CONTROL OF FIXED POINTS AND EXISTENCE AND UNIQUENESS OF CENTRIC LINKING SYSTEMS

GEORGE GLAUBERMAN AND JUSTIN LYND

Abstract. A. Chermak has recently proved that to each saturated fusion system over a finite p-group, there is a unique associated centric linking system. B. Oliver extended Chermak’s proof by showing that all the higher cohomological obstruction groups relevant to unique existence of centric linking systems vanish. Both proofs indirectly assume the classification of finite simple groups. We show how to remove this assumption, thereby giving a classification-free proof of the Martino-Priddy conjecture concerning the p-completed classifying spaces of finite groups. Our main tool is a 1971 result of the first author on control of fixed points by p-local subgroups. This result is directly applicable for odd primes, and we show how a slight variation of it allows applications for p = 2 in the presence of offenders.

1. Introduction

Given a saturated fusion system F over a finite p-group S, A. Chermak showed how to construct a centric linking system for F that is unique up to isomorphism [Che13]. His construction is made possible by a delicate filtration of the collection of F-centric subgroups, which makes use of the Thompson subgroup in a critical way, together with an iterative procedure for extending a linking system on a given collection of subgroups of S to a linking system on a larger collection. Within a given step of the procedure, one is working locally in the normalizer of a p-subgroup, and the problem of carrying out the inductive step is reduced to the problem of extending an automorphism of a linking system of a constrained group to an automorphism of the group. It is at this place where an appeal to the classification of finite simple groups is needed, in the form of the General FF-module Theorem of Meierfrankenfeld and Stellmacher [MS12].

In a companion paper [Oli13], B. Oliver showed how to interpret Chermak’s proof in terms of the established Broto-Levi-Oliver obstruction theory for the existence and uniqueness of centric linking systems [BLO03b, §3]. The obstruction groups appearing there are the higher derived limits over the orbit category of the fusion system, at the level of F-centric subgroups, of the (contravariant) center functor Z_{F}: O(F^{c}) → Ab which sends an Fcentric subgroup to its center. From the point of view of this obstruction theory, Chermak’s filtration gives a way of filtering the center functor Z_{F} so that the higher limits of each subquotient functor in the filtration can be shown to vanish. Within Oliver’s proof, the problem is again reduced to the case where F is the fusion system of a constrained group, and to showing that \underleftarrow{lim}^{1} (when p is odd) and \underleftarrow{lim}^{2} (when p = 2) of certain explicit subquotient functors of the center functor on the orbit category of this group vanish. For this, an appeal to the General FF-module Theorem gives a list of the possible groups, and then the proof is finished by examining these cases.

Date: June 3, 2015.

2000 Mathematics Subject Classification. Primary 20D20, Secondary 20D05.

The second author is supported by NSA Young Investigator Grant H98230-14-1-0312, and was supported by an AMS-Simons travel grant which allowed for travel related to this work.

We study the constrained case in Oliver’s proof of Chermak’s Theorem and give proofs of Proposition 3.3 of [Oli13] that do not depend on the classification of finite simple groups. When taken together with the reduction via Chermak’s filtration to this situation in [Oli13], we obtain a classification-free proof of existence and uniqueness of centric linking systems at all primes.

Theorem 1.1 (Oliver [Oli13]). Let F be a saturated fusion system over a finite p-group. Then \underleftarrow{lim}_{O(F^{c})}^{k} Z_{F} = 0 for all k ⩾ 1 if p is odd, and for all k ⩾ 2 if p = 2.

Proof. When p is odd, this follows from the proof of [Oli13, Theorem 3.4] and Proposition 3.11 below. When p = 2, it follows from the proof of [Oli13, Theorem 3.4] and Proposition 6.9 below. □

It was known very early on that \underleftarrow{lim}_{O(F^{c})}^{1} Z_{F} can be nonvanishing when p = 2. An example of this is given by the 2-fusion system F of the alternating group A_{6}, where \underleftarrow{lim}_{O(F^{c})}^{1} Z_{F} is of order 2 [Oli06, Proposition 1.6, Ch.10].

Theorem 1.2 (Chermak [Che13]). Each saturated fusion system has an associated centric linking system that is unique up to isomorphism.

