FINITE LOCALITIES I

Andrew Chermak

Kansas State University

June 2020

Introduction

This is the first of (at present) two papers concerning what might be thought of as “locally grouped spaces”, in loose analogy with the locally ringed spaces of algebraic geometry. The spaces that we have in mind are simplicial sets that generalize the simplicial sets that underly and determine the classifying spaces of finite (or compact) groups. If the analogy is pursued, then the role of “structure sheaf” is provided by the “fusion systems” associated with these spaces. Our approach here will be purely algebraic and combinatorial, so we will not be concerned with topological realizations. All of the groups to be considered will be finite; but a parallel series of papers, co-authored with Alex Gonzalez, will considerably broaden the scope.

Finite localities were introduced by the author in [Ch1], in order to give a positive solution to the question: Given a saturated fusion system F on a finite p-group, does there exist a “classifying space” for F, and if so, is such a space unique up to isomorphism ? The solution that was given in [Ch1] was closely tied to the specific goal, and did not allow for a complete development of ideas. The aim here is to provide such a development, and to thereby obtain the prerequisites for a version of finite group theory itself from a strictly “p-local” point of view. This entails the rigorous exclusion of everything that is extraneous to that point of view, including the notion that there be any ambient group whatsoever. Concommitantly, our aim is to supplement the theory saturated fusion systems over a finite p-group. As part of that program we shall need to establish a sort of dictionary that will establish an equivalence between such notions as “partial normal subgroup of a proper locality” and “normal subsystem of a saturated fusion system”. This will be done in a seperate paper co-authored with Ellen Henke [ChHe].

The division into parts closely parallels the extent to which fusion systems are drawn into the developing picture. This Part I can be characterized by its having no direct involvement with fusion systems at all, and by there being no mention in it of p′-elements

Typeset by AMS-TEX

or p′-subgroups of a group. We give a very brief overview of Part I in the following paragraphs.

Let G be a group, and let W(G) be the free monoid on G. Thus, W(G) is the set of all words in the alphabet G, with the binary operation given by concatenation of words. The product G × G → G extends, by generalized associativity, to a “product” Π : W(G) → G, whereby a word w = (g_{1}, ⋯, g_{n}) ∈ W(G) is mapped to g_{1} ⋯ g_{n}. The inversion map on G induces an “inversion” on W(G), sending w to (g_{n}^{−1}, ⋯, g_{1}^{−1}). In fact, one may easily replace the standard definition of “group” by a definition given in terms of Π and the inversion on W(G). One obtains the notion of partial group by restricting the domain of Π to a subset D of W(G), where D, the product, and the inversion, are required to satisfy conditions (see definition 1.1) that preserve the outlines of the strictly group-theoretic setup. When one looks at things in this way, a group is simply a partial group G having the property that D = W(G).

The notions of partial subgroup and homomorphism of partial groups immediately suggest themselves, and a partial subgroup of a partial group L may in fact be a group. We say that the partial group L is “objective” (see definition 2.1) provided that the domain D of the product is determined in a certain way by a collection Δ of subgroups of L (the set of “objects”), and provided that Δ has a certain “closure” property. If also L is finite, and there exists S ∈ Δ such that Δ is a collection of subgroups of S, where S is maximal in the set (partially ordered by inclusion) of p-subgroups of L, then (L, Δ, S) is a (finite) locality.

The basic properties of partial groups, objective partial groups, and localities, will be derived in sections 1 and 2. We then begin in section 3 to consider partial normal subgroups of localities in detail. One of the two key results in section 3 is the Frattini Lemma (3.11), which states that if N ⊴ L is a partial normal subgroup, then L = N_{L}(S ∩ N)N. The other is Stellmacher’s splitting lemma (3.12), which leads to the partition of L into a collection of “maximal cosets” of N, and to a partial group structure on the set L/N of maximal cosets. In section 4 it is shown that L/N is in fact a locality, and we obtain versions of the first Nöther isomorphism theorem and of its familiar consequences. In particular, the notions “partial normal subgroup” and “kernel of a projection” turn out to be equivalent. This may be compared with the situation in the theory of saturated fusion systems, where it is known that no such equivalence exists.

Section 5 concerns products of partial normal subgroups. The main result here (Theorem 5.1) has since been considerably strengthened by Ellen Henke [He], to the result that the product of any two partial normal subgroups of a locality is again a partial normal subgroup - where Henke’s proof relies on the special case given by 5.1. The paper ends with a result (Proposition 5.5) which provides an application of essentially all of the concepts and results from all of the earlier sections, and which will play a key role in Part III.

Composition of mappings will most often be written from left to right, and mappings which are likely to be composed with others will be written to the right of their arguments. In particular, this entails that conjugation within a group G be taken in the right-handed sense which is standard in finite group theory; so that x^{g} = g^{−1}xg for any x, g ∈ G.

The author extends his appreciation to Bernd Stellmacher for suggesting and proving the splitting lemma (3.12), and for helpful suggestions regarding definition 1.1. Special thanks are due to Ellen Henke for a detailed list of corrections, and for her improvements on the results in section 5.

Section 1: Partial groups

The reader is asked to forget what a group is, and to trust that what was forgotten will soon be recovered.

For any set X write W(X) for the free monoid on X. Thus, an element of W(X) is a finite sequence of (or word in) the elements of X, and the multiplication in W(X) consists of concatenation of words, to be denoted u ∘ v. The length of the word (x_{1}, ⋯, x_{n}) is n. The empty word is the word (∅) of length 0. We make no distinction between X and the set of words of length 1.

Definition 1.1. Let L be a non-empty set, let W = W(L) be the free monoid on L, and let D be a subset of W such that:

(1) L ⊆ D (i.e. D contains all words of length 1), and

u ∘ v ∈ D ⟹ u, v ∈ D.

Notice that since L is non-empty, (1) implies that also the empty word is in D. A mapping Π : D → L is a product if:

(2) Π restricts to the identity map on L, and

(3) u ∘ v ∘ w ∈ D ⟹ u ∘ Π(v) ∘ w ∈ D, and Π(u ∘ v ∘ w) = Π(u ∘ Π(v) ∘ w).

An inversion on L consists of an involutory bijection x ↦ x^{−1} on L, together with the mapping w ↦ w^{−1} on W given by

(x_{1}, ⋯, x_{n}) ↦ (x_{n}^{−1}, ⋯ x_{1}^{−1}).

We say that L, with the product Π : D → L and inversion (−)^{−1}, is a partial group if:

(4) w ∈ D ⟹ w^{−1} ∘ w ∈ D and Π(w^{−1} ∘ w) = 1, where 1 denotes the image of the empty word under Π. Notice that (1) and (4) yield w^{−1} ∈ D if w ∈ D. As (w^{−1})^{−1} = w, condition (4) is symmetric.

Example 1.2. Let L be the 3-element set {1, a, b} and let D be the subset of W(L) consisting of all words w such that the word obtained from w by deleting all entries equal to 1 is an alternating string of a’s and b’s (of odd or even length and which, if non-empty, may begin either with a or with b). Define Π : D → L by the formula: Π(w) = 1 if the number of a-entries in w is equal to the number of b’s; Π(w) = a if the number of a’s exceeds the number of b’s (necessarily by 1); and Π(w) = b if the number of b’s exceeds the number of a’s. Define inversion on L by 1^{−1} = 1, a^{−1} = b, and b^{−1} = a. It is then easy to check that L, with these structures, is a partial group. In fact, L is the “free partial group on one generator”, as will be made clear in 1.12 below.

It will be convenient to make the definition: a group is a partial group L in which W(L) = D. In order to distinguish between this definition and the usual one we shall use the expression “binary group” for a non-empty set G with an associative binary operation, identity element, and inverses, in the usual sense. The following lemma shows that the distinction is subtle.

Lemma 1.3.

(a) Let G be a binary group and let Π : W(G) → G be the “multivariable product” on G given by (g_{1}, ⋯, g_{n}) ↦ g_{1} ⋯ g_{n}. Then G, together with Π and the inversion in G, is a partial group, with D = W(G).

(b) Let L be a group; i.e. a partial group for which W(L) = D. Then L is a binary group with respect to the operation given by restricting Π to words of length 2, and with respect to the inversion in L. Moreover, Π is then the multivariable product on L defined as in (a).

Proof. Point (a) is given by generalized associativity in the binary group G. Point (b) is a straightforward exercise, and is left to the reader. □

Here are a few basic consequences of definition 1.1.

Lemma 1.4. Let L (with D, Π, and the inversion) be a partial group.

(a) Π is D-multiplicative. That is, if u ∘ v is in D then the word (Π(u), Π(v)) of length 2 is in D, and

Π(u ∘ v) = Π(u)Π(v),

where “Π(u)Π(v)” is an abbreviation for Π((Π(u), Π(v)).

(b) Π is D-associative. That is:

u ∘ v ∘ w ∈ D ⟹ Π(u ∘ v)Π(w) = Π(u)Π(v ∘ w).

(c) If u ∘ v ∈ D then u ∘ (1) ∘ v ∈ D and Π(u ∘ (1) ∘ v) = Π(u ∘ v).

(d) If u ∘ v ∈ D then both u^{−1} ∘ u ∘ v and u ∘ v ∘ v^{−1} are in D, Π(u^{−1} ∘ u ∘ v) = Π(v), and Π(u ∘ v ∘ v^{−1}) = Π(u).

(e) The cancellation rule: If both u ∘ v and u ∘ w are in D, and Π(u ∘ v) = Π(u ∘ w), then Π(v) = Π(w) (and similarly for right cancellation).

(f) If u ∈ D then u^{−1} ∈ D, and Π(u^{−1}) = Π(u)^{−1}. In particular, 1^{−1} = 1.

(g) The uncancellation rule: Suppose that both u ∘ v and u ∘ w are in D and that Π(v) = Π(w). Then Π(u ∘ v) = Π(u ∘ w). (Similarly for right uncancellation.)

Proof. Let u ∘ v ∈ D. Then 1.1(3) applies to (∅) ∘ u ∘ v and yields (Π(u)) ∘ v ∈ D with Π(u ∘ v) = Π((Π(u)) ∘ v). Now apply 1.1(3) to (Π(u)) ∘ v ∘ (∅), to obtain (a).

Let u ∘ v ∘ w ∈ D. Then u ∘ v and w are in D by 1.1(1), and D-multiplicativity yields Π(u ∘ v ∘ w) = Π(u ∘ v)Π(w). Similarly, Π(u ∘ v ∘ w) = Π(u)Π(v ∘ w), and (b) holds.

Since 1 = Π(∅), point (c) is immediate from 1.1(3).

Let u ∘ v ∈ D. Then v^{−1} ∘ u^{−1} ∘ u ∘ v ∈ D by 1.1(4), and then u^{−1} ∘ u ∘ v ∈ D by 1.1(1). Multiplicativity then yields

Π(u^{−1} ∘ u ∘ v) = Π(u^{−1} ∘ u)Π(v) = 1Π(v) = Π(∅)Π(v) = Π(∅ ∘ v) = Π(v).

As (w^{−1})^{−1} = w for any w ∈ W, one obtains w ∘ w^{−1} ∈ D for any w ∈ D, and Π(w ∘ w^{−1}) = 1. From this one easily completes the proof of (d).

Now let u ∘ v and u ∘ w be in D, with Π(u ∘ v) = Π(u ∘ w). Then (d) (together with multiplicativity and associativity, which will not be explicitly mentioned hereafter) yield

Π(v) = Π(u^{−1} ∘ u ∘ v) = Π(u^{−1})Π(u)Π(v) = Π(u^{−1})Π(u)Π(w) = Π(u^{−1} ∘ u ∘ w) = Π(w),

and (e) holds.

Let u ∈ D. Then u ∘ u^{−1} ∈ D, and then Π(u)Π(u^{−1}) = 1. But also (Π(u), Π(u)^{−1}) ∈ D, and Π(u)Π(u)^{−1} = 1. Now (f) follows by 1.1(2) and cancellation.

Let u, v, w be as in (g). Then u^{−1} ∘ u ∘ v and u^{−1} ∘ u ∘ w are in D by (d). By two applications of (d), Π(u^{−1} ∘u∘v) = Π(v) = Π(w) = Π(u^{−1} ∘u∘w), so Π(u∘v) = Π(u∘w) by (e), and (g) holds. □

It will often be convenient to eliminate the symbol “Π” and to speak of “the product g_{1} ⋯ g_{n}” instead of Π(g_{1}, ⋯, g_{n}). More generally, if {X_{i}}_{1≤i≤n} is a collection of subsets of L then the “product set X_{1} ⋯ X_{n}” is by definition the image under Π of the set of words (g_{1}, ⋯, g_{n}) ∈ D such that g_{i} ∈ X_{i} for all i. If X_{i} = {g_{i}} is a singleton then we may write g_{i} in place of X_{i} in such a product. Thus, for example, the product g^{−1}Xg stands for the set of all Π(g^{−1}, x, g) with (g^{−1}, x, g) ∈ D, and with x ∈ X.

A Word of Urgent Warning: In writing products in the above way one may be drawn into imagining that associativity holds in a stronger sense than that which is given by 1.4(b). This is an error that is to be avoided. For example one should not suppose, if (f, g, h) ∈ W, and both (f, g) and (fg, h) are in D, that (f, g, h) is in D. That is, it may be that “the product fgh” is undefined, even though the product (fg)h is defined. Of course, one is tempted to simply extend the domain D to include such triples (f, g, h), and to “define” the product fgh to be (fg)h. The trouble is that it may also be the case that gh and f(gh) are defined, but that (fg)h ≠ f(gh).

For L a partial group and g ∈ L, write D(g) for the set of all x ∈ L such that the product g^{−1}xg is defined. There is then a mapping

c_{g} : D(g) → L

given by x ↦ g^{−1}xg (and called conjugation by g). Our preference is for right-hand notation for mappings, so we write

x ↦ (x)c_{g} or x ↦ x^{g}

for conjugation by g.

The following result provides an illustration of the preceding notational conventions, and introduces a theme which will be developed further as we pass from partial groups to objective partial groups, localities, and (in Part III) regular localities.

Lemma 1.5. Let L be a partial group, and let f, g ∈ L.

(a) Suppose that the products fg and gf are defined and that fg = gf. Suppose further that f ∈ D(g). Then f^{g} = f.

(b) Suppose that f ∈ D(g), g ∈ D(f), and f^{g} = f. Then fg = gf and g^{f} = g.

Proof. (a): We are given (f, g) ∈ D, so (f^{−1}, f, g) ∈ D and Π(f^{−1}, f, g) = g, by 1.4(d) and D-associativity. We are given also f ∈ D(g) and fg = gf, so

f^{g} = Π(g^{−1}, f, g) = Π((g^{−1}, fg) = Π(g^{−1}, gf) = Π(g^{−1}, g, f) = f.

