1 Introduction
The problem of showing that a given fusion system is saturated arises in different contexts. For example, proving that certain subsystems of fusion systems are saturated is one of the major difficulties in developing a theory of saturated fusion systems in analogy to the theory of finite groups. When studying extensions of fusion systems, it is also crucial to understand under which conditions such extensions are saturated. In this paper, we seek to take up both themes simultaneously in a systematic way. To formulate and study extension problems, it is common to work with linking systems or, more generally, with transporter systems associated to fusion systems (see, e.g., [Reference Broto, Castellana, Grodal, Levi and Oliver6, Reference Oliver and Ventura25, Reference Oliver23]). The equivalent framework of localities (introduced by Chermak [Reference Chermak7]) was recently used by Chermak and the second named author of this paper to construct saturated subsystems of fusion systems, thereby enriching the theory of fusion systems by some new concepts (compare [Reference Chermak and Henke10, Theorem C] and [Reference Henke20]).
In the present paper (apart from the appendix), we work with localities rather than transporter systems. Our first main result is, however, formulated purely in terms of fusion systems. It gives a sufficient condition for a fusion system to be saturated. The proof generalises an argument used by Oliver [Reference Oliver23] to show that the fusion systems associated to certain extensions of groups by linking systems are saturated.
Our saturation criterion serves us as an important tool for studying kernels of localities as introduced in Definition 4 below. A kernel of a locality $\mathcal {L}$ is basically a partial normal subgroup $\mathcal {N}$ such that the factor locality $\mathcal {L}/\mathcal {N}$ is a group and $\mathcal {N}$ itself supports the structure of a locality. We show in an appendix that kernels of localities correspond to ‘normal pairs of transporter systems’ (compare Definition A.3).
In Section 6, we prove some results demonstrating that the theory of kernels can be used to construct saturated subsystems of fusion systems. More precisely, we study certain products in localities that give rise to ‘sublocalities’ whose fusion systems are saturated. As a special case, if $\mathcal {F}$ is a saturated fusion system over S, one can define a notion of a product of a normal subsystem with a subgroup of a model for $N_{\mathcal {F}}(S)$ (or equivalently with a saturated subsystem of $N_{\mathcal {F}}(S)$ ). In particular, our work generalises the notion of a product of a normal subsystem with a subgroup of S, which was introduced by Aschbacher [Reference Aschbacher3]. Our results on products are also used by the second named author of this article [Reference Henke19] to construct normalisers and centralisers of subnormal subsystems of saturated fusion systems.
For the remainder of this introduction, let $\mathcal {F}$ be a fusion system over a finite p group S.
We will adapt the terminology and notation regarding fusion systems from [Reference Aschbacher, Kessar and Oliver2, Chapter 1], except that we will write homomorphisms on the righthand side of the argument (similarly as in [Reference Aschbacher, Kessar and Oliver2, Chapter 2]) and that we will define centric radical subgroups of $\mathcal {F}$ differently, namely as follows.
Definition 1. Define a subgroup $P\leq S$ to be centric radical in $\mathcal {F}$ if

• P is centric: that is, $C_S(Q)\leq Q$ for every $\mathcal {F}$ conjugate Q of P; and

• $O_p(N_{\mathcal {F}}(Q))=Q$ for every fully $\mathcal {F}$ normalised $\mathcal {F}$ conjugate Q of P.
Write $\mathcal {F}^{cr}$ for the set of subgroups of S that are centric radical in $\mathcal {F}$ .
If $\mathcal {F}$ is saturated, then our notion of centric radical subgroups of $\mathcal {F}$ coincides with the usual notion (compare Lemma 2.6). However, defining centric radical subgroups as above is crucial if we want to conclude that the fusion systems of certain localities are saturated.
1.1 A saturation criterion
The following definition will be used to formulate the previously mentioned sufficient condition for a fusion system to be saturated.
Definition 2. Let $\Delta $ be a set of subgroups of S.

• The set $\Delta $ is called $\mathcal {F}$ closed if $\Delta $ is closed under $\mathcal {F}$ conjugacy and overgroupclosed in S.

• $\mathcal {F}$ is called $\Delta $ generated if every morphism in $\mathcal {F}$ can be written as a product of restrictions of $\mathcal {F}$ morphisms between subgroups in $\Delta $ .

• $\mathcal {F}$ is called $\Delta $ saturated if each $\mathcal {F}$ conjugacy class in $\Delta $ contains a subgroup that is fully automised and receptive in $\mathcal {F}$ (as defined in [Reference Aschbacher, Kessar and Oliver2, Definition I.2.2]).
Generalising arguments used by Oliver [Reference Oliver23], we prove the following theorem.
Theorem A. Let $\mathcal {E}$ be an $\mathcal {F}$ invariant saturated subsystem of $\mathcal {F}$ . Suppose $\mathcal {F}$ is $\Delta $ generated and $\Delta $ saturated for some $\mathcal {F}$ closed set $\Delta $ of subgroups of S with $\mathcal {E}^{cr}\subseteq \Delta $ . Then $\mathcal {F}$ is saturated.
1.2 Kernels of localities
The reader is referred to Section 3 for an introduction to partial groups and localities. We say that a locality $(\mathcal {L},\Delta ,S)$ is a locality over $\mathcal {F}$ to indicate that $\mathcal {F}=\mathcal {F}_S(\mathcal {L})$ . The set $\Delta $ is called the object set of $(\mathcal {L},\Delta ,S)$ . It follows from the definition of a locality that the normaliser $N_{\mathcal {L}}(P)$ of any object $P\in \Delta $ is a subgroup of $\mathcal {L}$ and thus a finite group. We will use the following definition.
Definition 3.

• A locality $(\mathcal {L},\Delta ,S)$ over $\mathcal {F}$ is called crcomplete if $\mathcal {F}^{cr}\subseteq \Delta $ .

• A finite group G is said to be of characteristic p if $C_G(O_p(G))\leq O_p(G)$ .

• A locality $(\mathcal {L},\Delta ,S)$ is called a linking locality if it is crcomplete and $N_{\mathcal {L}}(P)$ is of characteristic p for every $P\in \Delta $ .
The slightly nonstandard notion of centric radical subgroups introduced in Definition 1 ensures that the fusion system $\mathcal {F}_S(\mathcal {L})$ of a crcomplete locality $(\mathcal {L},\Delta ,S)$ is saturated (compare Proposition 3.18(c)). If $(\mathcal {L},\Delta ,S)$ is a crcomplete locality over $\mathcal {F}$ , then it gives rise to a transporter system $\mathcal {T}$ associated to $\mathcal {F}$ whose object set is $\Delta $ and thus contains the set $\mathcal {F}^{cr}$ . It follows from [Reference Oliver and Ventura25, Proposition 4.6] that the pcompleted nerve of such a transporter system $\mathcal {T}$ is homotopy equivalent to the pcompleted nerve of a linking system associated to $\mathcal {F}$ . The results we present next are centred around the following concept.
Definition 4. A kernel of a locality $(\mathcal {L},\Delta ,S)$ is a partial normal subgroup $\mathcal {N}$ of $\mathcal {L}$ such that $P\cap \mathcal {N}\in \Delta $ for every $P\in \Delta $ .
We show in Appendix A that kernels of localities correspond to ‘normal pairs of transporter systems’. In particular, the results presented below can be translated to results on transporter systems. The reader is referred to Definition A.3, Proposition A.4, Theorem A.7 and Remark A.8 for details.
If $\mathcal {N}$ is a kernel of a locality $(\mathcal {L},\Delta ,S)$ , then, setting
it is easy to see that $(\mathcal {N},\Gamma ,T)$ is a locality (compare Lemma 5.2). We also say in this situation that $(\mathcal {N},\Gamma ,T)$ is a kernel of $(\mathcal {L},\Delta ,S)$ .
Suppose now that $(\mathcal {N},\Gamma ,T)$ is a kernel of $(\mathcal {L},\Delta ,S)$ . Observe that T is an element of $\Gamma \subseteq \Delta $ , so $N_{\mathcal {L}}(T)$ is a subgroup of $\mathcal {L}$ . It follows therefore from [Reference Chermak8, Theorem 4.3(b), Corollary 4.5] that $\mathcal {L}/\mathcal {N}\cong N_{\mathcal {L}}(T)/N_{\mathcal {N}}(T)$ is a group. Thus, $\mathcal {L}$ can be seen as an extension of the group $\mathcal {L}/\mathcal {N}$ by the locality $(\mathcal {N},\Gamma ,T)$ .
If the kernel $(\mathcal {N},\Gamma ,T)$ is crcomplete, then the following theorem implies that $\mathcal {F}_S(\mathcal {L})$ is saturated. Its proof uses Theorem A.
Theorem B. Let $(\mathcal {N},\Gamma ,T)$ be a kernel of a locality $(\mathcal {L},\Delta ,S)$ . Then $(\mathcal {L},\Delta ,S)$ is crcomplete if and only if $(\mathcal {N},\Gamma ,T)$ is crcomplete. If so, then $\mathcal {F}_T(\mathcal {N})$ is a normal subsystem of $\mathcal {F}_S(\mathcal {L})$ .
Theorem C. Let $(\mathcal {L},\Delta ,S)$ be a locality with a kernel $(\mathcal {N},\Gamma ,T)$ . Then the following conditions are equivalent:

(i) $(\mathcal {L},\Delta ,S)$ is a linking locality;

(ii) $(\mathcal {N},\Gamma ,T)$ is a linking locality and $N_{\mathcal {L}}(T)$ is of characteristic p;

(iii) $(\mathcal {N},\Gamma ,T)$ is a linking locality and $C_{\mathcal {L}}(T)$ is of characteristic p.
We now want to consider special kinds of linking localities. The object set of any linking locality over $\mathcal {F}$ is always contained in the set $\mathcal {F}^s$ of $\mathcal {F}$ subcentric subgroups (defined in Definition 3.19). If $\mathcal {F}$ is saturated, then the existence and uniqueness of centric linking systems imply conversely that for every $\mathcal {F}$ closed set $\Delta $ of subgroups of $\mathcal {F}^s$ , there is an essentially unique linking locality over $\mathcal {F}$ with object set $\Delta $ . Chermak introduced an $\mathcal {F}$ closed set $\delta (\mathcal {F})\subseteq \mathcal {F}^s$ , which by [Reference Chermak and Henke10, Lemma 7.21] can be described as the set of all subgroups of S containing an element of $F^*(\mathcal {F})^s$ (where $F^*(\mathcal {F})$ is the generalised Fitting subsystem of $\mathcal {F}$ introduced by Aschbacher [Reference Aschbacher3]). Notice that there always exists an essentially unique linking locality over $\mathcal {F}$ whose object set is the set $\delta (\mathcal {F})$ . Such a linking locality is called a regular locality.
For an arbitrary locality $(\mathcal {L},\Delta ,S)$ , there is a largest subgroup R of S with $\mathcal {L}=N_{\mathcal {L}}(R)$ . This subgroup is denoted by $O_p(\mathcal {L})$ . Setting $\tilde {\Delta }:=\{P\leq S\colon PO_p(\mathcal {L})\in \Delta \}$ , the triple $(\mathcal {L},\tilde {\Delta },S)$ is also a locality (but with a possibly larger object set). It turns out that $(\mathcal {L},\Delta ,S)$ is a linking locality if and only if $(\mathcal {L},\tilde {\Delta },S)$ is a linking locality (compare [Reference Chermak and Henke10, Lemma 3.28]). This flexibility in the choice of object sets makes it possible to formulate a result similar to Theorem C for regular localities.
Theorem D. Let $(\mathcal {L},\Delta ,S)$ be a locality with a kernel $(\mathcal {N},\Gamma ,T)$ . Set
Then the following conditions are equivalent:

(i) $(\mathcal {L},\tilde {\Delta },S)$ is a regular locality;

(ii) $(\mathcal {N},\tilde {\Gamma },T)$ is a regular locality and $N_{\mathcal {L}}(T)$ is of characteristic p;

(iii) $(\mathcal {N},\tilde {\Gamma },T)$ is a regular locality and $C_{\mathcal {L}}(T)$ is of characteristic p.
Moreover, if these conditions hold, then $E(\mathcal {L})=E(\mathcal {N})$ .
In an unpublished preprint, Chermak defined a locality $(\mathcal {L},\Delta ,S)$ to be semiregular if (in our language) it has a kernel $(\mathcal {N},\Gamma ,T)$ that is a regular locality. He observed furthermore that a locality is semiregular if and only if it is an image of a regular locality under a projection of localities. As a consequence, images of semiregular localities under projections are semiregular. Moreover, since partial normal subgroups of regular localities form regular localities, it follows that partial normal subgroups of semiregular localities form semiregular localities. Thus, the category of semiregular localities and projections might provide a good framework to study extensions. This is one of our motivations to study kernels of localities more generally.
Remark. Extensions of partial groups and localities have already been studied by Gonzalez [Reference Gonzalez12]. He starts by giving important insights into the existence of extensions of partial groups. Basically, Gonzalez considers partial groups as simplicial sets and uses the concept of a simplicial fibre bundle. Gonzalez then states some results about extensions of localities in Section 7 of his paper. He calls a locality ‘saturated’ if it is crcomplete in our sense. Under certain conditions, it is shown that extensions of localities lead to (saturated) localities. To summarise, Gonzalez starts by defining isotypical extensions (compare [Reference Gonzalez12, Definition 7.1]) and shows that an isotypical extension of a locality $(\mathtt {L}",\Delta ",S")$ by a locality $(\mathtt {L}',\Delta ',S')$ leads to a locality $(\mathtt {T},\Delta ,S)$ (compare [Reference Gonzalez12, Proposition 7.6]). Slightly more precisely, we have $\mathtt {T}\subseteq \mathtt {L}$ for an extension $\mathtt {L}$ of the partial group $\mathtt {L}"$ by the partial group $\mathtt {L}'$ .
The situation Gonzalez studies is principally different from ours. However, in [Reference Gonzalez12, Example 7.9, Corollary 7.10], he considers a setup where $\mathtt {L}=\mathtt {T}$ (with $\mathtt {L}$ and $\mathtt {T}$ as above). In this situation, one can observe easily that $(\mathtt {L}',\Delta ',S')$ is a kernel of $(\mathtt {L},\Delta ,S)$ . Indeed, our Theorem B shows that the assumption in [Reference Gonzalez12, Corollary 7.10] that $\Delta '$ contains all $\mathcal {F}'$ centric subgroups is redundant. It would be the subject of further research to see how far our results have other interesting applications in the context of Gonzalez’s work.
1.3 Products in regular localities and fusion systems
We now demonstrate that the theory of kernels can be used to study certain products in regular localities and thereby construct saturated subsystems of saturated fusion systems.
We study regular localities rather than arbitrary (linking) localities, mainly because every partial normal subgroup of a regular locality can be given the structure of a regular locality. To be exact, if $(\mathcal {L},\Delta ,S)$ is a regular locality and $\mathcal {N}\unlhd \mathcal {L}$ , then $\mathcal {E}:=\mathcal {F}_{S\cap \mathcal {N}}(\mathcal {N})$ is saturated and $(\mathcal {N},\delta (\mathcal {E}),S\cap \mathcal {N})$ is a regular locality.
In localities or, more generally, in partial groups, there is a natural notion of products of subsets. More precisely, if $\mathcal {L}$ is a partial group with product $\Pi \colon \mathbf {D}\rightarrow \mathcal {L}$ , then for $\mathcal {X},\mathcal {Y}\subseteq \mathcal {L}$ , we set
The product of a partial normal subgroup with another partial subgroup is only in special cases known to be a partial subgroup. For example, the product of two partial normal subgroups of a locality $(\mathcal {L},\Delta ,S)$ is a partial normal subgroup (and thus forms a regular locality if $(\mathcal {L},\Delta ,S)$ is regular). Our next theorem gives a further example of a product that is a partial subgroup and can be given the structure of a regular locality.
For the theorem below to be comprehensible, a few preliminary remarks may be useful. For any linking locality $(\mathcal {L},\Delta ,S)$ , one can define the generalised Fitting subgroup $F^*(\mathcal {L})$ as a certain partial normal subgroup of $\mathcal {L}$ (see [Reference Henke18, Definition 3]). If $(\mathcal {L},\Delta ,S)$ is a regular locality, then $F^*(\mathcal {L})$ is a kernel of $(\mathcal {L},\Delta ,S)$ , and thus $T^*:=S\cap F^*(\mathcal {L})$ is an element of $\Delta $ . In particular, $N_{\mathcal {L}}(T^*)$ forms a group of characteristic p. If $\mathcal {N}\unlhd \mathcal {L}$ is a partial normal subgroup of $\mathcal {L}$ , then $N_{\mathcal {N}}(T^*)$ is a normal subgroup of $N_{\mathcal {L}}(T^*)$ . Hence, for any subgroup H of $N_{\mathcal {L}}(T^*)$ , the product $N_{\mathcal {N}}(T^*)H$ is a subgroup of $N_{\mathcal {L}}(T^*)$ .
Theorem E. Let $(\mathcal {L},\Delta ,S)$ be a regular locality. Moreover, fix
Then $\mathcal {N} H$ is a partial subgroup of $\mathcal {L}$ . Moreover, for every Sylow psubgroup $S_0$ of $N_{\mathcal {N}}(T^*)H$ with $T\leq S_0$ , the following hold:

(a) There exists a unique set $\Delta _0$ of subgroups of $S_0$ such that $(\mathcal {N} H,\Delta _0,S_0)$ is a crcomplete locality with kernel $(\mathcal {N},\delta (\mathcal {E}),T)$ .