Proof. This follows from Theorem 1.1 together with [BLO03b, Proposition 3.1]. □

For a finite group G, the canonical centric linking system for G at the prime p is sonamed because it provides a link between the fusion system of G and the p-completion, in the sense of Bousfield and Kan, of its classifying space. More precisely, Broto, Levi, and Oliver showed that two finite groups have homotopy equivalent p-completed classifying spaces if and only if there is an equivalence of categories between their centric linking systems [BLO03a]. The question of whether or not the fusion system alone is enough to recover the centric linking system, and thus the p-completed classifying space of the group, is known as the Martino-Priddy conjecture. A special case of Chermak’s Theorem, the Martino-Priddy conjecture was first proven by Oliver [Oli04, Oli06] by showing that Theorem 1.1 holds for the fusion system of a finite group using the classification of finite simple groups. Thus, one consequence of the results in this paper is a classification-free proof of this conjecture.

In addition to relying heavily on the reductions of Chermak and Oliver, our arguments use variations on techniques of the first author for studying when, for a finite group G acting on an abelian p-group D, some subgroup H controls fixed points in G on D – that is, when C_{D}(H) = C_{D}(G). In particular, very general conditions were given in [Gla71, Theorem A1.4] under which this holds for a suitable p-local subgroup H of G. This general result is the basis for the statement, also found in [Gla71], that the normalizer of the Thompson subgroup “controls weak closure of elements” when p is odd. We refer to §14 of [Gla71] for more details on this relationship.

In order to explain how control of fixed points is helpful in computing limits of constrained groups, we fix a finite group Γ having Sylow p-subgroup S and a normal p-subgroup Y containing its centralizer in Γ. The filtrations of the center functor that feature in Chermak’s proof correspond to objectwise filtrations of the collection of F-centric subgroups. A collection Q ⊆ F^{c} that is invariant under F-conjugacy and closed under passing to overgroups corresponds to a quotient Z_{F}^{Q} of the center functor. For such a collection, one can form the locality L_{Q}(Γ) (in the sense of [Che13, 2.10]) consisting of those g ∈ Γ that conjugate a subgroup in the collection Q to another subgroup in Q. Now let F = F_{S}(Γ). It is then known by work of Jackowski-McClure [JM92] that when Q is the set 𝒮 (S)_{⩾Y} of subgroups of S containing Y, the higher limit \underleftarrow{lim}^{k}Z_{F}^{𝒮 (S)_{⩾Y}} of the corresponding quotient functor vanishes if k > 0 and is isomorphic to C_{Z(Y)}(Γ) when k = 0. In general, if each member of Q contains Y, and R is the set of subgroups of S containing Y that are not in Q, the long exact sequence in cohomology for a short exact sequence of functors shows that \underleftarrow{lim}^{1}Z_{F}^{R} = 0 provided that the inclusion

\underleftarrow{lim}^{0}Z_{F}^{𝒮 (S)_{⩾Y}} ≅ C_{Z(Y)}(Γ) ↪ C_{Z(Y)}(L_{Q}(Γ)) ≅ \underleftarrow{lim}^{0}Z_{F}^{Q}

is an isomorphism. Hence \underleftarrow{lim}^{1}Z_{F}^{R} = 0 provided some normalizer in Γ of a subgroup in the collection Q controls fixed points in Γ on Z(Y). This observation essentially completes the picture when p is an odd prime, as is shown in §3.

For p = 2, the canonical obstruction to having control of fixed points by a 2-local subgroup is the symmetric group G = S_{3} acting on D = C_{2} × C_{2}. Here the 2-local subgroups are of order 2 and have nontrivial fixed points on D, while G does not. The aim for p = 2 is to isolate this obstruction in the case where D is an FF-module for G – that is, when G has nontrivial offenders on D (Definition 3.5 below). We define the notion of a solitary offender, which is an offender of order 2 having a specified embedding in G with respect to a Sylow 2-subgroup. The standard example of a solitary offender is generated by a transposition in an odd degree symmetric group acting on the natural module over the field with two elements, where the transposition is, in particular, a transvection. Away from the obstruction posed by solitary offenders, control of fixed points by the normalizer of a nonidentity normal – but perhaps not characteristic – subgroup of a Sylow 2-subgroup is obtained in §4.