(b): As (g^{−1}, f, g) ∈ D we obtain (f, g) ∈ D from 1.1(1), and (g, g^{−1}, f, g) ∈ D by 1.4(d). Then D-associativity yields fg = Π(g, g^{−1}, f, g) = gf^{g}. As f^{g} = f by hypothesis, we obtain fg = gf. Finally, since (f^{−1}, f, g) and (f^{−1}, g, f) are in D the uncancellation rule yields f^{−1}fg = f^{−1}gf, and so g^{f} = g. □

Notation. From now on, in any given partial group L, usage of the symbol “x^{g}” shall be taken to imply x ∈ D(g). More generally, for X a subset of L and g ∈ L, usage of “X^{g}” shall be taken to mean that X ⊆ D(g); whereupon X^{g} is by definition the set of all x^{g} with x ∈ X.

At this early point, and in the context of arbitrary partial groups, one can say very little about the maps c_{g}. The cancellation rule 1.4(e) implies that each c_{g} is injective, but beyond that the following lemma may be the best that can be obtained.

Lemma 1.6. Let L be a partial group and let g ∈ L. Then the following hold. (a) 1 ∈ D(g) and 1^{g} = 1.

(b) D(g) is closed under inversion, and (x^{−1})^{g} = (x^{g})^{−1} for all x ∈ D(g).

(c) c_{g} is a bijection D(g) → D(g^{−1}), and c_{g^{−1}} = (c_{g})^{−1}.

(d) L = D(1), and x^{1} = x for each x ∈ L.

Proof. By 1.1(4), g ∘ ∅ ∘ g^{−1} = g ∘ g^{−1} ∈ D, so 1 ∈ D(g) and then 1^{g} = 1 by 1.4(c). Thus (a) holds. Now let x ∈ D(g) and set w = (g^{−1}, x, g). Then w ∈ D, and w^{−1} = (g^{−1}, x^{−1}, g) by definition in 1.1. Then 1.1(4) yields w^{−1} ∘ w ∈ D, and so w^{−1} ∈ D by 1.1(1). This shows that D(g) is closed under inversion. Also, 1.1(4) yields 1 = Π(w^{−1} ∘ w) = (x^{−1})^{g}x^{g}, and then (x^{−1})^{g} = (x^{g})^{−1} by 1.4(f). This completes the proof of (b).

As w ∈ D, 1.4(d) implies that g ∘ w and then g ∘ w ∘ g^{−1} are in D. Now 1.1(3) and two applications of 1.4(d) yield

gx^{g}g^{−1} = Π(g, g^{−1}, x, g, g^{−1}) = Π((g, g^{−1}, x) ∘ g ∘ g^{−1}) = Π(g, g^{−1}, x) = x.

Thus x^{g} ∈ D(g^{−1}) with (x^{g})^{g^{−1}} = x, and thus (c) holds.

Finally, 1 = 1^{−1} by 1.4(f), and ∅ ∘ x ∘ ∅ = x ∈ D for any x ∈ L, proving (d). □

Definition 1.7. Let L be a partial group and let H be a non-empty subset of L. Then H is a partial subgroup of L (denoted H ≤ L) if H is closed under inversion (g ∈ H implies g^{−1} ∈ H) and closed with respect to products. The latter condition means, of course, that Π(w) ∈ H whenever w ∈ W(H) ∩ D. A partial subgroup N of L is a partial normal subgroup of L (denoted N ⊴ L) if x^{g} ∈ N for all x ∈ N and all g ∈ L for which x ∈ D(g). We say that H is a subgroup of L if H ≤ L and W(H) ⊆ D.

An equivalent way to state the condition for normality, which relies on the notational convention introduced above for interpreting product sets XY Z, is to say that the partial subgroup N of L is normal in L if g^{−1}N g ⊆ N for all g ∈ L.

We leave it to the reader to check that if H ≤ L then H is indeed a partial group, with D(H) = W(H) ∩ D(L).

Lemma 1.8. Let H and K be partial subgroups of a partial group L, and let {H_{i}}_{i∈I} be a set of partial subgroups of L.

(a) Each partial subgroup of H is a partial subgroup of L.

(b) Each partial subgroup of L which is contained in H is a partial subgroup of H.

(c) If H is a subgroup of L then H ∩ K is a subgroup of both H and K.

(d) Suppose K ⊴ L. Then H ∩ K ⊴ H. Moreover, H ∩ K is a normal subgroup of H if H is a subgroup of L.

(e) ⋂{H_{i} | i ∈ I} is a partial subgroup of L, and is a partial normal subgroup of L if H_{i} ⊴ L for all i.

Proof. One observes that in all of the points (a) through (e) the requisite closure with respect to inversion obtains. Thus, we need only be concerned with products.

(a) Let E ≤ H be a partial subgroup of H. Then

D(E) = W(E) ∩ D(H) = W(E) ∩ (W(H) ∩ D(L) = W(E) ∩ D(L),

and (a) follows.

(b) Suppose K ⊆ H and let w ∈ W(K) ∩ D(H). As D(H) ≤ D(L), and since K ≤ L by hypothesis, we obtain Π(w) ∈ K.

(c) Assuming now that H is a subgroup of L, we have W(H) ⊆ D(L), and then D(H ∩ K) ⊆ D(H) ∩ D(K), so that H ∩ K is a subgroup of both H and K.

(d) Let K ⊴ L and let x ∈ H ∩ K and h ∈ H with (h^{−1}, x, h) ∈ D(H). Then (h^{−1}, x, h) ∈ D(L), and x^{h} ∈ K. As H ≤ L we have also x^{h} ∈ H, and so H ∩ K ⊴ H. Now suppose further that H is a subgroup of L. That is, assume that W(H) ⊆ D(L). Then W(H ∩ K) ⊆ D(L), hence H ∩ K is a subgroup of H, and evidently a normal subgroup.

(e) Set X = ⋂{H_{i}}_{i∈I}. Then Π(w) ∈ X for all w ∈ W(X) ∩ D(L), and so X ≤ L. The last part of (e) may be left to the reader. □

For any subset X of a partial group L define the partial subgroup ⟨X⟩ of L generated by X to be the intersection of the set of all partial subgroups of L containing X. Then ⟨X⟩ is itself a partial subgroup of L by 1.8(e).

Lemma 1.9. Let X be a subset of L such that X is closed under inversion. Set X_{0} = X and recursively define X_{n} for n > 0 by

X_{n} = {Π(w) | w ∈ W(X_{n−1}) ∩ D}.

Then ⟨X⟩ = ⋃{X_{n}}_{n≥0}.

Proof. Let Y be the union of the sets X_{i}. Each X_{i} is closed under inversion by 1.4(f), and Y ≠ ∅ since 1 = Π(∅). Since Y is closed under products, by construction, we get Y ≤ ⟨X⟩, and then Y = ⟨X⟩ by the definition of ⟨X⟩. □

Lemma 1.10 (Dedekind Lemma). Let H, K, and A be partial subgroups of a partial group L, and assume A is a subset of HK.

(a) If K ≤ A then A = (A ∩ H)K.

(b) If H ≤ A then A = H(A ∩ K).

Proof. The proof is identical to the proof for binary groups, and is left to the reader. □

Definition 1.11. Let L and L′ be partial groups, let β : L → L′ be a mapping, and let β^{∗} : W → W′ be the induced mapping of free monoids. Then β is a homomorphism (of partial groups) if: (H1) Dβ^{∗} ⊆ D′, and (H2) (Π(w))β = Π′(wβ^{∗}) for all w ∈ D. The kernel of β is the set Ker(β) of all g ∈ L such that gβ = 1′. We say that β is an isomorphism if there exists a homomorphism β′ : L′ → L such that β ∘ β′ and β′ ∘ β are identity mappings. (Equivalently, β is an isomorphism if β is bijective and Dβ = D′.)

Example 1.12. Let L = {1, a, b} be the partial group from example 1.2, let L′ be any partial group, and let x ∈ L′. Then the mapping β : L → L′ given by

1 ↦ 1′, a ↦ x, b ↦ x^{−1}.

is a homomorphism. In fact, β is the unique homomorphism L → L′ which maps a to x, by the following lemma. Thus, L is the (unique up to a unique invertible homomorphism) free partial group on one generator. Free partial groups in general can be obtained as “free products” of copies of L (see 1.17 below).

Lemma 1.13. Let β : L → L′ be a homomorphism of partial groups. Then 1β = 1′, and (g^{−1})β = (gβ)^{−1} for all g ∈ L.

Proof. Since 11 = 1, (H1) and (H2) yield 1β = (11)β = (1β)(1β), and then 1β = 1′ by left or right cancellation. Since (g, g^{−1}) ∈ D for any g ∈ L by 1.4(d), (H1) yields (gβ, (g^{−1})β) ∈ D′, and then 1β = (gg^{−1})β = (gβ)((g^{−1})β) by (H2). As 1β = 1′ = (gβ)(gβ)^{−1}, left cancellation yields (g^{−1})β = (gβ)^{−1}. □

Lemma 1.14. Let β : L → L′ be a homomorphism of partial groups, and set N = Ker(β). Then N is a partial normal subgroup of L.

Proof. By 1.13 N is closed under inversion. For w in W(N) ∩ D the map β^{∗} : W → W′ sends w to a word of the form (1′, ⋯, 1′). Then Π′(wβ^{∗}) = 1′, and thus Π(w) ∈ N and N is a partial subgroup of L. Now let f ∈ L and let g ∈ N ∩ D(f). Then

(f^{−1}, g, f)β^{∗} = ((fβ)^{−1}, 1′, fβ) (by 1.13),

so that

(g^{f})β = Π′((f^{−1}, g, f)β^{∗}) = Π′(fβ)^{−1}, 1′, fβ) = 1′.

Thus N ⊴ L. □

It will be shown later (cf. 4.6) that partial normal subgroups of “localities” are always kernels of homomorphisms.

Lemma 1.15. Let β : L → L′ be a homomorphism of partial groups and let M be a subgroup of L. Then Mβ is a subgroup of L′. □

Proof. We are given W(M) ⊆ D(L), so β^{∗} maps W(M) into D(L′). (Note, however, example 1.12.) □

Lemma 1.16. Let G and G′ be groups (and hence also binary groups in the sense of 1.3). A map α : G → G′ is a homomorphism of partial groups if and only if α is a homomorphism of binary groups.

Proof. We leave to the reader the proof that if α is a homomorphism of partial groups then α is a homomorphism of binary groups. Now suppose that α is a homomorphism of binary groups. As W(G) = D(G) (and similarly for G′, it is immediate that α^{∗} maps D(G) into D(G). Assume that α is not a homomorphism of partial groups and let w ∈ D(G) be of minimal length subject to Π′(wα^{∗}) ≠ (Π(w))α. Then n > 1 and we can write w = u ∘ v with both u and v non-empty. Then

Π′(wα^{∗}) = Π′(uα^{∗} ∘ vα^{∗}) = Π′(uα^{∗})Π′(vα^{∗}) = ((Π(u))α)((Π(v))α) = (Π(u)Π(v))α,

as α is a homomorphism of binary groups. Since (Π(u)Π(v))α = (Π(w))α, the proof is complete. □

By a pointed set we mean a set with a distinguished base-point. There is then a category of pointed sets with base-point-preserving maps. This category is easily seen to possess all limits and colimits. There is then a forgetful functor P art → Set^{∗}, given by regarding a partial group as a pointed set having the identity element as its base-point.

Theorem 1.17. Let P art be the category of partial groups (and homomorphisms of partial groups), and let Set^{∗} be the category of pointed sets. Then P art has all limits and colimits, and these are preserved by the forgetful functor P art → Set^{∗}.

Proof. We outline a complete proof only for colimits, leaving the details concerning limits to the reader.

Let C be a small category, and let F : C → P art be a covariant functor. Write L_{a} for F (a), a an object of C, and write λ_{x} : L_{b} → L_{a} for F (x), where x : a → b is a morphism of C. Let L be the set-theoretic colimit of F. Thus, L is the disjoint union of the partial groups L_{a} taken over all a ∈ Ob(C), modulo the equivalence relation ∼ generated by the requirement:

(*) For each C-morphism x : a → b and each g ∈ L_{a}, we have g ∼ gλ_{x}. For each object a of C write D_{a} for D(L_{a}). Let D be the set of all w ∈ W(L) for which there exists a ∈ Ob(C) and w̃ = (g_{1}, ⋯, g_{n}) ∈ D_{a} such that w is the image of ṽ under the component-wise application of ∼. Then D contains all words of length 1 in W(L), and one observes that if w ∈ D then any segment of w is in D. That is, D satisfies the condition 1.1(1) in the definition of partial group.

There is a diagram

⨆\{D(L_{a})\} \underrightarrow{Π̃} ⨆\{L_{a}\} ⟶^{ρ} L

where Π̃ is the disjoint union of the products Π_{a} : D_{a} → L_{a}, and ρ is the quotient map given by ∼. Let

⨆{D(L_{a})} ⟶^{ρ^{∗}} D

be the component-wise quotient map. For x ∈ Mor_{C}(a, b), the condition that λ_{x} is a homomorphism of partial groups implies that Π̃ ∘ ρ factors through ρ^{∗} to a mapping Π : D → L. That Π satisfies 1.1(2) and 1.1(3) is then immediate.

The inversion maps on the partial groups L_{a} yield an involutory bijection on ⨆{L_{a}}, and 1.13 implies that this bijection factors through ρ to an involutory bijection g ↦ g^{−1} on L. The reader may now verify the condition 1.1(4) for L, in order to complete the proof that L, with Π and the inversion, is a partial group. One also verifies that the maps ρ_{a} : L_{a} → L given by restricting ρ are homomorphisms of partial groups. Since L is a colimit as a pointed set, we conclude that L is a colimit in the category of partial groups.

The situation for limits (i.e. where F : C → P art is contravariant) is entirely analogous to the fore-going. That is, one forms the set-theoretic limit L (the set of all tuples (g_{a})_{a∈Ob(C)} such that g_{b} = g_{a}λ_{x} for each C-morphism x : a → b), and one endows L with a product and an inversion map. The domain D is in this case the set of all words w ∈ W(L) such that, for each of the projection maps L ⟶^{π_{a}} L_{a}, we have wπ_{a}^{∗} ∈ D_{a}. One then defines Π : D → L by

((g_{a,1}, ⋯, (g_{a,n}) ↦ (Π_{a}(g_{a,1}, ⋯, g_{a,n}).

The inversion on L is of course given by (g_{a}) ↦ (g_{a}^{−1}), and then the verification of the conditions in definition 1.1 is straightforward. □

Section 2: Objective partial groups and localities

Recall the convention: if X is a subset of the partial group L, and g ∈ L, then any statement involving the expression “X^{g}” is to be understood as carrying the assumption that X ⊆ D(g). Thus, the statement “X^{g} = Y ” means: (g^{−1}, x, g) ∈ D for all x ∈ X, and Y is the set of products g^{−1}xg with x ∈ X.