(b) Let $\Delta _0$ be as in (a), and set $\tilde {\Delta }_0:=\{P\leq S\colon PO_p(\mathcal {N} H)\in \Delta _0\}$ . Then $(\mathcal {N} H,\tilde {\Delta }_0,S_0)$ is a regular locality if and only if $N_{\mathcal {N}}(T^*)H$ is a group of characteristic p.
If the hypothesis of Theorem E holds and $S\cap H$ is a Sylow psubgroup of H, then
is a Sylow psubgroup of $N_{\mathcal {N}}(T^*)H$ that is contained in S (compare Lemma 6.2). Thus, the crcomplete locality $(\mathcal {N} H,\Delta _0,S_0)$ from Theorem E(a) gives rise to a saturated subsystem $\mathcal {F}_{S_0}(\mathcal {N} H)=\mathcal {F}_{T(H\cap S)}(\mathcal {N} H)$ of $\mathcal {F}$ . This leads us to a statement that can be formulated purely in terms of fusion systems. We use here that, for every regular locality $(\mathcal {L},\Delta ,S)$ over $\mathcal {F}$ , there is by [Reference Chermak and Henke10, Theorem A] a bijection from the set of partial normal subgroups of $\mathcal {L}$ to the set of normal subsystems of $\mathcal {F}$ given by $\mathcal {N}\mapsto \mathcal {F}_{S\cap \mathcal {N}}(\mathcal {N})$ . By [Reference Chermak and Henke10, Theorem E(d)], this bijection takes $F^*(\mathcal {L})$ to $F^*(\mathcal {F})$ . In particular, $F^*(\mathcal {F})$ is a fusion system over $F^*(\mathcal {L})\cap S$ .
To formulate the result we obtain, we rely on the fact that every constrained fusion system is realised by a model: that is, by a finite group of characteristic p. Furthermore, if $\mathcal {F}$ is constrained and G is a model for $\mathcal {F}$ , then every normal subsystem of $\mathcal {F}$ is realised by a normal subgroup of G. We also use that, for every saturated fusion system $\mathcal {F}$ over S, every normal subsystem $\mathcal {E}$ of $\mathcal {F}$ and every subgroup R of S, there is a product subsystem $\mathcal {E} R$ defined (compare [Reference Aschbacher3, Chapter 8] or [Reference Henke13]).
Corollary F. Let $\mathcal {F}$ be a saturated fusion system over S and $\mathcal {E}\unlhd \mathcal {F}$ over $T\leq S$ . Let $T^*,T_0\leq S$ such that
Then $N_{\mathcal {F}}(T^*)$ is constrained and $N_{\mathcal {E}}(T_0)\unlhd N_{\mathcal {F}}(T^*)$ . Thus we may choose a model G for $N_{\mathcal {F}}(T^*)$ and $N\unlhd G$ with $\mathcal {F}_{S\cap N}(N)=N_{\mathcal {E}}(T_0)$ . Let $H\leq G$ with $S\cap H\in \mathrm {Syl}_p(H)$ . Set
Then the following hold:

(a) $\mathcal {E} H$ is a saturated fusion system over $S_0$ with $\mathcal {E}\unlhd \mathcal {E} H$ .

(b) If $\mathcal {D}$ is a saturated subsystem of $\mathcal {F}$ with $E(\mathcal {E})\unlhd \mathcal {D}$ and $\mathcal {F}_{S_0}(N H)\subseteq \mathcal {D}$ , then $\mathcal {E} H\subseteq \mathcal {D}$ .
For a saturated fusion system $\mathcal {F}$ with $\mathcal {E}\unlhd \mathcal {F}$ , it is shown, for example, in [Reference Chermak and Henke10, Lemma 7.13(a)] that $E(\mathcal {E})\unlhd \mathcal {F}$ . Hence, it makes sense in the situation above to form the product subsystem $E(\mathcal {E})S_0$ . Moreover, part (b) of Corollary F implies the following statement: If $\mathcal {D}$ is a saturated subsystem of $\mathcal {F}$ with $\mathcal {E}\unlhd \mathcal {D}$ and $\mathcal {F}_{S_0}(N H)\subseteq \mathcal {D}$ , then $\mathcal {E} H\subseteq \mathcal {D}$ .
Organisation of the paper
We start by proving our saturation criterion (Theorem A) in Section 2. An introduction to partial groups and localities is given in Section 3. After proving some preliminary results, we study kernels of localities in Section 5. This is used in Section 6 to prove Theorem E and Corollary F as well as some more detailed results on products.
2 Proving saturation
Throughout this section, let $\mathcal {F}$ be a fusion system over S.
In this section, we prove Theorem A. The reader is referred to [Reference Aschbacher, Kessar and Oliver2, Chapter I] for an introduction to fusion systems. We will adopt the notation and terminology from there with the following two caveats: firstly, we write homomorphisms on the righthand side of the arguments similarly as in [Reference Aschbacher, Kessar and Oliver2, Chapter II]. Secondly, we define the set $\mathcal {F}^r$ of $\mathcal {F}$ radical subgroups differently, namely as in Definition 2.5 below.
Definition 2.1. A subgroup $P\leq S$ is said to respect $\mathcal {F}$ saturation if there exists an element of $P^{\mathcal {F}}$ that is fully automised and receptive in $\mathcal {F}$ (as defined in [Reference Aschbacher, Kessar and Oliver2, Definition I.2.2]).
If $\Delta $ is a set of subgroups of S that is closed under $\mathcal {F}$ conjugacy, then notice that $\mathcal {F}$ is $\Delta $ saturated (as defined in the introduction) if and only if every element of $\Delta $ respects $\mathcal {F}$ saturation. On the other hand, P respects $\mathcal {F}$ saturation if $\mathcal {F}$ is $P^{\mathcal {F}}$ saturated. Observe also that a fusion system is saturated if every subgroup of S respects $\mathcal {F}$ saturation, or equivalently if $\mathcal {F}$ is $\Delta $ saturated, where $\Delta $ is the set of all subgroups of S. Roberts and Shpectorov [Reference Roberts and Shpectorov26] proved the following lemma, which we will use from now on, most of the time without reference.
Lemma 2.2. Let $\mathcal {C}$ be an $\mathcal {F}$ conjugacy class of $\mathcal {F}$ . Then $\mathcal {F}$ is $\mathcal {C}$ saturated if and only if the following two conditions hold:

(I) (Sylow axiom) Each subgroup $P\in \mathcal {C}$ that is fully $\mathcal {F}$ normalised is also fully $\mathcal {F}$ centralised and fully automised in $\mathcal {F}$ .