Theorem 1.3. Let G be a finite group, S a Sylow 2-subgroup of G, and D an abelian 2-group on which G acts faithfully. Assume that G has a minimal nontrivial offender on D that is not solitary. Then there is a subgroup J ⩽ S, generated by offenders and weakly closed in S with respect to G, such that C_{D}(N_{G}(J)) = C_{D}(G).

Proof. This follows from Theorem 4.1 via Lemmas 4.6 and 4.15. □

This could be viewed as the main result of §4, although the more detailed information contained in Lemmas 4.6 and 4.15 is needed for the remainder. When interpreted, Theorem 1.3 gives the vanishing of \underleftarrow{lim}^{1}Z_{F}^{R} for F the fusion system of any constrained group Γ as above with D = Z(Y) and G = Γ/C_{Γ}(D) having the prescribed action on D, and with R those subgroups containing Y whose images in G do not contain offenders on D.

The results of §4 are then applied to upgrade the vanishing of \underleftarrow{lim}^{1} to that of \underleftarrow{lim}^{2} away from the canonical obstruction, and this is carried out in Theorem 5.4. For this and for the remaining arguments we work with the bar resolution for these limits. As a result, the arguments involve questions about realizing an automorphism of a locality as an inner automorphism of a group, and thus begin to resemble those appearing in Chermak’s paper [Che13]. We hope that the preliminary lemmas of §5 will make more clear this connection for those who are more familiar with Chermak’s group theoretic approach.

With a little more work it is seen that in a minimal counterexample, G is generated by its solitary offenders on D, and in particular by transvections on Ω_{1}(D). In this paper, by a natural module for a symmetric group S_{m} (m ⩾ 3), we mean the lone nontrivial irreducible composition factor of the standard permutation module for S_{m} over the field with two elements. This is of dimension m − 1 when m is odd, and of dimension m − 2 when m is even.

The following is Theorem 6.2 below.

Theorem 1.4. Let G be a finite group and D an abelian 2-group on which G acts faithfully. Assume that G has no nontrivial normal 2-subgroups and that G is generated by its solitary offenders on D. Then G is a direct product of symmetric groups of odd degree at least three, and D/C_{D}(G) is a direct sum of natural modules.

Thus, ultimately, the General FF-module Theorem is replaced by an appeal (in the proof of Theorem 1.4) to McLaughlin’s classification of irreducible subgroups of SL_{n}(2) generated by transvections [McL69].

Notation. Conjugation maps and morphisms in fusion systems will be written on the right and in the exponent, while cocycles and cohomology classes for functors will be written on the left. Groups of cochains are written multiplicatively. However, on some occasions we express that a group, or a cocycle or a cohomology class, is trivial by saying that it is equal to 0.

Acknowledgements

It is our pleasure to thank the Department of Mathematics at the University of Chicago for supporting the visits of the second author, during which this work began. The second author thanks the department for its hospitality during his stays. We would also like to thank Andrew Chermak and Bob Oliver for their comments on an earlier draft of this paper.

2. Functors on orbit categories

In this section, we recall the terminology and constructions in homological algebra that are needed. Our choice of notation follows [Oli13], and we also recall here some of the preliminary lemmas on cohomology that we use from that paper. Since we will have very little explicit need for the theory of fusion or linking systems, we refer to [AKO11] for the definitions and basic properties.

Fix a saturated fusion system F over a finite p-group S. We say that subgroups P and Q of S are F-conjugate if they are isomorphic in F. A subgroup P of S is fully F-normalized if |N_{S}(P)| ⩾ |N_{S}(Q)| for every subgroup Q of S that is F-conjugate to P. The subgroup P is F-centric if C_{S}(Q) ⩽ Q for every subgroup Q of S that is F-conjugate to P. We write F^{f} and F^{c} for the collection of fully F-normalized and the collection of F-centric subgroups of S, respectively. We also use F^{c} to denote the full subcategory of F with objects the F-centric subgroups. Since morphisms are written on the right, Hom_{F}(P, Q) has a right action by Inn(Q) for each pair of subgroups P, Q ⩽ S. The orbit category O(F^{c}) of F-centrics has as objects the set F^{c} and as morphisms the orbits under this action:

Mor_{O(F^{c})}(P, Q) = Hom_{F}(P, Q)/ Inn(Q).

