textrm{Aut}_{\mathcal{F}}(S)$ by Lemma 3·2 and restrict faithfully to morphisms in $N_{\textrm{Aut}_{\mathcal{F}}(\textbf{J})}(\textrm{Aut}_S(\textbf{J}))$ . Comparing the orders of the normaliser of a Sylow 3-subgroup of $2.\textrm{M}_{12}$ with the normaliser of a Sylow 3-subgroup of $\textrm{Out}(\textbf{Q})\cong \textrm{Sp}_4(3).2$ , and applying the Frattini argument, we deduce that $\textrm{Aut}_{\mathcal{F}}(\textbf{Q})=\textrm{Aut}(\textbf{Q})=\textrm{Aut}_{\mathcal{G}}(\textbf{Q})\cong 3^4:(\textrm{Sp}_4(3)\;:\;2)$ and $\textrm{Aut}_{\mathcal{F}}(\textbf{J})\cong 2.\textrm{M}_{12}$ . Since $\textbf{J}$ admits $\textrm{Aut}_{\mathcal{F}}(S)$ faithfully, we deduce that $|\textrm{Aut}_{\mathcal{F}}(S)|=|\textrm{Aut}_{\mathcal{G}}(S)|$ . By Theorem 3·11, we conclude that there is $\beta\in\textrm{Aut}(S)$ with $N_{\mathcal{F}^\beta}(\textbf{Q})=N_{\mathcal{G}}(\textbf{Q})$ . Since $\textrm{Aut}_{\mathcal{G}}(S)=\textrm{Aut}_{N_{\mathcal{G}}(\textbf{Q})}(S)=\textrm{Aut}_{N_{\mathcal{F}^\beta}(\textbf{Q})}(S)$ and $|\textrm{Aut}_{\mathcal{F}^\beta}(S)|=|\textrm{Aut}_{\mathcal{F}}(S)|=|\textrm{Aut}_{\mathcal{G}}(S)|$ , we deduce that $\textbf{Q}$ is $\textrm{Aut}_{\mathcal{F}^\beta}(S)$ -invariant. A similar argument reveals that $\textbf{Q}$ is $\textrm{Aut}_{\mathcal{F}^\beta}(E_1)$ -invariant. As we are only interested in determining $\mathcal{F}$ up to isomorphism, we mat replace $\mathcal{F}$ by $\mathcal{F}^\beta$ so that $N_{\mathcal{G}}(\textbf{Q})=N_{\mathcal{F}}(\textbf{Q})$ , and $\textbf{Q}$ is $\textrm{Aut}_{\mathcal{F}}(S)$ -invariant and $\textrm{Aut}_{\mathcal{F}}(E_1)$ -invariant.

Now, $N_{\mathcal{G}}(E_1)=N_{N_{\mathcal{G}}(\textbf{Q})}(E_1)=N_{N_{\mathcal{F}}(\textbf{Q})}(E_1)=N_{\mathcal{F}}(E_1)$ . Then $N_{\mathcal{G}}(\textbf{J})\ge N_{\mathcal{G}}(E_1)\le N_{\mathcal{F}}(\textbf{J})$ and by [ Reference Oliver33 , proposition 2·11], it suffices to show that $\textrm{Aut}_{\mathcal{G}}(\textbf{J})=\textrm{Aut}_{\mathcal{F}}(\textbf{J})$ and that the homomorphism $H^1(\textrm{Out}_{\mathcal{G}}(\textbf{J}); \textbf{J}) \to H^1(\textrm{Out}_{N_{\mathcal{G}}(E_1)}(\textbf{J}); \textbf{J})$ induced by restriction is surjective. For the latter condition, we calculate in MAGMA (see [ Reference van Beek44 , appendix A]) that $H^1(\textrm{Out}_{N_{\mathcal{G}}(E_1)}(\textbf{J}); \textbf{J})=\{1\}$ and so the homomorphism is surjective.

Let $K\;:\!=\;\textrm{Aut}_{N_{\mathcal{G}}(E_1)}(\textbf{J})$ , $X\;:\!=\;\textrm{Aut}_{\mathcal{G}}(\textbf{J})$ and $Y\;:\!=\;\textrm{Aut}_{\mathcal{F}}(\textbf{J})$ so that $K\le X\cap Y\le \textrm{Aut}(\textbf{J})\cong \textrm{GL}_6(3)$ . We aim to show that $X=Y$ . Since there is only one conjugacy class of groups isomorphic to $2.\textrm{M}_{12}$ in $\textrm{GL}_6(3)$ , we may assume that there is $g\in \textrm{Aut}(\textbf{J})$ with $Y=X^g$ and $K\le X\cap Y$ . Hence, $K, K^g\le Y\cong 2.\textrm{M}_{12}$ . Now, K is the unique overgroup of $T\in\textrm{Syl}_3(X)$ of its isomorphism type whose largest normal 3-subgroup centralises only an element of order 3 in $\textbf{J}$ . Then, $K^g$ is the unique overgroup of $T^g\in\textrm{Syl}_3(X^g)$ with the same properties. Since $K\le X^g=Y$ , K is an overgroup of $P\in\textrm{Syl}_3(X^g)$ with $O_3(K)$ centralizing only an element of order 3 in $\textbf{J}$ . Thus, for $m\in X^g$ with $P^m=T^g$ , $K^m$ and $K^g$ are isomorphic overgroups of $T^g\in\textrm{Syl}_3(X)$ and by uniqueness, we deduce that $K^m=K^g$ . But now, $K=K^{gm^{-1}}$ so that $gm^{-1}\in N_{\textrm{GL}_6(3)}(K)$ and $X^g=X^{gm^{-1}}$ . However, one can calculate that $N_{\textrm{GL}_6(3)}(K)=N_{X}(K)$ so that $Y=X^g=X$ .