(II) (Extension axiom) Each subgroup $P\in \mathcal {C}$ that is fully $\mathcal {F}$ centralised is also receptive in $\mathcal {F}$ .
Furthermore, if $\mathcal {F}$ is $\mathcal {C}$ saturated, then for every fully $\mathcal {F}$ normalised $P\in \mathcal {C}$ and every $Q\in P^{\mathcal {F}}$ , there exists $\alpha \in \mathrm {Hom}_{\mathcal {F}}(N_S(Q),N_S(P))$ such that $Q\alpha =P$ .
Proof. If (I) and (II) hold, then every fully $\mathcal {F}$ normalised subgroup $P\in \mathcal {C}$ is fully automised and receptive, and thus $\mathcal {F}$ is $\mathcal {C}$ saturated. On the other hand, if $\mathcal {F}$ is $\mathcal {C}$ saturated, it follows from [Reference Aschbacher, Kessar and Oliver2, Lemma I.2.6(c)] that (I) and (II) and the statement of the lemma hold.
Corollary 2.3. If $\mathcal {F}$ is saturated and $P\leq S$ is fully $\mathcal {F}$ normalised, then for every $Q\in P^{\mathcal {F}}$ , there exists $\alpha \in \mathrm {Hom}_{\mathcal {F}}(N_S(Q),N_S(P))$ such that $Q\alpha =P$ .
Recall that a subgroup $P\leq S$ is called $\mathcal {F}$ centric if $C_S(Q)\leq Q$ for every $Q\in P^{\mathcal {F}}$ . Write $\mathcal {F}^c$ for the set of $\mathcal {F}$ centric subgroups of S. We will need the following lemma.
Lemma 2.4. Let $\mathcal {F}$ be a fusion system over S, and let $P\leq S$ be fully $\mathcal {F}$ centralised. Then P is $\mathcal {F}$ centric if and only if $C_S(P)\leq P$ .
Proof. If P is $\mathcal {F}$ centric, then clearly $C_S(P)\leq P$ . Suppose now that $C_S(P)\leq P$ . Then for any $Q\in P^{\mathcal {F}}$ , we have $C_S(Q)\geq Z(Q)\cong Z(P)=C_S(P)$ . Hence, as P is fully $\mathcal {F}$ centralised, we have $C_S(Q)=Z(Q)\leq Q$ for all $Q\in P^{\mathcal {F}}$ : that is, P is $\mathcal {F}$ centric.
We recall that $O_p(\mathcal {F})$ denotes the largest subgroup of S that is normal in $\mathcal {F}$ ([Reference Aschbacher, Kessar and Oliver2, Definition I.4.3]). Moreover, if $\mathcal {F}$ is saturated and $O_p(\mathcal {F}) \in \mathcal {F}^c$ , then $\mathcal {F}$ is called constrained ([Reference Aschbacher, Kessar and Oliver2, Definition I.4.8]). The Model Theorem for constrained fusion systems [Reference Aschbacher, Kessar and Oliver2, Theorem III.5.10] guarantees that every constrained fusion system $\mathcal {F}$ over S has a model: that is, there is a finite group G such that $S \in \mathrm {Syl}_p(G)$ , $\mathcal {F}_S(G) = \mathcal {F}$ and $C_G(O_p(G)) \leq O_p(G)$ .
Definition 2.5.

• A subgroup $P\leq S$ is called $\mathcal {F}$ radical if there exists a fully $\mathcal {F}$ normalised $\mathcal {F}$ conjugate Q of P such that $O_p(N_{\mathcal {F}}(Q)) = Q$ . We denote by $\mathcal {F}^r$ the set of $\mathcal {F}$ radical subgroups of S.

• Set $\mathcal {F}^{cr} = \mathcal {F}^c \cap \mathcal {F}^r$ , and call the elements of $\mathcal {F}^{cr}$ the $\mathcal {F}$ centric radical subgroups of S.

• A subgroup $P\leq S$ is called $\mathcal {F}$ critical if P is $\mathcal {F}$ centric and, for every $\mathcal {F}$ conjugate Q of P, we have
$$\begin{align*}\mathrm{Out}_S(Q)\cap O_p(\mathrm{Out}_{\mathcal{F}}(Q))=1.\end{align*}$$
As remarked before, our definition of radical subgroups differs from the usual one given, for example, in [Reference Aschbacher, Kessar and Oliver2, Definition I.3.1]. We show in part (b) of our next lemma that, for a saturated fusion system $\mathcal {F}$ , the set $\mathcal {F}^{cr}$ equals the set of $\mathcal {F}$ centric radical subgroups in the usual definition.
Lemma 2.6.

(a) For every $R\leq S$ , the following implications hold:
$$\begin{align*}R\in\mathcal{F}^c\mbox{ and }O_p(\mathrm{Aut}_{\mathcal{F}}(R))=\mathrm{Inn}(R)\Longrightarrow R\mbox{ is} \ \mathcal{F}\mbox{critical} \Longrightarrow R\in\mathcal{F}^{cr}. \end{align*}$$ 
(b) If $\mathcal {F}$ is saturated, then we have
$$\begin{align*}\mathcal{F}^{cr}=\{R\in\mathcal{F}^c\colon O_p(\mathrm{Aut}_{\mathcal{F}}(R))=\mathrm{Inn}(R)\}=\{R\leq S\colon R\mbox{ is} \ \mathcal{F}\mbox{critical}\}. \end{align*}$$
Proof. If $\mathrm {Inn}(R)=O_p(\mathrm {Aut}_{\mathcal {F}}(R))$ , then $\mathrm {Inn}(Q)=O_p(\mathrm {Aut}_{\mathcal {F}}(Q))=\mathrm {Aut}_S(Q)\cap O_p(\mathrm {Aut}_{\mathcal {F}}(Q))$ for every $\mathcal {F}$ conjugate Q of R, so R is $\mathcal {F}$ critical if in addition $R\in \mathcal {F}^c$ . This shows the first implication in (a).
Now let $R\in \mathcal {F}^c$ such that $R\not \in \mathcal {F}^r$ . If we pick a fully $\mathcal {F}$ normalised $\mathcal {F}$ conjugate Q of R, we have $Q<Q^*:=O_p(N_{\mathcal {F}}(Q))$ . So $\mathrm {Inn}(Q)<\mathrm {Aut}_{Q^*}(Q)$ as $Q\in \mathcal {F}^c$ . Moreover, $\mathrm {Aut}_{Q^*}(Q)$ is normal in $\mathrm {Aut}_{\mathcal {F}}(Q)$ , as by definition of $Q^*$ every element of $\mathrm {Aut}_{\mathcal {F}}(Q)$ extends to an element of $\mathrm {Aut}_{\mathcal {F}}(Q^*)$ . Hence, $\mathrm {Inn}(Q)<\mathrm {Aut}_{Q^*}(Q)\leq \mathrm {Aut}_S(Q)\cap O_p(\mathrm {Aut}_{\mathcal {F}}(Q))$ . This shows that R is not $\mathcal {F}$ critical, so (a) holds. In particular,
For the proof of (b), it is thus sufficient to show that $O_p(\mathrm {Aut}_{\mathcal {F}}(R))=\mathrm {Inn}(R)$ for every $R\in \mathcal {F}^{cr}$ . Now fix $R\in \mathcal {F}^{cr}$ . Since the property $O_p(\mathrm {Aut}_{\mathcal {F}}(R))=\mathrm {Inn}(R)$ is preserved if R is replaced by an $\mathcal {F}$ conjugate, we may assume without loss of generality that R is fully $\mathcal {F}$ normalised and $R=O_p(N_{\mathcal {F}}(R))$ . Note that $N_{\mathcal {F}}(R)$ is saturated. So as $R\in \mathcal {F}^c$ , the subsystem $N_{\mathcal {F}}(R)$ is constrained. Thus, we may choose a model G for $N_{\mathcal {F}}(R)$ . Then $O_p(G)=R=O_p(N_{\mathcal {F}}(R))$ and
Hence, $O_p(\mathrm {Aut}_{\mathcal {F}}(R))\cong O_p(G/Z(R))=R/Z(R)\cong \mathrm {Inn}(R)$ . Thus $O_p(\mathrm {Aut}_{\mathcal {F}}(R))= \mathrm {Inn}(R)$ . This proves that $\mathcal {F}^{cr} \subseteq \{R\in \mathcal {F}^c\colon O_p(\mathrm {Aut}_{\mathcal {F}}(R))=\mathrm {Inn}(R)\}\}$ , so (b) holds.
The $\mathcal {F}$ critical subgroups play a crucial role in showing that a fusion system is saturated. This is made precise in the following theorem, which we restate for the reader’s convenience.
Theorem 2.7. Suppose $\Delta $ is a set of subgroups of S that is closed under $\mathcal {F}$ conjugacy and contains every $\mathcal {F}$ critical subgroup. If $\mathcal {F}$ is $\Delta $ generated and $\Delta $ saturated, then $\mathcal {F}$ is saturated.
Proof. This is a reformulation of [Reference Broto, Castellana, Grodal, Levi and Oliver5, Theorem 2.2]. The reader might also want to note that the theorem follows from Lemma 2.9 below.
Later, we will need to prove saturation in a situation where it appears impossible to apply Theorem 2.7 directly. We will therefore have a closer look at the arguments used in the proof of that theorem.
Define a partial order $\preceq $ on the set of $\mathcal {F}$ conjugacy classes by writing $\mathcal {P}\preceq \mathcal {Q}$ if some (and thus every) element of $\mathcal {Q}$ contains an element of $\mathcal {P}$ .
Lemma 2.8. Let $\mathcal {H}$ be a set of subgroups of S closed under $\mathcal {F}$ conjugacy such that $\mathcal {F}$ is $\mathcal {H}$ generated and $\mathcal {H}$ saturated. Let $\mathcal {P}$ be an $\mathcal {F}$ conjugacy class that is maximal with respect to $\preceq $ among those $\mathcal {F}$ conjugacy classes that are not contained in $\mathcal {H}$ .
Write $\mathcal {S}_{\geq P}\supseteq \mathcal {S}_{>P}$ for the sets of subgroups of $N_S(P)$ that contain, or properly contain, an element $P\in \mathcal {P}$ . Then the following hold for every $P\in \mathcal {P}$ that is fully $\mathcal {F}$ normalised:

(a) The subsystem $N_{\mathcal {F}}(P)$ is $\mathcal {S}_{>P}$ generated and $\mathcal {S}_{>P}$ saturated.