The class of a morphism φ ∈ Hom_{F}(P, Q) is denoted by [φ].

The center functor is the contravariant functor Z_{F}: O(F^{c}) → Ab defined by Z_{F}(P) = Z(P) on objects. For a morphism φ: P → Q in F, Z_{F}([φ]) is the map from Z(Q) = C_{S}(Q) ⩽ C_{S}(P^{φ}) = Z(P^{φ}) to Z(P) induced by φ^{−1}.

Useful filtrations by subquotient functors of this functor often correspond to filtrations of the subgroups of S. Denote by 𝒮 (S) the set of subgroups of S, and by 𝒮 (S)_{⩾Y} the subset of those that contain a fixed subgroup Y ⩽ S.

Definition 2.1. A collection R ⊆ 𝒮 (S) is an interval if P ∈ R whenever P_{1}, P_{2} ∈ R and P_{1} ⩽ P ⩽ P_{2}. An interval R is F-invariant if P^{φ} ∈ R whenever P ∈ R and φ ∈ Hom_{F}(P, S).

Given an F-invariant interval R, define the functor Z_{F}^{R}: O(F^{c}) → Ab by Z_{F}^{R}(P) = Z(P) whenever P ∈ R, and by 1 otherwise. Then Z_{F}^{R} is a subfunctor of Z_{F} when R is closed under passing to (centric) subgroups, and a quotient functor of Z_{F} when R is closed under passing to overgroups in S – that is, when S ∈ R.

Following [Oli13], we write

L^{k}(F; R) := \underleftarrow{lim}_{O(F^{c})}^{k}Z_{F}^{R}

for the higher derived limits of these functors, and we think of them as cohomology groups of the category O(F^{c}) with coefficients in the functor Z_{F}^{R}. They are cohomology groups of a certain cochain complex C^{∗}(O(F^{c}); Z_{F}^{R}), in which k-cochains are maps from sequences of k composable morphisms in the category. A 0-cochain u is a map sending P ∈ F^{c} to an element u(P) ∈ Z_{F}^{R}(P), and a 1-cochain t is a map sending a morphism P ⟶^{[φ]} Q to an element t([φ]) ∈ Z_{F}^{R}(P). We will be working in §5 with cochains for Z_{F}^{Q} in the case where Q is closed under passing to overgroups. With our notational conventions, the coboundary maps on such 0- and 1-cohains in this special case are as follows:

(2.2)  du([φ]) = \begin{cases}u(Q)^{φ^{−1}}u(P)^{−1} & if P ∈ Q, and \\ 1 & otherwise; and \end{cases}

(2.3)  dt([φ][ψ]]) = \begin{cases}t([ψ])^{φ^{−1}}t([φψ])^{−1}t([φ]) & if P ∈ Q, and \\ 1 & otherwise, \end{cases}

for any sequence P ⟶^{φ} Q ⟶^{ψ} R of composable morphisms in F^{c}. We refer to [AKO11, §III.5.1] for more details on the bar resolution for these functors.

Definition 2.4. A general setup for the prime p is a triple (Γ, S, Y) where Γ is a finite group, S is a Sylow p-subgroup of Γ, and Y is a normal p-subgroup of Γ such that C_{Γ}(Y) ⩽ Y. A reduced setup is a general setup such that Y = C_{S}(Z(Y)) and O_{p}(Γ/C_{Γ}(Z(Y))) = 1.

We next state the three preliminary lemmas from [Oli13] that are needed later.

Lemma 2.5. Let F be a saturated fusion system over a p-group S, and let Q ⊆ F^{c} be an F-invariant interval such that S ∈ Q. Let F_{Q} be the full subcategory of F with object set Q.

(a) The inclusion F_{Q} → F^{c} induces an isomorphism of cochain complexes C^{∗}(O(F^{c}); Z_{Q}) ⟶^{≅} C^{∗}(O(F_{Q}); Z_{F}^{Q}|_{Q}), and in particular an isomorphism L^{∗}(F; Q) ⟶^{≅} L^{∗}(F_{Q}; Z_{Q}|_{Q}).