Definition 2.1. Let L be a partial group. For any collection Δ of subgroups of L define D_{Δ} to be the set of all w = (g_{1}, ⋯, g_{n}) ∈ W(L) such that:

(*) there exists (X_{0}, ⋯, X_{n}) ∈ W(Δ) with (X_{i−1})^{g_{i}} = X_{i} for all i (1 ≤ i ≤ n). Then L is objective if there exists a set Δ of subgroups of L such that the following two conditions hold. (O1) D = D_{Δ}. (O2) Whenever X and Y are in Δ and g ∈ L such that X^{g} is a subgroup of Y, then

every subgroup of Y containing X^{g} is in Δ. We say also that Δ is a set of objects) for L, if (O1) and (O2) hold.

It will often be convenient to somewhat over-emphasize the role of Δ in the above definition by saying that “(L, Δ) is an objective partial group”. What is meant by this is that L is an objective partial group and that Δ is a set (there will often be more than one) of objects for L.

We mention that the condition (O2) requires more than that X^{g} be a subset of Y, in order to conclude that overgroups of X in Y are objects. This is a non-vacuous distinction, since the conjugation map c_{g} : X → X^{g} need not send X to a subgroup of L, in a general partial group.

Example 2.2. Let G be a group, let S be a subgroup of G, and let Δ be a collection of subgroups of S such that S ∈ Δ. Assume that Δ satisfies (O2). That is, assume that Y ∈ Δ for every subgroup Y of S such that X^{g} ≤ Y for some X ∈ Δ and some g ∈ G. Let L be the set of all g ∈ G such that S ∩ S^{g} ∈ Δ, and let D be the subset D_{Δ} of W(L). Then L is a partial group (via the multivariable product in G and the inversion in G), and (L, Δ) is an objective partial group. Specifically:

(a) If Δ = {S} then L = N_{G}(S) (and so L is a group in this case).

(b) Take G = O_{4}^{+}(2). Thus, G is a semidirect product V ⋊ S where V is elementary abelian of order 9 and S is a dihedral group of order 8 acting faithfully on V. Let Δ be the set of all non-identity subgroups of S. One may check that S ∩ S^{g} ∈ Δ for all g ∈ G, and hence L = G (as sets). But L is not a group, as D_{Δ} ≠ W(G).

(c) Take G = GL_{3}(2), S ∈ Syl_{2}(G), and let M_{1} and M_{2} be the two maximal subgroups of G containing S. Set P_{i} = O_{2}(M_{i}) and set Δ = {S, P_{1}, P_{2}}. Then L = M_{1} ∪ M_{2} (in fact the “free amalgamated product” of M_{1} and M_{2} over S in the category of partial groups). On the other hand, if Δ is taken to be the set of all non-identity subgroups of S then L is somewhat more complicated. Its underlying set is M_{1}M_{2} ∪ M_{2}M_{1}.

In an objective partial group (L, Δ) we say that the word w = (g_{1}, ⋯, g_{n}) is in D via (X_{0}, ⋯, X_{n}) if the condition (*) in 2.1 applies specifically to w and (X_{0}, ⋯, X_{n}). We may also say, more simply, that w is in D via X_{0}, since the sequence (X_{0}, ⋯, X_{n}) is determined by w and X_{0}.

For any partial group L and subgroups X, Y of L, set

N_{L}(X, Y) = {g ∈ L | X ⊆ D(g), X^{g} ⊆ Y },

and set

N_{L}(X) = {g ∈ L | X^{g} = X}.

Lemma 2.3. Let (L, Δ) be an objective partial group.

(a) N_{L}(X) is a subgroup of L for each X ∈ Δ.

(b) Let g ∈ L and let X ∈ Δ with Y := X^{g} ∈ Δ. Then N_{L}(X) ⊆ D(g), and

c_{g} : N_{L}(X) → N_{L}(Y)

is an isomorphism of groups. More generally:

(c) Let w = (g_{1}, ⋯, g_{n}) ∈ D via (X_{0}, ⋯, X_{n}). Then

c_{g_{1}} ∘ ⋯ ∘ c_{g_{n}} = c_{Π(w)}

as isomorphisms from N_{G}(X_{0}) to N_{G}(X_{n}).

Proof. (a) Let X ∈ Δ and let u ∈ W(N_{L}(X)). Then u ∈ D via X, 1 ∈ N_{L}(X) (1.6(d)), and N_{L}(X)^{−1} = N_{L}(X) (1.6(c)).

(b) Let x, y ∈ N_{L}(X) and set v = (g^{−1}, x, g, g^{−1}, y, g). Then v ∈ D via Y, and then Π(v) = (xy)^{g} = x^{g}y^{g} (using points (a) and (b) of 1.4). Thus, the conjugation map c_{g} : N_{L}(X) → N_{L}(Y) is a homomorphism of binary groups (see 1.3), and hence a homomorphism of partial groups (1.16). Since c_{g^{−1}} = c_{g}^{−1} by 1.6(c), c_{g} is an isomorphism of groups.

(c) Let x ∈ N_{L}(X_{0}), set u_{x} = w^{−1} ∘ (x) ∘ w, and observe that u_{x} ∈ D via X_{n}. Then Π(u_{x}) can be written as (⋯ (x)^{g_{1}} ⋯)^{g_{n}}, and this yields (c). □

The next lemma provides two basic computational tools.

Lemma 2.4. Let (L, Δ) be an objective partial group.

(a) Let (a, b, c) ∈ D and set d = abc. Then bc = a^{−1}d and ab = dc^{−1} (and all of these products are defined).

(b) Let (f, g) ∈ D and let X ∈ Δ. Suppose that both X^{f} and X^{fg} are in Δ. Then X^{fg} = (X^{f})^{g}.

Proof. Point (a) is a fact concerning partial groups in general, and is immediate from 1.4(c). Now consider the setup in (b). As (f, g) ∈ Δ we have also (f^{−1}, f, g) ∈ Δ, and g = Π(f^{−1}, f, g) = f^{−1}(fg). Now observe that (f^{−1}, fg) ∈ D via P^{f}, and apply 2.3(c) to obtain P^{fg} = ((P^{f})^{f^{−1}})^{fg} = (P^{f})^{g}. □

The following result is a version of lemma 1.5(b) for objective partial groups. The hypothesis is weaker than that of 1.5(b), and the conclusion is stronger.

Lemma 2.5. Let (L, Δ) be an objective partial group and let f, g ∈ L. Suppose that f^{g} = f. Then fg = gf and g^{f} = g.

Proof. Suppose that (g^{−1}, f, g) ∈ D via (P_{0}, P_{1}, P_{2}, P_{3}). One then has the following commutative square of conjugation maps, in which the arrows are labeled by elements that perform the conjugation.

P_{2} ⟶^{g} P_{3}
f↑  ↑^{f^{g}}
P_{1} ⟶_{g} P_{0}

Now assume that f^{g} = f. Since any of the arrows in the diagram may be reversed using 1.6(c), one reads off that (f^{−1}, g, f) ∈ D via P_{2}. Then g^{f} = g and fg = gf by 1.5(b). □

The following result and its corollary are fundamental to the entire enterprise. The proof given here is due to Bernd Stellmacher.

Proposition 2.6. Let (L, Δ) be an objective partial group. Suppose that Δ is a collection of subgroups of some S ∈ Δ. For each g ∈ L define S_{g} to be the set of all x ∈ D(g) ∩ S such that x^{g} ∈ S. Then:

(a) S_{g} ∈ Δ. In particular, S_{g} is a subgroup of S.

(b) The conjugation map c_{g} : S_{g} → (S_{g})^{g} is an isomorphism of groups, and S_{g^{−1}} = (S_{g})^{g}.

(c) P^{g} is defined and is a subgroup of S, for every subgroup P of S_{g}. In particular, P^{g} ∈ Δ for any P ∈ Δ with P ≤ S_{g}.

Proof. Fix g ∈ L. Then the word (g) of length 1 is in D by 1.1(2), and since D = D_{Δ} by (O1) there exists X ∈ Δ such that Y := X^{g} ∈ Δ. Let a ∈ S_{g} and set b = a^{g}. Then X^{a} and X^{b} are subgroups of S (as a, b ∈ S), so X^{a} and Y^{b} are in Δ by (O2). Then (a^{−1}, g, b) ∈ D via X^{a}, so also (g, b) ∈ D. Also (a, g) ∈ D via X^{a^{−1}}. Since g^{−1}ag = b we get ag = gb by cancellation, and hence

a^{−1}gb = a^{−1}(gb) = a^{−1}(ag) = (a^{−1}a)g = g

by D-associativity. Since a^{−1}gb conjugates X^{a} to Y^{b}, we draw the following conclusion. (1) X^{a} ≤ S_{g} and (X^{a})^{g} ∈ Δ for all a ∈ S_{g}.

Now let c, d ∈ S_{g}. Then (1) shows that both X^{c} and X^{cd} are members of Δ which are conjugated to members of Δ by g. Setting w = (g^{−1}, c, g, g^{−1}, d, g), we conclude (by following X^{g} along the chain of conjugations given by w) that w ∈ D via X^{g}. Then D-associativity yields

(2)  Π(w) = (cd)^{g} = c^{g}d^{g}.

Since c^{g} and d^{g} are in S, we conclude that cd ∈ S_{g}. Since S_{g} is closed under inversion by 1.6(b), S_{g} is a subgroup of S. As X ≤ S_{g} ≤ S, where X and S are in Δ, (02) now yields S_{g} ∈ Δ. Thus (a) holds.

Since c_{g^{−1}} = (c_{g})^{−1} by 1.6(c), it follows that S_{g^{−1}} = (S_{g})^{g}. Points (b) and (c) are then immediate from (a) and 2.3(b). □

Corollary 2.7. Assume the hypothesis of 2.6, let w = (g_{1}, ⋯, g_{n}) ∈ W(L), and define S_{w} to be the set of all x ∈ S such that, for all k with 1 ≤ k ≤ n, the composition c_{g_{1}} ∘ ⋯ ∘ c_{g_{k}} is defined on x and maps x into S. Then S_{w} is a subgroup of S, and S_{w} ∈ Δ if and only if w ∈ D.

Proof. Let x_{0}, y_{0} ∈ S_{w} and let σ = (x_{0}, ⋯, x_{n}) and τ = (y_{0}, ⋯, y_{n}) be the corresponding sequences of elements of S, obtained from x_{0} and y_{0} via the sequence of maps c_{g_{1}} ∘ ⋯ ∘ c_{g_{k}}. Set S_{j} = S_{g_{j}} (1 ≤ j ≤ n). Then x_{i−1} and y_{i−1} are elements of S_{i−1}, and so x_{i−1}y_{i−1} ∈ S_{i−1} by 2.6(a). As c_{g_{i}} restricts to a homomorphism on S_{i−1} (see 2.3(b)) it follows that x_{i}y_{i} ∈ S_{i}. Thus S_{w} is closed under the binary product in S. That S_{w} is closed under inversion is given by 1.6(c), so S_{w} is a subgroup of S.

Suppose that S_{w} ∈ Δ, set P_{0} = S_{w}, and recursively define P_{k} for 0⟨k ≤ n by P_{k} = (P_{k−1})^{g_{k}}. Then P_{k} is a subgroup of S by 2.6(c) and induction on k. Then (O2) yields P_{k} ∈ Δ, and so w ∈ D by (O1). Conversely, if w ∈ D then (O1) shows that P ≤ S_{w} for some P ∈ Δ, and then S_{w} ∈ Δ by (O2). □

Henceforth our focus will be on finite objective partial groups of a certain kind. Here is the main definition.

Definition 2.8. Let p be a prime and let L be a finite partial group. Then L is a locality if there exists a p-subgroup S of L and a set Δ of subgroups of S such that:

(L1) (L, Δ) is objective.

(L2) S is in Δ, and S is maximal in the set (partially ordered by inclusion) of psubgroups of L.

As with definition 2.1, we shall tend to over-emphasize the roles of S and of Δ by saying that “(L, Δ, S) is a locality” when, strictly speaking, we mean only that L is a locality and that S and Δ fulfill the conditions (L1) and (L2). (The extent to which S and Δ are determined by L is explored in 2.16 and 2.17 below.)

Notice that if L is a locality then the hypothesis of 2.6 and of 2.7 is fulfilled, and we may therefore speak of the subgroups S_{g} and S_{w} of S for any g ∈ L and any w ∈ W(L).

Lemma 2.9. Let (L, Δ, S) be a locality, let (x_{1}, ⋯, x_{n}) ∈ D, and let (g_{0}, ⋯, g_{n}) ∈ W(N_{L}(S)). Then

((*))  (g_{0}, x_{1}, g_{1}^{−1}, g_{1}, x_{2}, ⋯, x_{n_{1}}, g_{n−1}^{−1}, g_{n−1}, x_{n}, g_{n}) ∈ D.

In particular, conjugation by g ∈ N_{L}(S) is an automorphism of the partial group L.

Proof. Set w = (x_{1}, ⋯, x_{n}) and set P = (S_{w})^{g_{0}^{−1}}. Then the word displayed in (*) is in D via P. □

Lemma 2.10. Let (L, Δ, S) be a locality and let P ∈ Δ. Then there exists g ∈ L such that N_{S}(P) ≤ S_{g} and such that N_{S}(P^{g}) is a Sylow p-subgroup of N_{L}(P^{g}).

Proof. Observe first of all that, by (L2) the lemma holds for P = S and g = 1. Among all P for which the lemma fails to hold, choose P so that first |P | and then |N_{S}(P)| is as large as possible. Set R = N_{S}(P) and let R^{∗} be a Sylow p-subgroup of N_{L}(P) containing R. Then R < R^{∗} (proper subgroup), and then also R < N_{R^{∗}}(R). We have P ≠ R as P ≠ S, and then the maximality of |P | yields the existence of an element f ∈ N_{L}(R, S) with N_{S}(R^{f}) ∈ Syl_{p}(N_{L}(R^{f})).

By 2.3(b) f-conjugation induces an isomorphism

N_{L}(R) ⟶^{c_{f}} N_{L}(R^{f}).

By Sylow’s theorem there exists x ∈ N_{L}(R^{f}) such that (N_{R^{∗}}(R)^{f})^{x} ≤ N_{S}(R^{f}). Here (f, x) ∈ D via R, so 2.3(c) yields (N_{R^{∗}}(R)^{f})^{x} = N_{R^{∗}}(R)^{fx}. Thus, by replacing f with fx, we may assume that f was chosen to begin with so that N_{R^{∗}}(R)^{f} ≤ N_{S}(R^{f}). Since R^{∗} normalizes P and c_{f} is an isomorphism, it follows that N_{R^{∗}}(R)^{f} normalizes P^{f}, and thus |N_{S}(P^{f})| > |N_{S}(P)|. The maximality of |N_{S}(P)| in the choice of P then implies that P^{f} is not a counter-example to the lemma. Set Q = P^{f}. Thus there exists h ∈ N_{L}(N_{S}(Q), S) such that N_{S}(Q^{h}) is a Sylow subgroup of N_{L}(Q^{h}). Here (f, h) ∈ D via R, so Q^{h} = P^{g} where g = fh, so P is not a counter-example to the lemma, and therefore no counter-example exists. □

Proposition 2.11. Let (L, Δ, S) be a locality and let H be a subgroup of L.