(b) If $N_{\mathcal {F}}(P)$ is $\mathcal {S}_{\geq P}$ saturated, then $\mathcal {F}$ is $\mathcal {H}\cup \mathcal {P}$ saturated.

(c) P is fully $\mathcal {F}$ centralised. Moreover, for every $Q\in \mathcal {P}$ , there exists $\alpha \in \mathrm {Hom}_{\mathcal {F}}(N_S(Q),N_S(P))$ such that $Q\alpha =P$ .

(d) If $C_S(P)\leq P$ , then P is $\mathcal {F}$ centric, and if $\mathrm {Out}_S(P)\cap O_p(\mathrm {Out}_{\mathcal {F}}(P))=1$ , then $\mathrm {Out}_S(Q)\cap O_p(\mathrm {Out}_{\mathcal {F}}(Q))=1$ for all $Q\in \mathcal {P}$ . In particular, if $C_S(P)\leq P$ and $\mathrm {Out}_S(P)\cap O_p(\mathrm {Out}_{\mathcal {F}}(P))=1$ , then P is $\mathcal {F}$ critical.
Proof. Basically, this follows from [Reference Broto, Castellana, Grodal, Levi and Oliver5, Lemma 2.4] and its proof. We take the opportunity to point out the following small error in the proof of part (b) of that lemma: in l.9 on p.334 of [Reference Broto, Castellana, Grodal, Levi and Oliver5] it says ‘replacing each $\varphi _i$ by $\chi _i\circ \varphi _i\circ \chi _i^{1}\in \mathrm {Hom}_{\mathcal {F}}(\chi _i(Q_i),S)$ ’. However, it should be ‘replacing each $\varphi _i$ by $\chi _{i+1}\circ \varphi _i\circ \chi _i^{1}\in \mathrm {Hom}_{\mathcal {F}}(\chi _i(Q_i),S)$ ’.
Let us now explain in detail how the assertion follows. By [Reference Broto, Castellana, Grodal, Levi and Oliver5, Lemma 2.4(a), (c)], $N_{\mathcal {F}}(P)$ is $\mathcal {S}_{>P}$ saturated and (b) holds. To prove (a), one needs to argue that $N_{\mathcal {F}}(P)$ is also $\mathcal {S}_{>P}$ generated. If $\varphi \in \mathrm {Hom}_{N_{\mathcal {F}}(P)}(A,B)$ , then $\varphi $ extends to a morphism $\hat {\varphi }\in \mathrm {Hom}_{\mathcal {F}}(AP,BP)$ that normalises P. If $A\not \leq P$ , then $AP\in \mathcal {S}_{>P}$ , and thus $\varphi $ is the restriction of a morphism between subgroups in $\mathcal {S}_{>P}$ . If $A\leq P$ , then $\hat {\varphi }\in \mathrm {Aut}_{\mathcal {F}}(P)$ , and by [Reference Broto, Castellana, Grodal, Levi and Oliver5, Lemma 2.4(b)], $\hat {\varphi }$ (and thus $\varphi $ ) is a composite of restrictions of morphisms in $N_{\mathcal {F}}(P)$ between subgroups in $\mathcal {S}_{>P}$ . Hence, $N_{\mathcal {F}}(P)$ is $\mathcal {S}_{>P}$ generated, and (a) holds.
The following property is shown in the proof of Lemma 2.4 in [Reference Broto, Castellana, Grodal, Levi and Oliver5] (see p.333, property (3)).

(*) There is a subgroup $\hat {P}\in \mathcal {P}$ fully $\mathcal {F}$ centralised such that, for all $Q\in \mathcal {P}$ , there exists a morphism $\varphi \in \mathrm {Hom}_{\mathcal {F}}(N_S(Q),N_S(\hat {P}))$ with $Q\varphi =\hat {P}$ .
In particular, there exists $\psi \in \mathrm {Hom}_{\mathcal {F}}(N_S(P),N_S(\hat {P}))$ such that $P\psi =\hat {P}$ . Since P is fully $\mathcal {F}$ normalised, the map $\psi $ is an isomorphism. In particular, P is fully $\mathcal {F}$ centralised as $\hat {P}$ is fully $\mathcal {F}$ centralised. Moreover, if $Q\in \mathcal {P}$ , then by (*), there exists $\varphi \in \mathrm {Hom}_{\mathcal {F}}(N_S(Q),N_S(\hat {P}))$ with $Q\varphi =\hat {P}$ , and we have $\alpha :=\varphi \psi ^{1}\in \mathrm {Hom}_{\mathcal {F}}(N_S(Q),N_S(P))$ , with $Q\alpha =\hat {P}\psi ^{1}=P$ . Hence, (c) holds.
It follows from Lemma 2.4 and the first part of (c) that P is $\mathcal {F}$ centric if $C_S(P)\leq P$ . If $Q\in \mathcal {P}$ , then the second part of (c) allows us to choose $\alpha \in \mathrm {Hom}_{\mathcal {F}}(N_S(Q),N_S(P))$ with $Q\alpha =P$ . Notice that the map $\alpha ^*\colon \mathrm {Aut}_{\mathcal {F}}(Q)\rightarrow \mathrm {Aut}_{\mathcal {F}}(P),\gamma \mapsto \alpha ^{1}\gamma \alpha $ is a group isomorphism with $\mathrm {Inn}(Q)\alpha ^*=\mathrm {Inn}(P)$ and $\mathrm {Aut}_S(Q)\alpha ^*\leq \mathrm {Aut}_S(P)$ . So $\alpha ^*$ induces an isomorphism from $\mathrm {Out}_{\mathcal {F}}(Q)$ to $\mathrm {Out}_{\mathcal {F}}(P)$ that maps $\mathrm {Out}_S(Q)\cap O_p(\mathrm {Out}_{\mathcal {F}}(Q))$ into $\mathrm {Out}_S(P)\cap O_p(\mathrm {Out}_{\mathcal {F}}(P))$ . This implies (d).
Lemma 2.9. Suppose that $\mathcal {H}$ is one of the following two sets:

(i) the set of all subgroups of S respecting $\mathcal {F}$ saturation; or

(ii) the set of all $P\leq S$ such that every subgroup of S containing an $\mathcal {F}$ conjugate of P respects $\mathcal {F}$ saturation.
Assume that $\mathcal {F}$ is $\mathcal {H}$ generated. Let $\mathcal {P}$ be an $\mathcal {F}$ conjugacy class, which is maximal with respect to $\preceq $ among the $\mathcal {F}$ conjugacy classes not contained in $\mathcal {H}$ . Then the elements of $\mathcal {P}$ are $\mathcal {F}$ critical.
Proof. Observe first that $\mathcal {H}$ is in either case closed under $\mathcal {F}$ conjugacy and $\mathcal {F}$ is $\mathcal {H}$ saturated. Moreover, note that $\mathcal {F}$ is not $\mathcal {H}\cup \mathcal {P}$ saturated; this is clear if $\mathcal {H}$ is as in (i); if $\mathcal {H}$ is as in (ii), then note that because of the maximal choice of $\mathcal {P}$ , $\mathcal {F}$ being $\mathcal {H}\cup \mathcal {P}$ saturated would imply that every subgroup containing an element of $\mathcal {P}$ would be either in $\mathcal {P}$ or in $\mathcal {H}$ and thus respect $\mathcal {F}$ saturation.
Now fix $P\in \mathcal {P}$ fully $\mathcal {F}$ normalised. By Lemma 2.8(a),(b) (using the notation introduced in that lemma), $N_{\mathcal {F}}(P)$ is $\mathcal {S}_{>P}$ generated and $\mathcal {S}_{>P}$ saturated, but not $\mathcal {S}_{\geq P}$ saturated as $\mathcal {F}$ is not $\mathcal {H}\cup \mathcal {P}$ saturated. Thus, by [Reference Broto, Castellana, Grodal, Levi and Oliver5, Lemma 2.5] applied with $N_{\mathcal {F}}(P)$ in place of $\mathcal {F}$ , we have $\mathrm {Out}_S(P)\cap O_p(\mathrm {Out}_{\mathcal {F}}(P))=1$ and $P\in N_{\mathcal {F}}(P)^c$ . It now follows from Lemma 2.8(d) that P is $\mathcal {F}$ critical.
Lemma 2.10. Let $\mathcal {E}$ be an $\mathcal {F}$ invariant subsystem of $\mathcal {F}$ over $T\leq S$ . Let $P\leq S$ , and set $P_0:=P\cap T$ . Then the following hold:

(a) If $C_S(P)\leq P$ and $\mathrm {Out}_S(P)\cap O_p(\mathrm {Out}_{\mathcal {F}}(P))=1$ , then $C_T(P_0)\leq P_0$ and $\mathrm {Out}_T(P_0)\cap O_p(\mathrm {Out}_{\mathcal {E}}(P_0))=1$ .