(b) If (Γ, S, Y) is a general setup, F = F_{S}(Γ), and Q = 𝒮 (S)_{⩾Y}, then

L^{k}(F; Q) = \begin{cases}C_{Z(Y)}(Γ) & if k = 0; and \\ 0 & otherwise. \end{cases}

Proof. This is Lemma 1.6 of [Oli13], with the additional information in part (a) shown in its proof. □

Part (b) of the following lemma gave us the first concrete indication that questions regarding control of fixed points by p-local subgroups would be relevant to Theorem 1.1. It is the starting point for nearly all the arguments to follow.

Lemma 2.6. Let F be a saturated fusion system over a p-group S. Let Q, R ⊆ F^{c} be F-invariant intervals such that

(i) Q ∩ R = ∅, (ii) Q ∪ R is an interval, (iii) Q ∈ Q, R ∈ R implies Q ≰ R.

Then Z_{F}^{R} is a subfunctor of Z_{F}^{Q∪R}, Z_{F}^{Q∪R}/Z_{F}^{R} ≅ Z_{F}^{Q}, and there is a long exact sequence

0 ⟶ L^{0}(F; R) ⟶ L^{0}(F; Q ∪ R) ⟶ L^{0}(F; Q) ⟶ ⋯
⟶ L^{k−1}(F; Q) ⟶ L^{k}(F; R) ⟶ L^{k}(F; Q ∪ R) ⟶ L^{k}(F; Q) ⟶ · · · .

In particular, if (Γ, S, Y) is a general setup, D = Z(Y), F = F_{S}(Γ) and Q∪R = 𝒮 (S)_{⩾Y}, then

(a) L^{k−1}(F; Q) ≅ L^{k}(F; R) for each k ⩾ 2, and

(b) there is a short exact sequence

1 ⟶ C_{D}(Γ) ⟶ C_{D}(Γ^{∗}) ⟶ L^{1}(F; R) ⟶ 1,

where Γ^{∗} is the set of g ∈ Γ such that there exists Q ∈ Q with Q^{g} ∈ Q.

Proof. This is nearly Lemma 1.7 of [Oli13]. Our part (a) is stated in the situation of a general setup, and it follows from the long exact sequence and Lemma 2.5(b). □

Lemma 2.7. Let (Γ, S, Y) be a general setup, Γ_{0} a normal subgroup of Γ containing Y, and S_{0} = S∩Γ_{0}. Set F = F_{S}(Γ) and F_{0} = F_{S_{0}}(Γ_{0}). Let Q ⊆ 𝒮 (S)_{Y} be an F-invariant interval such that S ∈ Q, and such that Γ_{0} ∩ Q ∈ Q whenever Q ∈ Q. Set Q_{0} = {Q ∈ Q | Q ⩽ Γ_{0}}. Then restriction induces an injection

L^{1}(F; Q) ⟶ L^{1}(F_{0}; Q_{0}).

Proof. This is Lemma 1.13 of [Oli13]. □

3. The norm argument and the odd case

For a finite group G with action on an abelian group V (written multiplicatively), and a subgroup H of G, the norm map 𝔑_{H}^{G}: C_{V} (H) → C_{V} (G) is defined by

𝔑_{H}^{G}(v) = ∏_{g∈[G/H]} v^{g}

for each v ∈ C_{V} (H), where v^{g} denotes the image of v under g, and where [G/H] is a set of right coset representatives for H in G. We say that 𝔑_{H}^{G} = 1 on V if this map is constant, equal to the identity of V. Since 𝔑_{H}^{G} = 𝔑_{K}^{G}𝔑_{H}^{K} whenever H ⩽ K ⩽ G, one sees that 𝔑_{H}^{G} = 1 on V whenever either of 𝔑_{K}^{G} or 𝔑_{H}^{K} is 1 on V.

In this section, we give some sufficient conditions for determining that the norm map 𝔑_{H}^{G} is constant for suitable p-local subgroups H of G, and then apply these results in Proposition 3.11 to obtain a proof of Theorem 1.1 for odd primes.

A subgroup A of G acts quadratically on V if [V, A, A] = 1 but [V, A] ≠ 1. In particular, when V is elementary abelian, each element of such a subgroup has quadratic or linear minimum polynomial in its action on V.

Lemma 3.1. Suppose that A is a p-group acting on an elementary abelian p-group V.