(a) There exists an object P ∈ Δ such that H ≤ N_{L}(P). Indeed, there exists a unique largest such P. (All subgroups are “local subgroups”.).

(b) Let P be the largest P ∈ Δ as in (a). Then there exists w ∈ W(H) such that P = S_{w}.

(c) If H is a p-subgroup of L then there exists g ∈ L such that H^{g} ≤ S.

Proof. For any w = (h_{1}, ⋯, h_{n}) ∈ W(H) let w′ be the word (g_{1}, ⋯, g_{n}) defined by g_{i} = h_{1} ⋯ h_{i}. As H is finite we may choose w so as to maximize the cardinality of the set X = {g_{1}, ⋯, g_{n}}. Supposing that X ≠ H, let g ∈ H − X, and set h = Π(w)^{−1}g. Then the set of entries of (w ∘ (h))′ is X ∪ {g}, contrary to the maximality of X. Thus X = H.

We have W(H) ⊆ D as H is a subgroup of L, and thus w ∈ D via some P ∈ Δ. Then P^{g_{i}} = P^{h_{1}···h_{i}} ≤ S for all i, and so P^{h} ≤ S for all h ∈ H. Set U = ⟨P^{h} | h ∈ H⟩ (the subgroup of S generated by the union of all P^{h} for h ∈ H). Then U ∈ Δ by (O2). In order to complete the proof of (a) and (b) it now suffices to show that H ≤ N_{L}(U), and for this it is enough to observe that, by 2.4(b), (P^{f})^{g} is defined and is equal to P^{fg} for all f, g ∈ H.

Next, by 2.10 there exists V ∈ Δ and g ∈ L such that V = U^{g} and such that N_{S}(V) ∈ Syl_{p}(N_{L}(V)). Let c_{g} : N_{L}(U) → N_{L}(V) be the isomorphism given by 2.3(b). Thus H^{g} is a p-subgroup of N_{L}(V), so there exists x ∈ N_{L}(V) with (H^{g})^{x} ≤ N_{S}(V). Since (g, x) ∈ D via U we may apply 2.3(c), obtaining (H^{g})^{x} = H^{gx}. Thus (c) holds with gx in the role of g. □

The theory being developed here purports to be “p-local”, so it is important to be able to analyze the structure of N_{L}(U) for U an arbitrary subgroup of S (not necessarily in Δ). Since our policy in this Part I is to avoid bringing in the language of fusion systems, we shall obtain for now only the following very limited result concerning such normalizers.

Lemma 2.13. Let (L, Δ, S) be a locality, let R ≤ S be a subgroup of S, and set Γ = {N_{P} (R) | P ∈ Δ}. Then:

(a) N_{L}(R) is a partial subgroup of L.

(b) If Γ ⊆ Δ then (N_{L}(R), Γ) is an objective partial group.

(c) If Γ ⊆ Δ and N_{S}(R) is a maximal p-subgroup of N_{L}(R) then (N_{L}(R), Γ, S) is a

locality. In particular, if R ⊴ S then (N_{L}(R), Δ, S) is a locality.

Proof. Set L_{R} = N_{L}(R). Then 2.3(c) shows that Π maps W(L_{R}) ∩ D into L_{R}, while 1.6(c) shows that L_{R} is closed under the inversion in L. Thus L_{R} is a partial subgroup of L.

Let D(L_{R}) be the domain of the product in the partial group L_{R}. Then D(L_{R}) = W(L_{R})∩D, by definition. Assume now that Γ ⊆ Δ, and let w ∈ D(L_{R}) via some P ∈ Δ. Then w ∈ D(L_{R}) via N_{P} (R), and thus L_{R} satisfies the condition (O1) for objectivity in definition 2.1. The set Γ is evidently closed in the sense of condition (O2) in definition 2.1, so we have (b). Point (c) is then immediate from definition 2.8. □

Lemma 2.14. Let (L, Δ, S) be a locality, and set

O_{p}(L) = ⋂{S_{w} | w ∈ W(L)}.

Then O_{p}(L) is the unique largest subgroup of S which is a partial normal subgroup of L.

Proof. Set Y = O_{p}(L), let g ∈ L, and let w ∈ W(L). Then Y ≤ S_{g} and Y ≤ S_{(g)∘w}, so Y^{g} ≤ S_{w}. Thus Y^{g} ≤ Y, and so Y ⊴ L.

Now let X ≤ S with X ⊴ L. Then X ⊴ S. Let g ∈ L, and set P = S_{g}. Then conjugation by g is defined on N_{L}(P) by 2.3(b), so N_{X}(P)^{g} is defined, and then N_{X}(P)^{g} ≤ X as X ⊴ L. Thus N_{X}(P)^{g} ≤ S, so N_{X}(P) ≤ S_{g}. As S_{g}X = P X is a subgroup of S, it follows that X ≤ P, so X^{g} = X, and X ≤ O_{p}(L). □

Given a locality L, there can be more than one choice for S and, after that, more than one choice for Δ, such that D = D_{Δ}.

Definition 2.15. Let L be a partial group. An automorphism α of L is inner if there exists g ∈ L such that α is given by conjugation by g. That is:

x ∈ L ⟹ x^{g} is defined and is equal to xα, and

(x_{1}, ⋯, x_{n}) ∈ D ⟹ (x_{1}^{g}, ⋯, x_{n}^{g}) ∈ D and x_{1}^{g} ⋯ x_{n}^{g} = (x_{1} ⋯ x_{n})^{g}. Write Inn(L) for the set of all inner automorphisms of L.

Definition 2.16. Let L be a locality and let S be a p-subgroup of L. Then S is a Sylow p-subgroup of L if there exists a set Δ of subgroups of S such that (L, Δ, S) satisfies the conditions (L1) and (L2) of definition 2.8. Write Syl_{p}(L) for the set of Sylow p-subgroups of L.

It will be convenient to introduce some notation regarding “conjugation” by words w ∈ (L), in the case of a locality (L, Δ, S). Thus, let w = (g_{1}, ⋯, g_{n}) ∈ W(L), and let X be a subgroup of S_{w}. For each x = x_{0} ∈ X and each index i from 1 to n there is then an element x_{i} ∈ S defined by x_{i} = (x_{i−1})^{g_{i}}, and we may write x^{w} for x_{n}. The mapping x ↦ x^{w} may be written as

c_{w} : X → X^{w}.

Proposition 2.17. Let (L, Δ, S) be a locality, and let K be the set of all g ∈ L such that conjugation by g is an inner automorphism of L.

(a) K is a subgroup of L.

(b) For each w = (x_{1}, ⋯, x_{n}) ∈ D and each (g_{0}, ⋯, g_{n}) ∈ W(K), we have

(*)  (g_{0}, x_{1}, g_{1}^{−1}, g_{1}, x_{2}, ⋯, g_{n−1}^{−1}, g_{n−1}, x_{n}, g_{n}) ∈ D.

(c) Set Q = O_{p}(K). Then Q ∈ Δ and K = N_{L}(Q). Moreover, for each w ∈ D there exists P ≤ Q ∩ S_{w} such that P ∈ Δ and P^{w} ≤ Q.

(d) Syl_{p}(L) = Syl_{p}(K).

(e) Inn(L) ⊴ Aut(L), and every automorphism of L can be factored as an inner automorphism followed by an automorphism which leaves S invariant.

Proof. Write S for Syl_{p}(L). For S′ ∈ S note that N_{L}(S′) is a subgroup of L. By 2.9 the condition (*) holds with N_{L}(S′) in the role of K. Let U be the union (taken over all S′ ∈ S) of the groups N_{L}(S′), and let H be the partial subgroup of L generated by U. The above observation concerning 2.9 together with a straightforward argument by induction on word-length then yields W(U) ⊆ D, and thus H is a subgroup of L. Moreover, W(U) has the following property: For any (g_{1}, ⋯, g_{n}) ∈ D(L), and any (u_{0}, ⋯, u_{n}) with each u_{i} ∈ W(U), the word

u_{0} ∘ g_{1} ∘ u_{1}^{−1} ∘ u_{1} ∘ g_{2} ∘ ⋯ ∘ u_{n−1}^{−1} ∘ u_{n−1} ∘ g_{n} ∘ u_{n}

is in D, by induction on the sum of the lengths of the words u_{i}. The condition (*) then holds with H in the role of K, by D-associativity. In particular, conjugation by g ∈ H is an automorphism of L, and thus H is a subset of K.

Now let g ∈ K. Then (L, Δ^{g}, S^{g}) is a locality, so S^{g} ∈ S. Then S^{g} ∈ Syl_{p}(H), and there exists h ∈ H with S^{g} = S^{h}. The product gh^{−1} is defined, and then gh^{−1} ∈ N_{L}(S). Thus gh^{−1} ∈ H, so g ∈ H, and we conclude that H = K. This completes the proof of (a), (b), and (d). Point (e) is immediate from (d), so it remains only to prove (c).

By 2.11(a), K normalizes a member of Δ. Then, since S ∈ Syl_{p}(K), we obtain Q := O_{p}(K) ∈ Δ. By 2.11(c) there exists a word u ∈ W(K) such that Q = S_{u}. Then also Q = S_{u^{−1}}. Let w ∈ D, and let f ∈ N_{L}(Q). The word v := u ∘ w ∘ u is in D by (*), and clearly S_{v} ≤ Q and S_{v^{−1}} ≤ Q. Set P = (S_{v})^{Π(u)}. Then P ≤ S_{w} ∩ Q and P^{Π(w)} ≤ Q. As S_{v} ∈ Δ we have P ∈ Δ. Write w = (x_{1}, ⋯, x_{n}). Then

(f^{−1}, x_{1}, f, f^{−1}, ⋯, f, f^{−1}, x_{n}, f) ∈ D via P ,

and this shows that conjugation by f is an inner automorphism of L. Thus K = N_{L}(Q), and the proof is complete. □

Lemma 2.18. Let L be a locality and let S be a Sylow p-subgroup of L. Then there is a unique smallest set Δ = Δ_{0} and a unique largest set Δ = Δ_{1} of subgroups of S such that the conditions (L1) and (L2) of 2.8 are satisfied by (L, Δ, S).

Proof. Set Take Δ_{0} to be the overgroup-closure in S of the set of all S_{w} for w ∈ D. Take Δ_{1} to be the union of all the the sets Γ of subgroups of S which fulfill (L1) and (L2). □

Section 3: Partial normal subgroups

Throughout this section we fix a locality (L, Δ, S) and a partial normal subgroup N ⊴ L. Recall that this means that N ≤ L is a partial subgroup of L and that Π(g^{−1}, x, g) ∈ N for all x ∈ N and all g ∈ L for which (g^{−1}, x, g) ∈ D. Set T = S ∩ N.

Lemma 3.1.

(a) (T ∩ S_{g})^{g} ≤ T for all g ∈ L.

(b) Let x ∈ N and let P be a subgroup of S_{x}. Then P T = P^{x}T.

(c) T is maximal in the poset of p-subgroups of N.

Proof. (a) Let g ∈ L and let t ∈ S_{g} ∩ T. Then t^{g} ∈ S, and t^{g} ∈ N as N ⊴ L. Thus t^{g} ∈ T.

(b) Let a ∈ P. Then (P^{x})^{a} ≤ S and P^{a} = P. Setting w = (a^{−1}, x^{−1}, a, x) we then have w ∈ D via P^{xa}. Now Π(w) = a^{−1}a^{x} ∈ S, while also Π(w) = (x^{−1})^{a}x ∈ N, and so Π(w) ∈ T. Then a^{x} ∈ aT, and we have thus shown that P^{x} ≤ P T. Then P^{x}T ≤ P T. The equality P^{x}T = P T can then be deduced from (a) (which implies that |P^{x} ∩ T | = |P ∩ T |), or from symmetry with x^{−1} and P^{x} in place of x and P.

(c) Let R be a p-subgroup of N containing T. By 2.9(b) there exists g ∈ L with R^{g} ≤ S, and then R^{g} ≤ S ∩ N = T. As T^{g} = T and conjugation by g is injective, we conclude that R = T. □

Lemma 3.2. Let x, y ∈ N and let f ∈ N_{L}(T).

(a) If (x, f) ∈ D then (f, f^{−1}, x, f) ∈ D, xf = fx^{f}, and S_{(x,f)} = S_{(f,x^{f})} = S_{x} ∩ S_{f}.

(b) If (f, y) ∈ D then (f, y, f^{−1}, f) ∈ D, fy = y^{f^{−1}}f, and S_{(f,y)} = S_{(y^{f^{−1}} ,f)} = S_{y^{f^{−1}}} ∩ S_{f}.

Proof. Set Q = S_{(x,f)} and P = S_{x} ∩ S_{f}. As x ∈ N we have Q^{x}T = QT by 3.1(b). Then since f ∈ N_{L}(T) we obtain QT ≤ S_{f}. Thus Q ≤ P. But also P^{x}T = P T, so P = Q. Now (f, f^{−1}, x, f) ∈ D via Q, and Π(f, f^{−1}, x, f) = xf = fx^{f}. As

Q = S_{f} ∩ S_{x} = S_{(f,f^{−1},x,f)} ≤ S_{(f,x^{f})},

we obtain S_{(x,f)} ≤ S_{(f,x^{f})}. Thus, in order to complete the proof of (a), it remains to show that S_{(f,x^{f})} ≤ Q.

On the other hand, in addressing (b), set R = S_{(f,y)}. Then R^{fy}T = R^{f}T ≤ S_{f^{−1}}, so (f, y, f^{−1}, f) ∈ D via R, and hence R = S_{(f,y,f^{−1},f)}. Take y = x^{f}. Thus

R = S_{(f,x^{f})} = S_{(f,x^{f} ,f^{−1},f)} ≤ S_{(x,f)} = Q,

completing the proof of (a). Moreover, we have shown that fy = y^{f^{−1}}f. The remainder of (b) now follows as an application of (a) to (y^{f^{−1}}, f). The remainder of (b) now follows as an application of (a) to (y^{f^{−1}}, f). □

Lemma 3.3. Let w ∈ W(N_{L}(T)), set g = Π(w), and let x, y ∈ N.

(a) Suppose that (x) ∘ w ∈ D and set P = S_{(x)∘w}. Then u := w^{−1} ∘ (x) ∘ w ∈ D, and S_{u} = P^{g}.

(b) Suppose that w ∘ (y) ∈ D and set Q = S_{w∘(y)}. Then v := w ∘ (y) ∘ w^{−1} ∈ D, and S_{v} = Q.

Proof. We prove only (a), leaving it to the reader to supply a similar argument for (b). As P^{x}T = P T by 3.1(b), and since both P^{x} and T are contained in S_{w}, we have P ≤ S_{w}. Then P^{g} ≤ S by 2.3(c), and P^{g} ≤ S_{u}. In particular u ∈ D via P^{g}. As S_{u} ≤ S_{w^{−1}}, and (S_{u})^{g^{−1}} ≤ P, we obtain S_{u} ≤ P^{g}. Thus S_{u} = P^{g}. □

Lemma 3.4. Let w = (f_{1}, g_{1}, ⋯, f_{n}, g_{n}) ∈ D, with f_{i} ∈ N_{L}(T) and with g_{i} ∈ N for all i. Set u = (f_{1}, ⋯, f_{n}). Then there exists g ∈ N such that

(*)  S_{w} ≤ S_{u∘(g)} and Π(w) = Π(u ∘ (g)).