(b) If $\mathcal {E}$ is saturated, $P_0$ is fully $\mathcal {E}$ normalised and P is $\mathcal {F}$ critical, then $P_0\in \mathcal {E}^{cr}$ .
Proof. Assume that $C_T(P_0)\not \leq P_0$ or $\mathrm {Out}_T(P_0)\cap O_p(\mathrm {Out}_{\mathcal {E}}(P_0))\neq 1$ . Let Q be the preimage of $\mathrm {Out}_T(P_0)\cap O_p(\mathrm {Out}_{\mathcal {E}}(P_0))$ in $N_T(P_0)$ . Note that $C_T(P_0)\leq Q$ , so our assumption implies in any case that $P_0<Q$ and thus $Q\not \leq P$ . As P normalises $P_0$ , P also normalises Q. Hence, $PQ$ is a pgroup with $P<PQ$ . This yields $P<N_{PQ}(P)=PN_Q(P)$ , and thus $N_Q(P)\not \leq P$ . Note that $[P,N_Q(P)]\leq P\cap T=P_0$ . So $X:=\langle \mathrm {Aut}_Q(P)^{\mathrm {Aut}_{\mathcal {F}}(P)}\rangle $ acts trivially on $P/P_0$ . At the same time, by definition of Q, the elements of $\mathrm {Aut}_Q(P)$ , and thus the elements of X, restrict to automorphisms of $P_0$ that lie in $O_p(\mathrm {Out}_{\mathcal {E}}(P_0))\unlhd \mathrm {Aut}_{\mathcal {F}}(P_0)$ . Hence, a $p^{\prime }$ element of X acts trivially on $P/P_0$ and on $P_0$ , and therefore it is trivial by properties of coprime action (compare [Reference Kurzweil and Stellmacher21, 8.2.2(b)] or [Reference Aschbacher, Kessar and Oliver2, Lemma A.2]). This shows that X is a pgroup, so $\mathrm {Aut}_Q(P)\leq \mathrm {Aut}_S(P)\cap O_p(\mathrm {Aut}_{\mathcal {F}}(P))$ . As $N_Q(P)\not \leq P$ , this implies either $C_S(P)\not \leq P$ or $1\neq \mathrm {Out}_Q(P)\leq \mathrm {Out}_S(P)\cap O_p(\mathrm {Out}_{\mathcal {F}}(P))$ . This proves (a).
Assume now that $\mathcal {E}$ is saturated, $P_0$ is fully $\mathcal {E}$ normalised and P is $\mathcal {F}$ critical. The latter condition implies $C_S(P)\leq P$ and $\mathrm {Out}_S(P)\cap O_p(\mathrm {Out}_{\mathcal {F}}(P))=1$ . So by (a), $C_T(P_0)\leq P_0$ and $\mathrm {Out}_T(P_0)\cap O_p(\mathrm {Out}_{\mathcal {E}}(P_0))=1$ . As $\mathcal {E}$ is saturated and $P_0$ is fully $\mathcal {E}$ normalised, $P_0$ is fully $\mathcal {E}$ centralised and fully $\mathcal {E}$ automised. Hence, $P_0$ is $\mathcal {E}$ centric by Lemma 2.4 and $O_p(\mathrm {Out}_{\mathcal {E}}(P_0))\leq \mathrm {Out}_T(P_0)$ . The latter condition implies $O_p(\mathrm {Out}_{\mathcal {E}}(P_0))=1$ . Using Lemma 2.6(a), we conclude that $P_0\in \mathcal {E}^{cr}$ .
The basic idea in the proof of the following theorem is taken from Step 5 of the proof of [Reference Oliver23, Theorem 9]. As we argue afterwards, it easily implies Theorem A. If $P,Q\leq S$ , $\varphi \in \mathrm {Hom}_{\mathcal {F}}(P,Q)$ and $A\leq \mathrm {Aut}_{\mathcal {F}}(P)$ , note that $A^{\varphi }:=\varphi ^{1}A\varphi \leq \mathrm {Aut}_{\mathcal {F}}(P\varphi )$ .
Theorem 2.11. Let $\mathcal {H}$ be the set of all subgroups $P\leq S$ such that every subgroup of S containing an $\mathcal {F}$ conjugate of P respects $\mathcal {F}$ saturation. Assume that $\mathcal {F}$ is $\mathcal {H}$ generated. Suppose furthermore that there exists an $\mathcal {F}$ invariant saturated subsystem $\mathcal {E}$ of $\mathcal {F}$ with $\mathcal {E}^{cr}\subseteq \mathcal {H}$ . Then $\mathcal {F}$ is saturated.
Proof. Let $T\leq S$ be such that $\mathcal {E}$ is a subsystem of $\mathcal {F}$ over T. By $\mathcal {H}_0$ , denote the set of subgroups of T that are elements of $\mathcal {H}$ . Write $\mathcal {H}^{\perp }$ for the set of subgroups of S not in $\mathcal {H}$ and $\mathcal {H}_0^{\perp }$ for the set of elements of $\mathcal {H}^{\perp }$ that are subgroups of T. If $P\leq S$ , write $P_0$ for $P\cap T$ , and similarly for subgroups of S with different names.
For the proof of the assertion, we will proceed in three steps. In the first two steps, we will argue that property (2.1) below implies the assertion, and in the third step, we will prove that (2.1) holds.
Step 1: We argue that (2.1) implies the following property:
This can be seen as follows. If P is as in (2.2) and $P_0$ is fully $\mathcal {E}$ normalised, then we are done. If not, then by (2.1), we can pass on to an $\mathcal {F}$ conjugate $P"$ of P such that $N_T(P_0)<N_T(P_0")$ . Again, if $P_0"$ is fully $\mathcal {E}$ normalised, then we are done; otherwise, we can repeat the process. Since T is finite, we will eventually end up with an $\mathcal {F}$ conjugate Q of P such that $Q_0$ is fully $\mathcal {E}$ normalised. So (2.1) implies (2.2).
Step 2: We show that property (2.2) implies the assertion. Assume that (2.2) holds and $\mathcal {F}$ is not saturated. Then $\mathcal {H}^{\perp }\neq \emptyset $ . By construction, $\mathcal {H}\supseteq \mathcal {H}_0$ is $\mathcal {F}$ closed. So, for every subgroup $P\leq S$ , $P_0\in \mathcal {H}_0$ implies $P\in \mathcal {H}$ . In particular, as $\mathcal {H}^{\perp }\neq \emptyset $ , we can conclude $\mathcal {H}_0^{\perp }\neq \emptyset $ . Set
For any $Q\in \mathcal {H}_0^{\perp }$ with $Q=m$ , we have $Q=Q_0$ , and thus $Q\in \mathcal {M}$ . Hence, $\mathcal {M}\neq \emptyset $ . As T is strongly closed, $\mathcal {M}$ is closed under $\mathcal {F}$ conjugacy. Thus we can choose an $\mathcal {F}$ conjugacy class $\mathcal {P}\subseteq \mathcal {M}$ such that the elements of $\mathcal {P}$ are of maximal order among the elements of $\mathcal {M}$ .
We argue first that $\mathcal {P}$ is maximal with respect to $\preceq $ among the $\mathcal {F}$ conjugacy classes contained in $\mathcal {H}^{\perp }$ . For that, let $P\in \mathcal {P}$ and $P\leq R\leq S$ . We need to show that $R=P$ or $R\in \mathcal {H}$ . Notice that $m=P_0\leq R_0$ . If $m<R_0$ , then by definition of m, we have $R_0\in \mathcal {H}_0\subseteq \mathcal {H}$ , and thus $R\in \mathcal {H}$ . If $m=R_0$ , then $R\in \mathcal {M}$ or $R\in \mathcal {H}$ . So the maximality of $P$ yields $R=P$ or $R\in \mathcal {H}$ . Hence, $\mathcal {P}$ is maximal with respect to $\preceq $ among the $\mathcal {F}$ conjugacy classes contained in $\mathcal {H}^{\perp }$ . Thus Lemma 2.9 implies that the elements of $\mathcal {P}$ are $\mathcal {F}$ critical. By (2.2), there exists $P\in \mathcal {P}$ such that $P_0$ is fully $\mathcal {E}$ normalised. Then by Lemma 2.10(b), we have $P_0\in \mathcal {E}^{cr}$ . Since by assumption $\mathcal {E}^{cr}\subseteq \mathcal {H}$ , it follows that $P_0\in \mathcal {H}$ , so $P\in \mathcal {H}$ . This contradicts the choice of $\mathcal {P}\subseteq \mathcal {M}\subseteq \mathcal {H}^{\perp }$ . Hence, (2.2) implies that $\mathcal {F}$ is saturated.
Step 3: We complete the proof by proving (2.1). Whenever we have subgroups $Q_1\leq Q\leq S$ , we set $\mathrm {Aut}_{\mathcal {F}}(Q:Q_1):=N_{\mathrm {Aut}_{\mathcal {F}}(Q)}(Q_1)$ , $\mathrm {Aut}_S(Q:Q_1):=N_{\mathrm {Aut}_S(Q)}(Q_1)$ , $N_S(Q:Q_1):=N_S(Q)\cap N_S(Q_1)$ , and so on. Now let $P\leq S$ be such that $P_0$ is an element of $\mathcal {H}_0^{\perp }$ of maximal order that is not fully $\mathcal {E}$ normalised. Note that $T\in \mathcal {E}^{cr}\subseteq \mathcal {H}$ and thus $P_0\neq T$ . Hence, $P_0<N_T(P_0)$ . So the maximality of $P_0$ yields $R:=N_T(P_0)\in \mathcal {H}_0$ . In particular, every $\mathcal {F}$ conjugate of R respects $\mathcal {F}$ saturation. Since $\mathcal {E}$ is saturated, there exists $\rho \in \mathrm {Hom}_{\mathcal {E}}(R,T)$ such that $P_0':=P_0\rho $ is fully $\mathcal {E}$ normalised. As $R':=R\rho $ respects $\mathcal {F}$ saturation, there exists $\sigma \in \mathrm {Hom}_{\mathcal {F}}(R',S)$ such that $R":=R'\sigma $ is fully automised and receptive. Set $P_0":=P_0' \sigma =P_0 \rho \sigma $ . As $R"$ is fully automised, by Sylow’s Theorem, there exists $\xi \in \mathrm {Aut}_{\mathcal {F}}(R")$ such that
is a Sylow psubgroup of $\mathrm {Aut}_{\mathcal {F}}(R":P_0")^{\xi }=\mathrm {Aut}_{\mathcal {F}}(R": P_0" \xi )$ . So replacing $\sigma $ by $\sigma \xi $ , we may assume
Notice that $\mathrm {Aut}_S(R:P_0)^{\rho \sigma }$ and $\mathrm {Aut}_S(R':P_0')^{\sigma }$ are psubgroups of $\mathrm {Aut}_{\mathcal {F}}(R":P_0")$ . Hence, again by Sylow’s Theorem, there exist $\gamma ,\delta \in \mathrm {Aut}_{\mathcal {F}}(R":P_0")$ such that $\mathrm {Aut}_S(R:P_0)^{\rho \sigma \gamma }$ and $\mathrm {Aut}_S(R':P_0')^{\sigma \delta }$ are contained in $\mathrm {Aut}_S(R":P_0")$ . This implies $N_S(R:P_0)\leq N_{\rho \sigma \gamma }$ and $N_S(R':P_0')\leq N_{\sigma \delta }$ . Hence, as $R"$ is receptive, the map $\rho \sigma \gamma \in \mathrm {Hom}_{\mathcal {F}}(R,R")$ extends to $\varphi \in \mathrm {Hom}_{\mathcal {F}}(N_S(R:P_0),S)$ , and similarly $\sigma \delta \in \mathrm {Hom}_{\mathcal {F}}(R',R")$ extends to $\psi \in \mathrm {Hom}_{\mathcal {F}}(N_S(R':P_0'),S)$ . Notice that
Moreover, since $P_0=P\cap T$ , $R=N_T(P_0)$ and T is strongly closed, we have $P\leq N_S(P_0)=N_S(R:P_0)$ . Hence
is a welldefined $\mathcal {F}$ conjugate of P, for which we have indeed that $P"\cap T=P \varphi \cap T=(P\cap T)\varphi =P_0 \varphi =P_0"$ as our notation suggests. So it only remains to show that $N_T(P_0)<N_T(P_0")$ . We will show this by arguing that
Recall that $P_0$ is by assumption not fully $\mathcal {E}$ normalised, whereas $P_0'=P_0 \rho $ is fully $\mathcal {E}$ normalised. Therefore, we have $R'=R \rho =N_T(P_0) \rho <N_T(P_0')$ . Thus $R'<N_{N_T(P_0')}(R')=N_T(R':P_0')$ . As $R=R'$ , this shows the first inequality in (2.3). As $P_0' \psi =P_0"$ and T is strongly closed, we have $N_T(R':P_0') \psi \leq N_T(P_0")$ . Since $\psi $ is injective, this shows the second inequality in (2.3). So the proof of (2.1) is complete. As argued in Steps 1 and 2, this proves the assertion.
Proof of Theorem A.
Write $\mathcal {H}$ for the set of all subgroups $P\leq S$ such that every subgroup of S containing an $\mathcal {F}$ conjugate of P respects $\mathcal {F}$ saturation. Since $\Delta $ is $\mathcal {F}$ closed and $\mathcal {F}$ is $\Delta $ saturated, it follows that $\Delta \subseteq \mathcal {H}$ . Hence, as $\mathcal {F}$ is $\Delta $ generated, $\mathcal {F}$ is also $\mathcal {H}$ generated. Moreover, $\mathcal {E}^{cr}\subseteq \Delta \subseteq \mathcal {H}$ . Thus the assertion follows from Theorem 2.11 above.
3 Partial groups and localities
In this section, we introduce some basic definitions and notations that will be used in the remainder of the paper. We refer the reader to [Reference Chermak8] for a more comprehensive introduction to partial groups and localities.
3.1 Partial groups
Following the notation introduced in [Reference Chermak8], we will write $\mathbf {W}(\mathcal {L})$ for the set of words in a set $\mathcal {L}$ . The elements of $\mathcal {L}$ will be identified with the words of length one, and $\emptyset $ denotes the empty word. The concatenation of words $u_1,u_2,\dots ,u_k\in \mathbf {W}(\mathcal {L})$ will be denoted $u_1\circ u_2\circ \cdots \circ u_k$ .
Definition 3.1 [Reference Chermak8, Definition 1.1].
Suppose $\mathcal {L}$ is a nonempty set and $\mathbf {D}\subseteq \mathbf {W}(\mathcal {L})$ . Let $\Pi \colon \mathbf {D} \longrightarrow \mathcal {L}$ be a map, and let $()^{1} \colon \mathcal {L} \longrightarrow \mathcal {L}$ be an involutory bijection, which we extend to a map
Then $\mathcal {L}$ is called a partial group with product $\Pi $ and inversion $()^{1}$ if the following hold for all words $u,v,w\in \mathbf {W}(\mathcal {L})$ :