(a) if p is odd and A acts quadratically on V, then 𝔑_{A_{0}}^{A} = 1 on V for every proper subgroup A_{0} of A.

(b) if p = 2, then 𝔑_{A_{0}}^{A} = 1 on V for every subgroup A_{0} of A satisfying one of the following conditions:

(i) |A : A_{0}| ⩾ 2 and C_{V} (A_{0}) = C_{V} (A), or (ii) |A : A_{0}| ⩾ 4 and A acts quadratically on V.

Proof. We view elements of A as endomorphisms of V. Suppose first that p is odd. Let A_{0} ⩽ A_{1} ⩽ A with A_{1} of index p in A, and a ∈ A − A_{1}. Then (1 − a)^{2} = 0 in End(V) by assumption and pa = 0 in End(V) since V is elementary abelian. Hence

𝔑_{A_{1}}^{A}(v) = v^{1+a+···+a^{p−1}} = v^{(1−a)^{p−1}} = 1

for all v ∈ C_{V} (A_{1}), since p − 1 ⩾ 2.

Let p = 2. Under assumption (i), choose coset representatives {1, a} for a maximal subgroup A_{1} of A containing A_{0}, and then 𝔑_{A_{1}}^{A}(v) = vv^{a} = v^{2} = 1 for all v ∈ C_{V} (A_{1}) = C_{V} (A), since V is elementary abelian. Under assumption (ii), fix a subgroup A_{1} containing A_{0} with index 4 in A and choose coset representatives for A_{1} in A as {1, a, b, ab} for suitable a, b ∈ A − A_{1}. Then 1 + a + b + ab = (1 − a)(1 − b) = 0 in End(V) since the action is quadratic, and so 𝔑_{A_{1}}^{A}(v) = v^{1+a+b+ab} = 1 for all v ∈ C_{V} (A_{1}) as required. □

Theorem 3.2. Suppose G is a finite group, S ∈ Syl_{p}(G), and D is a p-group on which G acts. Let A be a nonempty set of subgroups of S, and set J = ⟨A⟩. Assume that J is weakly closed in S with respect to G and that

whenever A ∈ A, g ∈ G, A ≰ S^{g}, and V is a composition factor of D under G, then 𝔑_{A∩S^{g}}^{A} = 1 on V.

(3.3)

Then C_{D}(N_{G}(J)) = C_{D}(G).

Proof. This is Theorem A1.4 of [Gla71]. □

We refer to Theorem 3.2, and also to Theorem 4.1 below, as the norm argument for short. It is usually applied with p odd and in the presence of quadratic elements in G on D (cf. Lemma 3.1(a)).

Definition 3.4. For a general setup (Γ, S, Y), set D = Z(Y) and G = Γ/C_{Γ}(D). Let A be a set of abelian subgroups of G that is invariant under G-conjugation. For any subgroup H of G, set A ∩ H = {A ∈ A | A ⩽ H} and J_{A}(H) = ⟨A ∩ H⟩. For a subgroup H of Γ, we let J_{A}(H, D) be the preimage in H of J_{A}(HC_{Γ}(D)/C_{Γ}(D)). Often J_{A}(H, D) will be abbreviated to J_{A}(H) when D is understood.

The collection A of Definition 3.4 will generally be some subset of the collection of nontrivial best offenders in G on D, as defined below.

Definition 3.5. Let G be a finite group and let D be an abelian p-group on which G acts faithfully. An abelian p-subgroup A ⩽ G is an offender on D if

|A||C_{D}(A)| ⩾ |D|

and a best offender if

|A||C_{D}(A)| ⩾ |B||C_{D}(B)|

for every subgroup B ⩽ A. An offender A is nontrivial if A ≠ 1. We write A_{D}(G) for the collection of nontrivial best offenders in G on D.

A best offender is, in particular, an offender, as can be seen by taking B = 1 in the above definition. Conversely, each nontrivial offender A contains a nontrivial best offender, which can be obtained as a subgroup B such that the quantity |B||C_{D}(B)| is maximal among all nontrivial subgroups of A. In turn, by the Timmesfeld Replacement Theorem [Tim82], each nontrivial best offender contains a nontrivial quadratic best offender, namely a best offender that acts quadratically on D. We include a short proof of this, using the Thompson Replacement Theorem, in the form which is needed here.