Similarly, there exists g′ ∈ N such that

(**)  S_{w} ≤ S_{(g′)∘u} and Π(w) = Π((g′) ∘ u).

Proof. The case n = 1 is given by 3.2. For the general case, write

w = (f_{1}, g_{1}) ∘ w_{1},

and set u_{1} = (f_{2}, ⋯, f_{n}). Induction on n implies that there exists x_{1} ∈ N such that S_{w_{1}} ≤ S_{u_{1}∘(x_{1})} and Π(w_{1}) = Π(u_{1} ∘ (x_{1})). Set

w′ = (f_{1}, g_{1}) ∘ u_{1} ∘ (x_{1}).

Thus S_{w} ≤ S_{w′}, and we have Π(w) = Π(w′) by D-associativity. Apply 3.3(a) to the word (g_{1})∘u_{1} to obtain (*). A similar argument in which one begins by writing w = w_{n}∘(f_{n}, g_{n}) yields (**). □

The following result will be of fundamental importance in Part III.

Lemma 3.5. Suppose that

(*)  C_{N_{L}(P)}(O_{p}(N_{L}(P)) ≤ O_{p}(N_{L}(P) ∀ P ∈ Δ.

Then N_{N} (T) ≤ N_{L}(C_{S}(T)T) ≤ N_{L}(C_{S}(T)).

Proof. Let g ∈ N_{N} (T) and set P = S_{g}. Then P^{g} = P by 3.1(b). Set M = N_{L}(P), K = N ∩ M, and D = N_{C_{S}(T)}(P). Here M is a subgroup of L as P ∈ Δ, and then K is a normal subgroup of M by 1.7(d). We have T ⊴ M by 3.1(a), so D ≤ C_{M}(T), and then [K, D] ≤ C_{K}(T). Notice that T ∈ Syl_{p}(K) by 3.1(c), so that K/T is a p′-group. Then C_{K}(T) = Z(T) × Y where Y is a p′-group; and thus Y is a normal p′-subgroup of M. Then [O_{p}(M), Y] = 1, so Y ≤ O_{p}(M) by hypothesis, and thus Y = 1. Thus [K, D] ≤ T, so [g, D] ≤ T, and D ≤ P. As C_{S}(T)P is a p-group we thereby obtain C_{S}(T) ≤ P. This shows that C_{S}(T) is g-invariant for all g ∈ N_{N} (T), and yields the lemma. □

Definition 3.6. Let L ∘ Δ be the set of all pairs (f, P) ∈ L × Δ such that P ≤ S_{f}. Define a relation ↑ on L ∘ Δ by (f, P) ↑ (g, Q) if there exist elements x ∈ N_{N} (P, Q) and y ∈ N_{N} (P^{f}, Q^{g}) such that xg = fy.

This relation may be indicated by means of a commutative square:

(*)  \begin{matrix}Q & ⟶^{g} & Q^{g} \\ ^{x}↑ & & ↑^{y} \\ P & ⟶^{f} & P^{f} \end{matrix}

of conjugation maps, labeled by the conjugating elements, and in which the horizontal arrows are isomorphisms and the vertical arrows are injective homomorphisms. The relation (f, P) ↑ (g, Q) may also be expressed by:

w := (x, g, y^{−1}, f^{−1}) ∈ D via P, and Π(w) = 1.

It is easy to see that ↑ is reflexive and transitive. We say that (f, P) is maximal in L∘Δ if (f, P) ↑ (g, Q) implies that |P | = |Q|. As S is finite there exist maximal elements in L ∘ Δ. Since (f, P) ↑ (f, S_{f}) for (f, P) ∈ L ∘ Δ, we have P = S_{f} for every maximal (f, P). For this reason, we will say that f is ↑-maximal in L (with respect to N) if (f, S_{f}) is maximal in L ∘ Δ.

Lemma 3.7. Let f ∈ L.

(a) If f ∈ N_{L}(S) then f is ↑-maximal.

(b) If f is ↑-maximal then so is f^{−1}.

(c) If f is ↑-maximal and (f, S_{f}) ↑ (g, Q), then g is ↑-maximal and Q = S_{g}.

Proof. Point (a) is immediate from definition 3.6. Now suppose that f is ↑-maximal, and let g ∈ L with (f^{−1}, S_{f^{−1}}) ↑ (g^{−1}, S_{g^{−1}}). Since S_{f^{−1}} = (S_{f})^{f} and S_{g^{−1}} = (S_{g})^{g}, one obtains a diagram

(S_{g})^{g} ⟶^{g^{−1}} S_{g}
x↑  ↑^{y}
(S_{f})^{f} ⟶^{f^{−1}} S_{f}

as in definition 3.6, from which it is easy to read off the relation (f, S_{f}) ↑ (g, S_{g}). Then |S_{f}| = |S_{g}| as f is ↑-maximal, and then also |S_{f^{−1}}| = |S_{g^{−1}}|. Thus f^{−1} is ↑-maximal, and (b) holds. Point (c) is immediate from the transitivity of ↑. □

Lemma 3.8. Let (g, Q), (h, R) ∈ L ∘ Δ with(g, Q) ↑ (h, R), and suppose that T ≤ R. Then there exists a unique y ∈ N with g = yh. Moreover:

(a) Q^{y} ≤ R, and Q ≤ S_{(y,h)}.

(b) If N_{T} (Q^{g}) ∈ Syl_{p}(N_{N} (Q^{g})), then N_{T} (Q^{y}) ∈ Syl_{p}(N_{N} (Q^{y})).

Proof. By definition of the relation ↑, there exist elements u ∈ N_{N} (Q, R) and v ∈ N_{N} (Q^{g}, R^{h}) such that (u, h, v^{−1}, g^{−1}) ∈ D via Q, and such that Π(w) = 1.

R ⟶^{h} R^{h}
u↑  ↑^{v}
Q ⟶_{g} Q^{g}

In particular, uh = gv. Since T ≤ R, points (a) and (b) of 3.1 yield

T = T^{h}, Q^{u}T = QT ≤ R, and Q^{g}T = Q^{gv}T ≤ R^{h}.

Then

w := (u, h, v^{−1}, h^{−1}) ∈ D via (Q, Q^{u}, Q^{uh}, Q^{uhv^{−1}} = Q^{g}, Q^{gh^{−1}}).

Set y = Π(w). Then y = u(v^{−1})^{h^{−1}} ∈ N_{N} (Q, R). Since (u, h, v^{−1}, h^{−1}, h) and (g, v, v^{−1}) are in D (as L is a partial group), we get yh = uhv^{−1} = g. This yields (a). The uniqueness of y is given by right cancellation.

Suppose now that N_{T} (Q^{g}) ∈ Syl_{p}(N_{N} (Q^{g})). As N_{T} (Q^{y})^{h} = N_{T} (Q^{g}), it follows from 2.3(b) that N_{T} (Q^{y}) ∈ Syl_{p}(N_{N} (Q^{y})). □

Proposition 3.9. Let g ∈ L and suppose that g is ↑-maximal with respect to N. Then T ≤ S_{g}.

Proof. Set P = S_{g} and Q = P^{g}. We first show:

(1) Let y ∈ N_{N} (P, S). Then |T ∩P | = |T ∩P^{y}|, and (g, P) ↑ (y^{−1}g, P^{y}). In particular,

y^{−1}g is ↑-maximal. As P^{y}T = P T we obtain

|P^{y} : P^{y} ∩ T | = |P^{y}T : T | = |P T : T | = |P : P ∩ T |,

and so |T ∩ P | = |T ∩ P^{y}|. The following diagram

(*)  \begin{matrix}P^{y} & ⟶^{y^{−1}g} & P^{g} \\ ^{y}↑ & & ↑^{1} \\ P & ⟶_{g} & P^{g} \end{matrix}

shows that (g, P) ↑ (y^{−1}g, P^{y}) and completes the proof of (1).

Suppose next that N_{T} (P) ∈ Syl_{p}(N_{N} (P)). Then N_{T} (P)^{g} ∈ Syl_{p}(N_{N} (Q)), by 2.3(b), and there exists x ∈ N_{N} (Q) such that N_{T} (Q)^{x} ≤ N_{T} (P)^{g}. Here (x, g^{−1}) ∈ D via Q, and we get N_{T} (Q)^{xg^{−1}} ≤ N_{T} (P) ≤ S. Thus (g^{−1}, Q) ↑ (xg^{−1}, N_{T} (Q)Q). As g^{−1} is ↑-maximal by 3.7, it follows that N_{T} (Q) ≤ Q. Hence T ≤ Q and then T ≤ P. We have thus shown:

(2) If T ≰ P then N_{T} (P) ∉ Syl_{p}(N_{N} (P)).

We next show:

(3) Suppose that there exists y ∈ N such that P ≤ S_{y} and such that N_{T} (P^{y}) ∈

Syl_{p}(N_{N} (P^{y})). Then T ≤ P. Indeed, under the hypothesis of (3) we have (g, P) ↑ (y^{−1}g, P^{y}) by (1). Then (y^{−1}g, P^{y}) is ↑-maximal and P^{y} = S_{y^{−1}g}. If T ≰ P then T ≰ P^{y}, and then (2) applies to (y^{−1}g, P^{y}) in the role of (g, P) and yields a contradiction. So, (3) holds.

Among all counter-examples, let g be chosen so that |P | is as large as possible. By 2.10 there exists f ∈ L so that Q^{f} ≤ S and so that N_{S}(Q^{f}) ∈ Syl_{p}(N_{L}(Q^{f})). Set h = gf (where the product is defined via P) and set R = P^{h}. Let (h′, P′) be maximal in L ∘ Δ with (h, P) ↑ (h′, P′), and set R′ = R^{h′}. If T ≤ P′ then 3.8 yields h = yh′ for some y ∈ N such that P ≤ S_{y} and such that N_{T} (P^{y}) ∈ Syl_{p}(N_{N} (P^{y})). The existence of such an element y contradicts (3), so we conclude that T ≰ P′. Then (h′, P′) is a counterexample to the proposition, and the maximality of |P | yields |P | = |P′|. Then (h, P) is maximal in L ∘ Δ, as is (h^{−1}, R) by 3.7(b), and so T ≰ R. But N_{T} (R) ∈ Syl_{p}(N_{N} (R)) since R = Q^{f}, so (2) applies with (h^{−1}, R) in the role of (g, P), and yields T ≤ R. Then T ≤ P, and the proof is complete. □

Lemma 3.10. Suppose that N_{N} (T) ≤ N_{L}(S). Then every element of N_{L}(T) is ↑- maximal with respect to N.

Proof. Let f ∈ N_{L}(T) and set P = S_{f}. Then T ≤ P and T ≤ P^{f}. Let (g, Q) ∈ L ∘ Δ with (f, P) ↑ (g, Q), and let x, y ∈ N be chosen as in definition 3.6. Then P ≤ Q by 3.9, and we have P = P^{x} and (P^{f})^{y} = P^{f} ≤ Q^{g} by 3.1(b). In order to show that f is ↑-maximal it suffices now to show that P = Q, and hence it suffices to show that N_{Q}(P) ≤ P.

Set D = N_{Q}(P). As x ∈ N_{N} (T) ≤ N_{L}(S) we have [D, x] ≤ S ∩ N = T, and similarly [D^{g}, y] ≤ T. This shows that (x, g, y^{−1}) ∈ D via D. As xg = fy by the setup of definition 3.6, we have xgy^{−1} = f by 1.3(g), and thus D ≤ S_{f}. That is, we have N_{Q}(P) ≤ P, as required. □

If X and Y are subsets of L then one has the notion of the product XY, introduced in section 1, as the set of all Π(x, y) with (x, y) ∈ D ∩ (X × Y).

Corollary 3.11 (Frattini Lemma). Let (L, Δ, S) be a locality, let N ⊴ L be a partial normal subgroup, and let Λ be the set of ↑-maximal elements of L with respect to N. Then L = N Λ = ΛN. In particular, we have L = N_{L}(T)N = N N_{L}(T).

Proof. Let f ∈ L, set P = S_{f}, and choose (g, Q) ∈ L ∘ Δ so that (f, P) ↑ (g, Q) and so that g is ↑-maximal. Then 3.8 yields f = xg for some x ∈ N, and then 3.2 shows that f = gy where y = x^{g}. □

The following result will be seen to play a crucial role in the theory being developed here. It was discovered and proved by Bernd Stellmacher, in his reading of an early draft of [Ch1]. The proof given here is his.

Lemma 3.12 (Splitting Lemma). Let (x, f) ∈ D with x ∈ N and with f ↑-maximal with respect to N. Then S_{(x,f)} = S_{xf} = S_{(f,x^{f})}.

Proof. Appealing to 3.2: Set y = x^{f} and g = xf (so that also g = fy), and set Q = S_{(x,f))} (so that also Q = S_{(f,y)}). Thus Q ≤ S_{f} ∩ S_{g}. Also, 3.2(a) yields Q = S_{f} ∩ S_{x}. Set

P_{0} = N_{S_{f}} (Q), P_{1} = N_{S_{g}}(Q), P = ⟨P_{0}, P_{1}⟩,

and set R = P_{0} ∩ P_{1}. Then Q ≤ R. In fact, 2.3(b) shows that y = f^{−1}g and that (R^{f})^{y} = R^{g}, so R ≤ Q, and thus P_{0} ∩ P_{1} = Q. Assume now that (x, f) is a counterexample to the lemma. That is, assume Q < S_{g} (proper inclusion). Then Q < P_{1} and so P_{1} ≰ P_{0}. Thus:

(1) P_{1} ≰ S_{f}. Among all counter-examples, take (x, f) so that |Q| is as large as possible. We consider two cases, as follows.

CASE 1: x ∈ N_{L}(T).

As f ∈ N_{L}(T) (3.5) we have T ≤ Q, and then x ∈ N_{L}(Q) by 3.1(b). Thus Q^{g} = Q^{xf} = Q^{f}. Set Q′ = Q^{g}. Then 2.2(b) yields an isomorphism c_{f} : N_{L}(Q) → N_{L}(Q′).

Here f = x^{−1}g so c_{f} = c_{x^{−1}} ∘ c_{g} by 2.2(c). As x ∈ N_{N} (Q) ⊴ N_{L}(Q), we obtain (P_{1})^{x^{−1}} ≤ N_{N} (Q)P_{1}, and then

(P_{1})^{f} = ((P_{1})^{x^{−1}})^{g} ≤ (N_{N} (Q)P_{1})^{g} ≤ N_{N} (Q′)N_{S}(Q′).