• $\mathcal {L} \subseteq \mathbf {D}$ and
$$\begin{align*}u \circ v \in \mathbf {D} \Longrightarrow u,v \in \mathbf {D}.\end{align*}$$(So in particular, $\emptyset \in \mathbf {D}$ .) 
• $\Pi $ restricts to the identity map on $\mathcal {L}$ .

• $u \circ v \circ w \in \mathbf {D} \Longrightarrow u \circ (\Pi (v)) \circ w \in \mathbf {D}$ , and $\Pi (u \circ v \circ w) = \Pi (u \circ (\Pi (v)) \circ w)$ .

• $w \in \mathbf {D} \Longrightarrow w^{1} \circ w\in \mathbf {D}$ and $\Pi (w^{1} \circ w) = \operatorname {\mathbf {1}}$ , where $\operatorname {\mathbf {1}}:=\Pi (\emptyset )$ .
For the remainder of this section, let ${\mathcal {L}}$ always be a partial group with product ${\Pi \colon \mathbf {D}\rightarrow \mathcal {L}}$ . As above, set ${\operatorname {\mathbf {1}}:=\Pi (\emptyset )}$ .
If $w=(f_1,\dots ,f_n)\in \mathbf {D}$ , then we sometimes write $f_1f_2\cdots f_n$ for $\Pi (f_1,\dots ,f_n)$ . In particular, if $(x,y)\in \mathbf {D}$ , then $xy$ denotes the product $\Pi (x,y)$ .
Notation 3.2.

• For every $f\in \mathcal {L}$ , write $\mathbf {D}(f):=\{x\in \mathcal {L}\colon (f^{1},x,f)\in \mathbf {D}\}$ for the set of all x such that the conjugate $x^f:=\Pi (f^{1},x,f)$ is defined.

• By $c_f$ , denote the conjugation map $c_f\colon \mathbf {D}(f)\rightarrow \mathcal {L},x\mapsto x^f$ .

• For $f\in \mathcal {L}$ and $\mathcal {X}\subseteq \mathbf {D}(f)$ , set $\mathcal {X}^f:=\{x^f\colon x\in \mathcal {X}\}$ .

• Given $\mathcal {X}\subseteq \mathcal {L}$ , set
$$\begin{align*}N_{\mathcal{L}}(\mathcal{X}):=\{f\in\mathcal{L}\colon \mathcal{X}\subseteq\mathbf {D}(f)\mbox{ and }\mathcal{X}^f=\mathcal{X}\}\end{align*}$$and$$\begin{align*}C_{\mathcal{L}}(\mathcal{X}):=\{f\in\mathcal{L}\colon x\in\mathbf {D}(f)\mbox{ and }x^f=x\mbox{ for all }x\in\mathcal{X}\}.\end{align*}$$ 
• For $\mathcal {X},\mathcal {Y}\subseteq \mathcal {L}$ , define $N_{\mathcal {Y}}(\mathcal {X})=\mathcal {Y}\cap N_{\mathcal {L}}(\mathcal {X})$ and $C_{\mathcal {Y}}(\mathcal {X})=\mathcal {Y}\cap C_{\mathcal {L}}(\mathcal {X})$ .
We call $N_{\mathcal {Y}}(\mathcal {X})$ the normaliser of $\mathcal {X}$ in $\mathcal {Y}$ and $C_{\mathcal {Y}}(\mathcal {X})$ the centraliser of $\mathcal {X}$ in $\mathcal {Y}$ .
Definition 3.3.

• A subset $\mathcal {H}\subseteq \mathcal {L}$ is called a partial subgroup of $\mathcal {L}$ if $h^{1}\in \mathcal {H}$ for all $h\in \mathcal {H}$ , and moreover $\Pi (w)\in \mathcal {H}$ for all $w\in \mathbf {D}\cap \mathbf {W}(\mathcal {H})$ .

• A partial subgroup $\mathcal {H}$ of $\mathcal {L}$ is a called a subgroup of $\mathcal {L}$ if $\mathbf {W}(\mathcal {H})\subseteq \mathbf {D}(\mathcal {L})$ .

• By a psubgroup of $\mathcal {L}$ , we mean a subgroup S of $\mathcal {L}$ such that $S$ is a power of p.

• Let $\mathcal {N}$ be a partial subgroup of $\mathcal {L}$ . Then we call $\mathcal {N}$ a partial normal subgroup of $\mathcal {L}$ (and write $\mathcal {N}\unlhd \mathcal {L}$ ) if $n^f\in \mathcal {N}$ for all $f\in \mathcal {L}$ and all $n\in \mathcal {N}\cap \mathbf {D}(f)$ .

• A partial subgroup $\mathcal {H}$ of $\mathcal {L}$ is called a partial subnormal subgroup of $\mathcal {L}$ if there exists a series $\mathcal {H}=\mathcal {H}_0\unlhd \mathcal {H}_1\unlhd \cdots \unlhd \mathcal {H}_k=\mathcal {L}$ of partial subgroups of $\mathcal {L}$ .
If $\mathcal {H}$ is a partial subgroup of $\mathcal {L}$ , notice that $\mathcal {H}$ is itself a partial group with product $\Pi _{\mathbf {W}(\mathcal {H})\cap \mathbf {D}}$ . If $\mathcal {H}$ is a subgroup of $\mathcal {L}$ , then $\mathcal {H}$ forms a group with binary product defined by $x\cdot y:=\Pi (x,y)$ for all $x,y\in \mathcal {H}$ . In particular, every psubgroup of $\mathcal {L}$ forms a pgroup. The following definition will be crucial in the definition of a locality.
Definition 3.4. Let $\mathcal {L}$ be a partial group, and let $\Delta $ be a collection of subgroups of $\mathcal {L}$ . Define $\mathbf {D}_{\Delta }$ to be the set of words $w=(g_1, \dots , g_k) \in \mathbf {W}(\mathcal {L})$ such that there exist $P_0, \dots ,P_k \in \Delta $ with $P_{i1} \subseteq \mathbf {D}(g_i)$ and $P_{i1}^{g_i} = P_i$ for all $1 \leq i \leq k$ . If such w and $P_0,\dots ,P_k$ are given, then we also say that $w\in \mathbf {D}_{\Delta }$ via $P_0,P_1,\dots ,P_k$ , or just that $w\in \mathbf {D}_{\Delta }$ via $P_0$ .
Lemma 3.5. Let $\mathcal {L}$ be a partial group and $\mathcal {N}\unlhd \mathcal {L}$ .

(a) If $f\in \mathcal {L}$ and $P\subseteq \mathbf {D}(f)$ , then $(P\cap \mathcal {N})^f=P^f\cap \mathcal {N}$ .

(b) Let S be a subgroup of $\mathcal {L}$ and $\Gamma $ a set of subgroups of $\mathcal {N}\cap S$ . Set
$$\begin{align*}\Delta:=\{P\leq S\colon P\cap \mathcal{N}\in\Gamma\}.\end{align*}$$Then $\mathbf {D}_{\Gamma }=\mathbf {D}_{\Delta }$ .
Proof. (a) Notice that $(P\cap \mathcal {N})^f\subseteq P^f\cap \mathcal {N}$ as $\mathcal {N}\unlhd \mathcal {L}$ . Since $(c_f)^{1}=c_{f^{1}}$ by [Reference Chermak8, Lemma 1.6(c)], we have $(P^f)^{f^{1}}=P$ . So, similarly, $(P^f\cap \mathcal {N})^{f^{1}}\subseteq P\cap \mathcal {N}$ , and thus $P^f\cap \mathcal {N}\subseteq (P\cap \mathcal {N})^f$ . This shows $(P\cap \mathcal {N})^f=P^f\cap \mathcal {N}$ as required.
(b) As $\Gamma \subseteq \Delta $ , we have $\mathbf {D}_{\Gamma }\subseteq \mathbf {D}_{\Delta }$ . Conversely, if $w=(f_1,\dots ,f_n)\in \mathbf {D}_{\Delta }$ via $P_0,P_1,\dots ,P_n\in \Delta $ , then (a) yields that $w\in \mathbf {D}_{\Gamma }$ via $P_0\cap \mathcal {N},P_1\cap \mathcal {N},\dots ,P_n\cap \mathcal {N}$ . Thus $\mathbf {D}_{\Delta } \subseteq \mathbf {D}_{\Gamma }$ , implying $\mathbf {D}_{\Gamma }=\mathbf {D}_{\Delta }$ .
3.2 Localities
Definition 3.6. Let $\mathcal {L}$ be a partial group, and let S be a psubgroup of $\mathcal {L}$ . For $f\in \mathcal {L}$ , set
More generally, if $w=(f_1,\dots ,f_k)\in \mathbf {W}(\mathcal {L})$ , then write $S_w$ for the set of $s\in S$ such that there exists a sequence $s=s_0,\dots ,s_k$ of elements of S with $s_{i1}\in \mathbf {D}(f_i)$ and $s_{i1}^{f_i}=s_i$ for $i=1,\dots ,k$ .
Definition 3.7. Let $\mathcal {L}$ be a finite partial group with product $\Pi \colon \mathbf {D}\rightarrow \mathcal {L}$ , let S be a psubgroup of $\mathcal {L}$ , and let $\Delta $ be a nonempty set of subgroups of S. We say that $(\mathcal {L}, \Delta , S)$ is a locality if the following hold:

1. S is maximal with respect to inclusion among the psubgroups of $\mathcal {L}$ ;

2. $\mathbf {D} = \mathbf {D}_{\Delta }$ ;

3. $\Delta $ is closed under passing to $\mathcal {L}$ conjugates and overgroups in S; that is, $\Delta $ is overgroupclosed in S and $P^f\in \Delta $ for all $P\in \Delta $ and $f\in \mathcal {L}$ with $P \subseteq S_f$ .
If