Also (P_{0})^{f} ≤ N_{S}(Q′), so

(2) P^{f} ≤ N_{N} (Q′)N_{S}(Q′). Since T ≤ Q′, T is a Sylow p-subgroup of N_{N} (Q′) by 3.1(c), and thus N_{S}(Q′) is a Sylow p-subgroup of N_{N} (Q′)N_{S}(Q′). By (2) and Sylow’s Theorem there is then an element v ∈ N_{N} (Q′) such that P^{fv} ≤ N_{S}(Q′). In particular, we have:

(3) P_{0} ≤ S_{(f,v)} and P ≤ S_{fv}.

Set u = v^{f^{−1}}. Then (u, f) ∈ D and we have uf = fv and S_{(u,f)} = S_{(f,v)} by 3.2. If S_{(f,v)} = S_{fv} then (f, v) ∈ D via P, so that P ≤ S_{f}, contrary to (1). Thus S_{(f,v)} ≠ S_{fv}, and so (u, f) is a counter-example to the lemma. Then (3) and the maximality of |Q| in the choice of (x, f) yields Q = P_{0} = N_{S_{f}} (Q), and so Q = S_{f}. As (f, Q) ↑ (g, P) via (x^{−1}, 1), we have contradicted the ↑-maximality of f.

CASE 2: The case x ∉ N_{L}(T).

Let h be ↑-maximal, with (g, S_{g}) ↑ (h, S_{h}). Then T ≤ S_{h} by 3.9, and there exists r ∈ N with g = rh by 3.8. Note that 3.8 yields also S_{g} = S_{(r,h)} ≥ Q = S_{(x,g)}.

Set w = (f^{−1}, x^{−1}, r, h), observe that w ∈ D via Q^{g}, and find

Π(w) = (f^{−1}x^{−1})(rh) = g^{−1}g = 1.

Then 2.3 yields h = r^{−1}xf. Since both f and h are in N_{L}(T), 2.2(c) yields r^{−1}x ∈ N_{L}(T), and so r^{−1}x ∈ N_{N} (T). Then Case 1 applies to (r^{−1}x, f), and thus S_{h} = S_{(r^{−1}x,f)} ≤ S_{f} (using 3.2). By definition of ↑ there exist a, b ∈ N such that one has the usual “commutative diagram”:

\begin{matrix}S_{h} & ⟶^{h} & S_{h^{−1}} \\ ^{a}↑ & & ↑^{b} \\ S_{g} & ⟶_{g} & S_{g^{−1}} \end{matrix}.

As T ≤ S_{h}, 3.1(b) yields

S_{g} ≤ S_{g}T = (S_{g})^{a}T ≤ S_{h},

and so S_{g} ≤ S_{f}. This again contradicts (1), and completes the proof. □

The splitting lemma yields a useful criterion for partial normality, as follows.

Corollary 3.13. Let L be a locality, let N ⊴ L, and let K ⊴ N be a partial normal subgroup of N. Suppose that K is N_{L}(T)-invariant. I.e. suppose that x^{h} ∈ K for all (h^{−1}, x, h) ∈ D such that x ∈ K and h ∈ N_{L}(T).) Then K ⊴ L.

Proof. Let x ∈ K and let f ∈ L such that x^{f} is defined. By the Frattini Lemma we may write f = yg with y ∈ N and with g ↑-maximal, and then the splitting lemma yields S_{f} = S_{(y,g)}. Set u = (f^{−1}, x, f) and v = (g^{−1}, y^{−1}, x, y, g). Then S_{u} = S_{v} ∈ Δ, and x^{f} = Π(u) = Π(v) = (x^{y})^{g}. Thus x^{f} ∈ K, and K ⊴ L. □

A subset X of L of the form N f, f ∈ L, will be called a coset of N. A coset N f is maximal if it not a proper subset of any coset of N.

Proposition 3.14. The following hold.

(a) N f = fN for all f ∈ N_{L}(T), and if f is ↑-maximal with respect to N then N f = N fN = fN.

(b) Let f, g ∈ L and assume that f is ↑-maximal. Then

(g, S_{g}) ↑ (f, S_{f}) ⟺ N g ⊆ N f ⟺ g ∈ N g.

(c) If g ∈ L is ↑-maximal relative to N then N g is a maximal coset of N. Moreover, every maximal coset of N is of the form N f for some ↑-maximal element f.

(d) L is partitioned by the set L/N of maximal cosets of N.

(e) Let u := (g_{1}, ⋯, g_{n}) ∈ D and let v := (f_{1}, ⋯, f_{n}) be a sequence of ↑-maximal elements of L such that g_{i} ∈ N f_{i} for all i. Then T S_{u} ≤ S_{v}, and Π(u) ∈ N Π(v).

Proof. (a): That N f = fN for f ∈ N_{L}(T) is given by 3.2. Now let f be ↑-maximal relative to N. Then f ∈ N_{L}(T) by 3.9. Let x, y ∈ N such that (x, f, y) ∈ D. Then (x, fy) ∈ D, and (x, fy) = (x, y′f) where y′ = fyf^{−1}. The splitting lemma (3.12) then yields (x, y′, f) ∈ D, and thus N fN ⊆ N f. The reverse inclusion is obvious, and yields (a).

(b): We have g = xf for some x ∈ N by 3.11, and then 3.12 yields S_{g} = S_{(x,f)} ≤ S_{f}. Let y ∈ N such that (y, g) ∈ D. Since (S_{(y,g)})^{y} ≤ S_{g} = S_{(x,f)} we get (y, x, f) ∈ D, and yg = (yx)f ∈ N f. Thus (g, S_{g}) ↑ (f, S_{f}) ⟹ N g ⊆ N f. Clearly N g ⊆ N f ⟹ g ∈ N f. The required circle of implications is then completed by 3.12.

(c): Immediate from (b).

(d): Let f and g be ↑-maximal, and let h ∈ N f ∩ N g. Thus there exist x, y ∈ N with (x, f) ∈ D, (y, g) ∈ D, and with h = xf = yg. Then also (x^{−1}, x, f) ∈ D, so (x^{−1}, yg) = (x^{−1}, xf) ∈ D. The splitting lemma (3.12) then yields (x^{−1}, y, g) ∈ D, and we thereby obtain f = x^{−1}yg ∈ N g. Now (b) implies that N f ⊆ N g, and symmetry gives the reverse inclusion. Thus N f = N g if N f ∩ N g ≠ ∅.

(e): Let x_{i} ∈ N with g_{i} = x_{i}f_{i}. Set Q_{0} = S_{u}, and set Q_{i} = (Q_{i−1})^{g_{i}} for 1 ≤ i ≤ n. Then Q_{i−1} ≤ S_{g_{i}} = S_{(x_{i},f_{i})} ≤ S_{f_{i}} by 3.12, and then also Q_{i−1}T = (Q_{i−1})^{x_{i}}T ≤ S_{f_{i}}, by 2.11(b) and 3.9. Thus:

(Q_{i−1}T)^{f_{i}} = (Q_{i−1})^{x_{i},f_{i}}T = Q_{i}T ≤ S_{f_{i+1}},

and v ∈ D via Q_{0}T. Now Π(u) ∈ N Π(v) by 3.4. □

Let ≡ be the equivalence relation on L defined by the partition in 3.14(d). In view of 3.14 we may refer to the ≡-classes as the maximal cosets of N in L.

Lemma 3.15. Let H be a partial subgroup of the locality L, containing the partial normal subgroup N ⊴ L. Then H is the disjoint union of the maximal cosets of N contained in H.

Proof. Let f ∈ H. Apply the Frattini lemma (3.11) to obtain f = xh for some x ∈ N and some h ∈ N_{L}(T) such that h is ↑-maximal with respect to N. Then h = x^{−1}f by 1.3(d), and thus h ∈ H as N ≤ H. Then also N h ⊆ H, where N h is a maximal coset of N by 3.14(b). □

The set L/N of maximal cosets of N may also be denoted \overline{L}. Let ρ : L → \overline{L} be the mapping which sends g ∈ L to the unique maximal coset of N containing g. Set W := W(L) and \overline{W} = W(\overline{L}), and let ρ^{∗} : W → \overline{W} be the induced mapping of free monoids. For any subset or element X of W, write \overline{X} for the image of X under ρ^{∗}, and similarly if Y is a subset or element of L write \overline{Y} for the image of Y under ρ. In particular, \overline{D} denotes the image of D under ρ^{∗}. Set \overline{Δ} = \{\overline{P} | P ∈ Δ\}.

For w ∈ W, we shall say that w is ↑-maximal if every entry of w is ↑-maximal.

Lemma 3.16. There is a unique mapping \overline{Π} : \overline{D} → \overline{L}, a unique involutory bijection \overline{g} ↦ \overline{g}^{−1} on \overline{L}, and a unique element \overline{1} of \overline{L} such that \overline{L}, with these structures, is a partial group, and such that ρ is a homomorphism of partial groups. Moreover, the homomorphism of free monoids ρ^{∗} : W(L) → W(\overline{L}) maps D(L) onto D(\overline{L}).

Proof. Let u = (g_{1}, ⋯, g_{n}) and v = (h_{1}, ⋯, h_{n}) be members of D such that \overline{u} = \overline{v}. By 3.14(d) there exists, for each i, an ↑-maximal f_{i} ∈ L with g_{i}, h_{i} ∈ N f_{i}. Set w = (f_{1}, ⋯, f_{n}). Then w ∈ D by 3.14(e), and then 3.3(a) shows that Π(u) and Π(v) are elements of N Π(w). Thus \overline{Π(u)} = \overline{Π(w)} = \overline{Π(v)}, and there is a well-defined mapping \overline{Π} : \overline{D} → \overline{L} given by

(*)  \overline{Π}(w) = \overline{Π(w)}.

For any subset X of L write X^{−1} for the set of inverses of elements of X. For any f ∈ L we then have (N f)^{−1} = f^{−1}N^{−1} by 1.1(4). Here N^{−1} = N as N is a partial group, and then (N f)^{−1} = N f^{−1} by 3.14(a). The inversion map N f ↦ N f^{−1} is then well-defined, and is an involutory bijection on \overline{L}. Set \overline{1} = N.

We now check that the axioms in 1.1, for a partial group, are satisfied by the above structures. Since \overline{D} is the image of D under ρ^{∗}, we get \overline{L} ⊆ \overline{D}. Now let \overline{w} = \overline{u}∘\overline{v} ∈ \overline{D}, let u, v be ↑-maximal pre-images in W of \overline{u}, and \overline{v}, and set w = u ∘ v. Then w is ↑-maximal, and so w ∈ D by 3.14(e). Then u and v are in D, and so \overline{u} and \overline{v} are in \overline{D}. Thus \overline{D} satisfies 1.1(1). Clearly, (*) implies that \overline{Π} restricts to the identity on \overline{L}, so \overline{Π} satisfies 1.1(2).

Next, let \overline{u} ∘ \overline{v} ∘ \overline{w} ∈ \overline{D}, and choose corresponding ↑-maximal pre-images u, v, w. Set g = Π(v). Then \overline{g} = \overline{Π}(\overline{v}) by (*). By 1.1(3) we have both u ∘ v ∘ w and u ∘ (g) ∘ w in D, and these two words have the same image under Π. Applying ρ^{∗} we obtain words in \overline{D} having the same image under \overline{Π}, and thus \overline{Π} satisfies 1.1(3). By definition, \overline{Π}(∅) = \overline{1}, and then the condition 1.1(4) is readily verified. Thus, \overline{L} is a partial group.

By definition, \overline{D} is the image of D under ρ^{∗}. So, in order to check that ρ is a homomorphism of partial groups it suffices to show that if w ∈ D then \overline{Π}(wρ^{∗}) = Π(w)ρ. But this is simply the statement (*). Moreover, it is this observation which establishes that the given partial group structure on \overline{L} is the unique one for which ρ is a homomorphism of partial groups. We have f ∈ Ker(ρ) if and only if fγ = \overline{1} = N. Since N f ⊆ N implies f ∈ N, and since N is the maximal coset of L containing 1, we obtain Ker(ρ) = N. □

Section 4: Quotient localities

We continue the setup in which (L, Δ, S) is a fixed locality and N ⊴ L is a partial normal subgroup. We have seen in 3.16 that the set L/N of maximal cosets of N inherits from L a partial group structure via the projection map ρ : L → L/N. The aim now is to go further, and to show that L/N is a locality. The argument for this involves some subleties: the main problem lies in showing that D(L/N) contains D\overline{Δ} (see 2.1), where \overline{Δ} is the the set of all P ρ with P ∈ Δ. The following three lemmas are intended as steps toward addressing this point.

Lemma 4.1. Let (L, Δ, S) be a locality and let N ⊴ L. Then (N S, Δ, S) is a locality.

Proof. By 2.9 N S is a partial subgroup of L. One observes that D(N S) is the subset D_{Δ} of W(N S), as defined in (2.1), and this suffices to show that (N S, Δ) is objective. As S is a maximal p-subgroup of N S there is nothing more that needs to be shown. □

Lemma 4.2. Let P be a subgroup of S, let Γ be a non-empty set of S-conjugates of P, and set X = ⋃ Γ. Assume that P^{x} ∈ Γ for all x ∈ X. Then either Γ = {P } or N_{S}(P) ∩ X ⊈ P.

Proof. Let Σ be the set of overgroups Q of P in S such X ∩ Q = P. Thus P ∈ Σ. Regard Σ as a poset via inclusion, and let Q be maximal in Σ. If Q = S then X = P and Γ = {P }. On the other hand, suppose that Q ≠ S. Then Q is a proper subgroup of N_{S}(Q), and the maximality of Q implies that there exists x ∈ N_{S}(Q) ∩ X with x ∉ P. Since P^{x} ≤ Q, and since P^{x} ⊆ X by hypothesis, we conclude that P^{x} = P. Thus N_{S}(P) ∩ X ⊈ P. □

Theorem 4.3. Let (L, Δ, S) be a locality, let \overline{L} be a partial group, and let β : L → \overline{L} be a homomorphism of partial groups such that the induced map β^{∗} : W(L) → W(\overline{L}) sends D(L) onto D(\overline{L}). Set N = Ker(β) and T = S ∩ N. Further, set D = D(L), \overline{D} = D(\overline{L}), \overline{S} = Sβ, and \overline{Δ} = \{P β | P ∈ Δ\}. Then (\overline{L}, \overline{Δ}, \overline{S}) is a locality. Moreover:

(a) The fibers of β are the maximal cosets of N.

(b) For each \overline{w} ∈ W(\overline{L}) there exists w ∈ W(L) such that \overline{w} = wβ^{∗} and such that each entry of w is ↑-maximal relative to the partial normal subgroup N of L. For any such w we then have w ∈ D if and only if \overline{w} ∈ \overline{D}.

(c) Let P, Q ∈ Δ with T ≤ P ∩ Q. Then β restricts to a surjection N_{L}(P, Q) → N\overline{L}(P β, Qβ), and to a surjective homomorphism if P = Q.

(d) β is an isomorphism if and only if N = 1.

(e) We have (S_{g})β = \overline{S}_{gβ} for each g ∈ L such that g is ↑-maximal with respect to N.

Proof. The hypothesis that Dβ^{∗} = \overline{D} implies that β^{∗} maps the set of words of length 1 in L onto the set of words of length 1 in \overline{L}. Thus β is surjective.

Let M be a subgroup of L. The restriction of β to M is then a homomorphism of partial groups, and hence a homomorphism M → Mβ of groups by 1.13. In particular, \overline{S} is a p-group, and \overline{Δ} is a set of subgroups of \overline{S}.

We have N ⊴ L by 1.14. Let Λ be the set of elements g ∈ L such that g is ↑-maximal relative to N. For any g ∈ Λ, β is constant on the maximal coset N g (see 3.14) of N, so β restricts to a surjection of Λ onto \overline{L}. This shows that β^{∗} restricts to a surjection of W(Λ) onto W(\overline{L}). If \overline{w} ∈ \overline{D} then there exists w ∈ D with wβ^{∗} = \overline{w}, and then 3.14(e) shows that such a w may be chosen to be in W(Λ). Set D(Λ) = D ∩ W(Λ). Thus:

(1) β^{∗} maps D(Λ) onto \overline{D}.

Let \overline{w} ∈ \overline{D}, let w ∈ D(Λ) with wβ^{∗} = \overline{w}, let \overline{a}, \overline{b} ∈ \overline{S}, and let a, b ∈ S with aβ = \overline{a} and bβ = \overline{b}. Then a, b ∈ Λ by 3.7(a), and (a) ∘ w ∘ (b) ∈ D(Λ) by 2.9. Then (\overline{a}) ∘ \overline{w} ∘ (\overline{b}) ∈ \overline{D}. This shows:

(2) \overline{D} is a \overline{S}-biset (as in 2.9).

Let \overline{a} ∈ \overline{S}, let a ∈ S be a preimage of \overline{a}, and let h ∈ L be any preimage of \overline{a}. Then (a, h^{−1}) ∈ D, and ah^{−1} ∈ N, so:

(3) The β-preimage of an element \overline{a} ∈ \overline{S} is a maximal coset N a, where a ∈ S.

Fix \overline{g} ∈ \overline{L}, let g ∈ Λ with gβ = \overline{g}, set P = S_{g}, and set

\overline{S}\overline{g} = \{\overline{x} ∈ \overline{S} | \overline{x}\overline{g} ∈ \overline{S}\}.

Let \overline{a} ∈ \overline{S}\overline{g} and set \overline{b} = \overline{a}\overline{g}. As in the proof of 2.6 we may then show that \overline{P} \overline{a} ⊆ \overline{S}\overline{g}. Namely, from (\overline{g}^{−1}, \overline{a}, \overline{g}) ∈ \overline{D} and \overline{Π}(\overline{g}^{−1}, \overline{a}, \overline{g}) = \overline{b} we obtain (from two applications of (2)):

(*)  (\overline{P} \overline{a})\overline{g} = \overline{P}^{\overline{g}\overline{b}} ≤ \overline{S}.

Thus, the set Γ of all \overline{S}\overline{g}-conjugates of \overline{P} is a set of subgroups of the set \overline{S}\overline{g}. Setting X = ⋃ Γ we thus have the setup of lemma 4.2, with \overline{S}\overline{g} in the role of R.

Assume now that \overline{P} ≠ \overline{S}\overline{g}. Then 4.2 yields an element \overline{x} ∈ X − \overline{P} such that \overline{x} normalizes \overline{P}. Let Q be the β-preimage of \overline{P} ⟨\overline{x}⟩ in S. Then Q^{g} is defined (and is a subgroup of N_{L}(P^{g})) by 2.3(b). As \overline{Q}\overline{g} ≤ \overline{S} we obtain Q^{g} ≤ N S. As N S is a locality by 4.1, it follows from 2.11(b) that there exists f ∈ N with (Q^{g})^{f} ≤ S. Here (g, f) ∈ D via P, so Q ≤ S_{gf}, and this contradicts the ↑-maximality of g. We conclude:

(4) (S_{g})β = \overline{S}_{gβ} for each g ∈ Λ.

Thus (e) holds.

For \overline{w} ∈ W(\overline{L}) define \overline{S}\overline{w} to be the set of all \overline{x} ∈ \overline{S} such that \overline{x} is conjugated successively into \overline{S} by the entries of \overline{w}. An immediate consequence of (4) is then:

(5) (S_{w})β = \overline{S}_{wβ^{∗}} for all w ∈ W(Λ).

Notice that (5) implies point (b).

We may now verify that (\overline{L}, \overline{Δ}) is objective. Thus, let \overline{w} ∈ W(\overline{L}) with \overline{S}\overline{w} ∈ \overline{Δ}. Let w ∈ W(Λ) with wβ^{∗} = \overline{w}. Then (5) yields S_{w} ∈ Δ, so w ∈ D, and hence \overline{w} ∈ \overline{D}. Thus (\overline{L}, \overline{Δ}) satisfies the condition (O1) in definition 2.1 of objectivity. Now let \overline{P} ∈ \overline{Δ}, let \overline{f} ∈ N\overline{L}(\overline{P} , \overline{S}), and set \overline{Q} = (\overline{P})\overline{f}. Then (4) yields \overline{Q} = Qβ for some Q ∈ Δ, and thus \overline{Q} ∈ \overline{Δ}. Any overgroup of \overline{Q} in \overline{S} is the image of an overgroup of Q in S as β maps S onto \overline{S}, so (\overline{L}, \overline{Δ}) satisfies (O2) in 2.1. Thus (\overline{L}, \overline{Δ}) is objective.

Let P, Q ∈ Δ with T ≤ P ∩ Q, set \overline{P} = P β and \overline{Q} = Qβ, and let \overline{g} ∈ \overline{L} such that \overline{P} \overline{g} is defined and is a subset of \overline{Q}. Let g be a preimage of \overline{g} in Λ. Then P ≤ S_{g} by (4). As β is a homomorphism, (P^{g})β is a subgroup of \overline{Q}. Then P^{g} ≤ Q since β restricts to an epimorphism S → \overline{S} with kernel T. As β maps subgroups of L homomorphically to subgroups of \overline{L} (by 1.13) we obtain (c). In the special case that P = Q = S we obtain in this way an epimorphism from N_{L}(S) to N\overline{L}(\overline{S}). As S is a Sylow subgroup of N_{L}(S) it follows that \overline{S} is a maximal p-subgroup of \overline{L}, and so (\overline{L}, \overline{Δ}, \overline{S}) is a locality.

Let g, h ∈ Λ with γβ = hβ. Then S_{g} = S_{h} by (4), so (g, h^{−1}) ∈ D, and (gh^{−1})β = 1. Thus g ∈ N h, and then N g = N h by 3.14(b). This yields (a), and it remains only to prove (d).

If β is an isomorphism then β is injective, and N = 1. On the other hand, suppose that N = 1. Then (a) shows that β is injective, and so β is a bijection. That β^{−1} is then a homomorphism of partial groups is given by (5). Thus (d) holds, and the proof is complete. □

Definition 4.4. Let L and L′ be partial groups and let β : L → L′ be a homomorphism. Then β is a projection if Dβ^{∗} = D′.

Corollary 4.5. Let (L, Δ, S) be a locality, let N ⊴ L be a partial normal subgroup and let ρ : L → L/N be the mapping which sends g ∈ L to the unique maximal coset of N containing g. Set \overline{L} = L/N, set \overline{S} = Sρ, and let \overline{Δ} be the set of images under ρ of the members of Δ. Regard \overline{L} as a partial group in the unique way (given by 3.16) which makes ρ into a homomorphism of partial groups. Then (\overline{L}, \overline{Δ}, \overline{S}) is a locality, and ρ is a projection.

Proof. Immediate from 3.16 and 4.3. □

Theorem 4.6 (“First Isomorphism Theorem”). Let (L, Δ, S) and (L′, Δ′, S′) be localities, let β : L → L′ be a projection, and let N ⊴ L be a partial normal subgroup of L contained in Ker(β). Let ρ : L → L/N be the projection given by 4.5. Then there exists a unique homomorphism

γ : L/N → L′

such that ρ ∘ γ = β, and γ is a projection. Moreover, γ is an isomorphism if and only if N = Ker(β).

Proof. Set M = Ker(β), and let h ∈ L be ↑-maximal relative to M. Then Mh is a maximal coset of M in L by 3.14(b), and Mh = MhM by 3.14(a). Let g ∈ Mh. As N ≤ M by hypothesis, the splitting lemma (3.12, as applied to M and h) yields

N gN ⊆ N (Mh)N ⊆ M(Mh)M = MhM.

The definition 3.6 of the relation ↑ on L ∘ Δ shows that N gN contains an element f which is ↑-maximal with respect to N. Then f ∈ MhM, so f ∈ Mh, and another application of the splitting lemma yields N f ⊆ Mh. Here N f is the maximal coset of N containing f. We have thus shown:

(*) The partition L/N of L is a refinement of the partition L/M.

By 4.3(a) β induces a bijection L/M → L′. Set \overline{L} = L/N. Then (*) implies that there is a mapping γ : \overline{L} → L′ which sends the maximal coset N fN to fβ. Clearly, γ is the unique mapping \overline{L} → L′ such that ρ ∘ γ = β.

Let \overline{w} ∈ D(\overline{L}). Then 4.3(b) yields a word w ∈ D such that wρ^{∗} = \overline{w} and such that the entries of w are ↑-maximal relative to N. We have \overline{w}γ^{∗} = wβ^{∗}, so γ^{∗} maps D(\overline{L}) into D(L′). Let Π′ and \overline{Π} be the products in L′ and \overline{L}, respectively. As β and ρ are homomorphisms we get

Π′(\overline{w}γ^{∗}) = Π′(wβ^{∗}) = (Π(w))β = (\overline{Π(w)})γ = (\overline{Π}(\overline{w}))γ,

and thus γ is a homomorphism. As β = ρ ∘ γ is a projection, and ρ is a projection, one verifies that γ^{∗} maps D(\overline{L}) onto D(L′) and that γ maps \overline{Δ} onto Δ′. Thus γ is a projection.

We have M = N if and only if Ker(γ) = 1. Then 4.3(d) shows that γ is an isomorphism if and only if M = N; completing the proof. □

Proposition 4.7 (Partial Subgroup Correspondence). Let (L, Δ, S) and (\overline{L}, \overline{Δ}, \overline{S}) be localities, and let β : L → \overline{L} be a projection. Set N = Ker(β) and set T = S ∩ N. Then β induces a bijection σ from the set ℌ of partial subgroups H of L containing N to the set \overline{ℌ} of partial subgroups \overline{H} of \overline{L}. Moreover, for any H ∈ ℌ, we have Hβ ⊴ L′ if and only if H ⊴ L.

Proof. Any partial subgroup of L containing N is a union of maximal cosets of N by 3.15. Then 4.3(a) enables the same argument that one has for groups, for proving that ρ induces a bijection ℌ → \overline{ℌ}. Since each maximal coset of N contains an element which is ↑-maximal with respect to N, one may apply 4.3(b) in order to show that a partial subgroup H ∈ ℌ is normal in L if and only if its image is normal in \overline{L}. The reader should have no difficulty with the details of the argument. □

4.8 Remark. A comprehensive “second isomorphism theorem” appears to be out of reach for two reasons. First, given a partial subgroup H ≤ L and a partial normal subgroup N ⊴ L, there appears to be no reason for the image of H under the projection ρ : L → L/N to be a partial subgroup of L/N, other than in special cases. Second, there seems to be no way, in general, to define the quotient of H over the partial normal subgroup H ∩ N of H. On the other hand, a “third isomorphism theorem” may easily be deduced from 4.3 and from the trivial observation that a composition of projections is again a projection.

Lemma 4.9. Let N ⊴ L and let ρ : L → L/N be the canonical projection. Further, let H be a partial subgroup of L containing N and let X be an arbitrary subset of L. Then (X ∩ H)ρ = Xρ ∩ Hρ.

Proof. By 3.15, H is a union of maximal cosets of N, and then Hρ is the set of those maximal cosets. On the other hand Xρ is the set of all maximal cosets N g of N such that X ∩ N g ≠ ∅. Thus Xρ ∩ Hρ ⊆ (X ∩ H)ρ. The reverse inclusion is obvious. □

Corollary 4.10. Let N ⊴ L, and let M be a partial normal subgroup of L containing N. Let ρ : L → L/N be the canonical projection. Then (S ∩ M)ρ is a maximal p-subgroup of Mρ.

Proof. Write (\overline{L}, \overline{Δ}, \overline{S}) for the quotient locality given by 4.5, and set \overline{M} = Mρ. Applying 4.9 with S in the role of X, we obtain (S ∩ M)ρ = \overline{S} ∩ \overline{M}. Since \overline{M} ⊴ \overline{L}, it follows from 2.11(c) that \overline{S} ∩ \overline{M} is maximal in the poset of p-subgroups of \overline{M}, completing the proof. □

Proposition 4.11. Let N ⊴ L, set T = S ∩ N, and set L_{T} = N_{L}(T). Set \overline{L} = L/N and let ρ : L → \overline{L} be the canonical projection. Then the partial subgroup L_{T} of L is a locality (L_{T} , Δ, S), and the restriction of ρ to L_{T} is a projection L_{T} → \overline{L}.

Proof. That L_{T} is a partial subgroup of L having the structure of a locality (L_{T} , Δ, S) is given by 2.13. Let ρ_{T} be the restriction of ρ to L_{T}. Then ρ_{T} is a homomorphism of partial groups, and 3.14(e) shows that ρ_{T} maps D(L_{T}) onto D(\overline{L}). That is, ρ_{T} is a projection. □

We end this section with an application. For G a finite group, O_{p′}(G) denotes the largest normal subgroup of G having order prime to p, and G is of characteristic p if C_{G}(O_{p}(G)) ≤ O_{p}(G).

Proposition 4.12. Let (L, Δ, S) be a locality. For each P ∈ Δ set Θ(P) = O_{p′}(N_{L}(P)), and set Θ = ⋃{Θ(P)}_{P ∈Δ}. Assume:

(*) P ∈ Δ ⟹ N_{L}(P)/Θ(P) is of characteristic p. Then Θ ⊴ L, S ∩ Θ = 1, and the canonical projection ρ : L → L/Θ resricts to an isomorphism S → Sρ. Moreover, upon identifying S with Sρ:

(a) (L/Θ, Δ, S) is a locality.

(b) F_{S}(L/Θ) = F_{S}(L).

(c) For each P ∈ Δ, the restriction

ρ_{P} : N_{L}(P) → N_{L/Θ}(P)

of ρ induces an isomorphism

N_{L/Θ}(P) ≅ N_{L}(P)/Θ(P),

and N_{L/Θ}(P) is of characteristic p.

Proof. Let x ∈ Θ. Then there exists Q ∈ Δ with x ∈ Θ(Q). Choose such a Q so that |Q| is as large as possible, and set R = N_{S_{x}}(Q). Then [R, x] ≤ RR^{x} ≤ S. But also

[R, x] ≤ [N_{L}(Q), Θ(Q)] ≤ Θ(Q),

and so [R, x] = 1. Then x ∈ Θ(R) by (*) (with R in the role of Q), and the maximality of |Q| yields Q = R. Thus Q = S_{x}, and we have thus shown that x ∈ Θ(S_{x}). Now let P ∈ Δ with P ≤ S_{x}. Then [P, x] ≤ [S_{x}, x] = 1, and (*) yields x ∈ Θ(P). Thus:

(**) Let x ∈ Θ, and let P ∈ Δ with P ≤ S_{x}. Then x ∈ Θ(P).

Clearly, 1 ∈ Θ, and Θ is closed under inversion. Let

w = (x_{1}, ⋯, x_{n}) ∈ W(Θ) ∩ D,

and set P = S_{w}. By (**), and by induction on n, we obtain x_{i} ∈ Θ(P) for all i, and hence Π(w) ∈ Θ(P). Thus Θ is a partial subgroup of L. Now let x ∈ Θ and let g ∈ L be given such that (g^{−1}, x, g) ∈ D via some Q ∈ Δ. Then Q^{g^{−1}} ≤ S_{x}, so (**) yields x ∈ Θ(Q^{g^{−1}}), and then x^{g} ∈ Θ(Q) by 2.3(b). This completes the proof that Θ ⊴ L.

Set \overline{L} = L/Θ and adopt the usual “bar”-convention for images of elements, subgroups, and collections of subgroups under the quotient map ρ : L → \overline{L}. Since Θ is a set of p′- elements of L we have S ∩ Θ = 1, and we may therefore identify S with \overline{S}, and Δ with \overline{Δ}. Point (a) is then given by 4.5.

For each P ∈ Δ let ρ_{P} be the restriction of ρ to N_{L}(P). Then ρ_{P} is an epimorphism N_{L}(P) → N\overline{L}(P) by 4.3(c), with kernel Θ(P). This yields point (c).

By 4.3(c) the conjugation maps c_{g} : P → Q in F, with P, Q ∈ Δ and with g ∈ L, are the same as the conjugation maps c\overline{g} : P → Q with \overline{g} ∈ L/Θ. Since F_{S}(L) is Δ-generated (by definition 2.12), we obtain F_{S}(L) = F_{S}(L/Θ). That is, (b) holds, and the proof is complete. □

Section 5: Products of partial normal subgroups

There are two main results in this section. The first (Theorem 5.1) concerns products of partial normal subgroups in a locality. The second (Proposition 5.6) is an application essentially all of the results preceding it, and will play a vital role in the successor to this work.

Theorem 5.1. Let (L, Δ, S) be a locality, and let M ⊴ L and N ⊴ L be partial normal subgroups. Set U = S ∩ M and V = S ∩ N, and assume:

(*) M normalizes V, and N normalizes U. Then MN = N M ⊴ L, and S ∩ MN = UV.

The proof will require the following version of the Splitting Lemma (3.12).

Lemma 5.2. Assume the hypothesis of 5.1, and let g ∈ MN. Then there exists (x, y) ∈ D with x ∈ M, y ∈ N, g = xy, and S_{g} = S_{(x,y)}.

Proof. Consider the set of all triples (g, x, y) ∈ MN × M × N such that g is a counterexample to the lemma, and g = xy. Among all such triples, let (g, x, y) be chosen so that |S_{(x,y)}| is as large as possible. Set Q = S_{(x,y)} and set P = N_{S_{g}}(Q). It suffices to show that P = Q in order to obtain the lemma.

By 3.2 we have (y, y^{−1}, x, y) ∈ D and g = yx^{y}, with S_{(x,y)} = S_{(y,x^{y})}. Suppose that P ≤ S_{y}. Then P^{y} ≤ S, and since P^{g} ≤ S we conclude that P ≤ S_{(y,x^{y})}, and hence P = Q, as desired. Thus we may assume:

(1) P ≰ S_{y}. Let h be ↑-maximal in the maximal coset of M containing g. Then 3.9 yields an element r ∈ M such that g = rh, and 3.12 yields S_{g} = S_{(r,h)}. Then Q ≤ S_{(r,h)}, so (y^{−1}, x^{−1}, r, h) ∈ D via Q^{g} and Π(y^{−1}, x^{−1}, r, h) = Π(g^{−1}, g) = 1. Thus:

(*)  y = x^{−1}rh and h = r^{−1}xy.

Since y, h ∈ N_{L}(U), it follows that r^{−1}x ∈ N_{M}(U), and then that h = (r^{−1}x)y ∈ MN.

Suppose that h does not provide a counter-example to the lemma. That is, suppose that there exists x′ ∈ M and y′ ∈ N such that (x′, y′) ∈ D, x′y′ = h, and S_{(x′,y′}) = S_{h}. As r^{−1}xy = h = x′y′ we get xy = rx′y′, and (rx′, y′) ∈ D with rx′y′ = rh = g. The idea now is to replace (x, y) with (rx′, y′) and to contradict the assumption that S_{g} ≠ Q. In order to achieve this, observe first of all that S_{g} ≤ S_{r} since S_{(r,h)} = S_{rh} = S_{g}. Then observe that (S_{g})^{r} ≤ S_{h}, and that S_{h} = S_{(x′,y′)} ≤ S_{x′}. Thus (S_{g})^{r} ≤ S_{x′}, and so S_{g} ≤ S_{rx′}. As rx′y′ = g we conclude that S_{g} ≤ S_{(rx′,y′)}, which yields the desired contradiction. We conclude that h is itself a counter-example to the lemma.

Since h = r^{−1}xy by (*), and since h and y are in N_{L}(U), we have r^{−1}x ∈ N_{M}(U), and then U ≤ S_{(r^{−1}x,y)} since h ∈ N_{L}(U). Note furthermore that Q = S_{(x,y)} ≤ S_{g} = S_{(r,h)} ≤ S_{r}, and thus Q^{r}U ≤ S_{(r^{−1}x,y)}. The maximality of |Q| in our initial choice of (g, x, y) then yields Q^{r} = Q^{r}U = S_{(r^{−1}x,y)}. Thus U^{r} ≤ Q^{r}, and conjugation by r^{−1} yields U ≤ Q. A symmetric argument yields V ≤ Q. Setting H = N_{L}(Q), it now follows from 3.1(b) that x, y ∈ H. Then Q = O_{p}(H).

Set X = H ∩ M and Y = H ∩ N. Then X, Y, and UV are normal subgroups of H, and XY/UV is a p′-group. Set \overline{H} = H/(X ∩ Y)UV. Here P ≤ H and [P, g] ≤ S. Since g ∈ XY we obtain

[\overline{P} , \overline{g}] = [\overline{P} , \overline{x}][\overline{P} , \overline{y}] ≤ \overline{X}\overline{Y} ,

33 and since \overline{X}\overline{Y} is a p′-group we get [\overline{P} , \overline{g}] = 1. As \overline{X} ∩ \overline{Y} = 1 we have C_{\overline{X}\overline{Y}} (\overline{P}) = C\overline{X}(\overline{P}) × C\overline{Y} (\overline{P}). As \overline{g} = \overline{x}\overline{y} it follows that \overline{x} and \overline{y} centralize \overline{P}. Thus P^{x} ≤ (X ∩ Y)P and P ∈ Syl_{p}((X ∩ Y)P). By Sylow’s Theorem there exists z ∈ X ∩ Y with P^{x} = P^{z}. Replacing (x, y) with (xz^{−1}, zy) we get g = (xz^{−1})(zy) and P ≤ S_{(xz^{−1},zy)}. This contradicts the maximality of Q and yields a final contradiction, proving the lemma. □

Proof of 5.1. Let w = (g_{1}, ⋯, g_{n}) ∈ W(MN) ∩ D via Q ∈ Δ. By 4.11 we may write g_{i} = x_{i}y_{i} with x_{i} ∈ M, y_{i} ∈ N, and with S_{g_{i}} = S_{(x_{i},y_{i})}. Set w′ = (x_{1}, y_{1}, ⋯, x_{n}, y_{n}). Then w′ ∈ D via Q and Π(w) = Π(w′). Since each y_{i} normalizes U, it follows from 3.4 that Π(w′) = Π(w′′) for some w′′ such that w′′ = u ∘ v ∈ D, where u ∈ W(M), and where v ∈ W(N). Thus MN is closed under Π. In order to show that MN = (MN)^{−1} we note that if (x, y) ∈ D ∩ (M × N) then (y^{−1}, x^{−1}) ∈ D and that y^{−1}x^{−1} ∈ MN by 3.2. Thus MN is a partial subgroup of L. Moreover, we have shown that MN = N M.

Let g ∈ MN and let f ∈ L with (f^{−1}, g, f) ∈ D. As usual we may write f = hr with r ∈ N, h ∈ N_{L}(V), and S_{f} = S_{(h,r)}. Write g = xy as in 5.2. By assumption we have (f^{−1}, g, f) ∈ D via some P ∈ Δ. Setting v = (r^{−1}, h^{−1}, x, y, h, r) it follows that v ∈ D via P and that g^{f} = Π(v). Here (h^{−1}, h, y, h) ∈ D via S_{(y,h)} by 3.2, so v′ := (r^{−1}, h^{−1}, x, h, h^{−1}, y, h, r) ∈ D via P. Then

g^{f} = Π(v) = Π(v′) = (x^{h}y^{h})^{r} ∈ (MN)^{r}.

Since r ∈ N, and MN is a partial group, we conclude that g^{f} ∈ MN. Thus MN ⊴ L.

Set M = N_{M}(S), N = N_{N} (S), and let s ∈ S ∩ MN. Then 5.2 yields s = fg with f ∈ M, g ∈ N, and with S = S_{(f,g)}. Thus f ∈ M and g ∈ N, where M and N are normal subgroups of the group N_{L}(S). Then UV is a normal Sylow p-subgroup of MN, and since s = fg ∈ MN we obtain s ∈ UV. Thus S ∩ MN = UV, and the proof is complete. □

Lemma 5.3. Let (L, Δ, S) be a locality, let M and N be partial normal subgroups of L, and set U = S ∩ M and V = S ∩ N. Suppose that M ∩ N ≤ S. Then M ≤ N_{L}(V) and N ≤ N_{L}(U).

Proof. Let g ∈ M, set P = S_{g}, and let x ∈ N_{V} (P). Then (x^{−1}, g^{−1}, x, g) ∈ D via P^{gx}, and then x^{−1}g^{−1}xg ∈ M ∩ N. The hypothesis then yields x^{g} ∈ S, and thus N_{V} (P) ≤ P. Since P V is a subgroup of S we conclude that V ≤ P, and then V^{g} = V by 3.1(a). Thus M ≤ N_{L}(V), and the lemma follows. □

Corollary 5.4. Let M, N ⊴ L and suppose that M ∩ N ≤ S. Then MN ⊴ L, and S ∩ MN = (S ∩ M)(S ∩ N).

Proof. Immediate from 5.1 and 5.3. □

The proof of the following result (which will play an essential role in Part III) uses essentially everything that has preceded it.

Proposition 5.5. Let (L, Δ, S) be a locality, let N ⊴ L be a partial normal subgroup, set T = S ∩ N, set L_{T} = N_{L}(T), and let K be a partial normal subgroup of L_{T}. Assume:

(1) C_{N_{L}(P)}(O_{p}(N_{L}(P)) ≤ O_{p}(N_{L}(P) for all P ∈ Δ, and

(2) K ≤ C_{L}(T). Then ⟨K, N ⟩ ⊴ L, and S ∩ ⟨K, N ⟩ = (S ∩ K)T. Moreover, if S = C_{S}(T)T then ⟨K, N ⟩ = KN = N K.

Proof. Set N_{T} = N_{N} (T). Then N_{T} ⊴ L_{T} by 1.8. View L_{T} as a locality (L_{T} , Δ, S) is in 2.13. The condition (1) allows us to apply 3.7, and to thereby conclude that N_{T} normalizes C_{S}(T)T. Set U = S ∩ K. Then U ≤ C_{S}(T) by (2), and then [U, N_{T}] ≤ C_{S}(T)T ∩ K = U. Thus N_{T} normalizes U. Since K normalizes T, we conclude from 5.1 that KN_{T} ⊴ L_{T}.

Let \overline{L} be the quotient locality L/N and let ρ : L → \overline{L} be the canonical projection. Then the restriction of ρ to L_{T} is a projection L_{T} → \overline{L} by 4.11, and (KN_{T})ρ = Kρ. Then Kρ ⊴ \overline{L} by partial subgroup correspondence (4.7). The ρ-preimage of Kρ in L is then a partial normal subgroup of L, containing the partial subgroup ⟨K, N ⟩ of L generated by K and N.

By 1.9, ⟨K, N ⟩ is the union of its subsets Y_{i}, where Y_{0} = K ∪ N and (for k > 0) Y_{k} is the set of all Π(w) with w ∈ W(Y_{k−1}) ∩ D. Clearly ρ maps Y_{0} into Kρ, and a straight-forward induction on k then shows that ρ maps each Y_{k} into Kρ. Thus ⟨K, N ⟩ is mapped onto Kρ, and partial subgroup correspondence then implies that ⟨K, N ⟩ is the preimage of Kρ. As Kρ ⊴ \overline{L}, a further application of partial subgroup correspondence yields ⟨K, N ⟩ ⊴ L.

Set U = S ∩ K and set V = S ∩ ⟨K, N ⟩. Then the restriction of ρ to V is a homomorphism of groups by 1.16, with kernel T. Here U is a maximal p-subgroup of K by 3.1(c), and both V ρ and Uρ are maximal p-subgroups of Kρ by 4.10. Then V ρ = Uρ and V = UT.

Suppose now that S = C_{S}(T)T. Then each element of K is ↑-maximal with respect to N, by 3.7. Let w ∈ W(KN) ∩ D, write w = (x_{1}y_{1}, ⋯, x_{n}y_{n} with x_{i} ∈ K and y_{i} ∈ N, and set w′ = (x_{1}, y_{1}, ⋯, x_{n}, y_{n}). Then w′ ∈ D by the Splitting Lemma (3.12), and Π(w′) = Π(w′). Here Π(w′) ∈ KN by 3.4, so ⟨K, N ⟩ = KN in this case. One similarly has ⟨K, N ⟩ = N K, completing the proof. □

Remark 5.6. In the proof of 5.5 hypothesis (2) serves no other purpose than to guarantee that KN_{T} is a partial normal subgroup of L_{T}. In fact, by [Theorem A in He], the product of partial normal subgroups of a locality is always a partial normal subgroup, and so (2) is redundant.

References

[BLO] C. Broto, R. Levi, and Bob Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16 (2003), 779–856.

[Ch1] Andrew Chermak, Fusion systems and localities, Acta Math. 211 (2013), 47-139.

[He1] Ellen Henke, Products of partial normal subgroups, (arXiv:1506.01459) (2015).

[O1] Bob Oliver, Equivalences of classifying spaces completed at odd primes, Math. Proc. Camb. Phil. Soc. 137 (2004), 321-347.

[O2] , Equivalences of classifying spaces completed at the prime two, Amer. Math. Soc. Memoirs 848 (2006).

Manhattan Kansas

E-mail address: chermak@math.ksu.edu

36