Lemma 3.6. Suppose A is a nontrivial offender on D. Let B be a nontrivial subgroup of A that is minimal under inclusion subject to |B||C_{D}(B)| ⩾ |A||C_{D}(A)|. Then B is a quadratic best offender on D.

Proof. It follows from the choice of B that B is a best offender, so we need only show that it acts quadratically on D. We work in the semidirect product DB, where we set C = BC_{D}(B).

We first show that C is an abelian subgroup of DB of maximum possible order. Suppose that C_{1} is an abelian subgroup of DB such that |C_{1}| ⩾ |C|, and let B_{1} be the image of C_{1} under the projection of DB onto B. Then C_{1} ∩ D ⩽ C_{D}(C_{1}) = C_{D}(B_{1}), and so

|C| ⩽ |C_{1}| = |C_{1}/(C_{1} ∩ D)||C_{1} ∩ D| = |B_{1}||C_{1} ∩ D| ⩽ |B_{1}||C_{D}(B_{1})| ⩽ |B||C_{D}(B)| = |C|

with the last inequality since B is a best offender on D. Therefore equality holds everywhere, which yields

C_{1} ∩ D = C_{D}(B_{1}) and |C_{1}| = |C| = |B_{1}C_{D}(B_{1})| = |B||C_{D}(B)|. (3.7)

This shows that C is an abelian subgroup of maximal order in DB.

Note that if D normalizes C, then [D, C] ⩽ C and so [D, B, B] = [D, C, C] ⩽ [C, C] = 1 since C is abelian. Hence B acts quadratically on D in this case.

Suppose that D does not normalize C. Then by Thompson’s Replacement Theorem [Gor80, Theorem 8.2.5], there exists an abelian subgroup C_{1} of DB such that |C_{1}| = |C|, C_{1} ∩ D > C ∩ D, and C_{1} normalizes C. Take B_{1} as above. Then by (3.7),

|B_{1}||C_{D}(B_{1})| = |B||C_{D}(B)|

and

|B_{1}| = |B||C_{D}(B)|/|C_{D}(B_{1})| = |B||C ∩ D|/|C_{1} ∩ D| < |B|.

By the minimal choice of B, we have that B_{1} = 1 and therefore that C_{1} = C_{D}(B_{1}) = D. This shows that D normalizes C, a contradiction which completes the proof of the lemma. □

Lemma 3.8. Let (Γ, S, Y) be a general setup for the prime p, set D = Z(Y), and use bar notation for images modulo C_{Γ}(D). If Q is a subgroup of S containing C_{S}(D), then \overline{N_{Γ}(Q)} = NΓ̄(Q̄).

Proof. Let N be the preimage of NΓ̄(Q̄). Then C_{Γ}(D)N_{Γ}(Q) ⩽ N and we must show that N ⩽ C_{Γ}(D)N_{Γ}(Q). Now QC_{Γ}(D) is normal in N, and QC_{Γ}(D) ∩ S = QC_{S}(D) = Q is a Sylow p-subgroup of N. By the Frattini argument,

N = (QC_{Γ}(D))N_{N}(Q) ⩽ QC_{Γ}(D)N_{Γ}(Q) = C_{Γ}(D)N_{Γ}(Q)

as desired. □

The following proposition is a generalization of [Oli13, Proposition 3.2] for odd primes.

Proposition 3.9. Let (Γ, S, Y) be a general setup for the prime p. Set D = Z(Y), G = Γ/C_{Γ}(D), and F = F_{S}(Γ). Let A be a G-invariant collection of p-subgroups of G, each of which acts nontrivially and quadratically on D. Let R ⊆ 𝒮 (S)_{⩾Y} be an F-invariant interval such that Y ∈ R and J_{A}(S) ∉ R. If p is odd, then L^{1}(F; R) = 0.

Proof. Set Q = 𝒮 (S)_{⩾Y} − R. Let Γ^{∗} be the subset of Γ consisting of those g ∈ Γ for which there is Q ∈ Q with Q^{g} ∈ Q. Then Q and R are F-invariant intervals that satisfy the hypotheses of Lemma 2.6, so

(3.10) it suffices to show that C_{D}(Γ) = C_{D}(Γ^{∗})

by part (b) of that lemma.

As each element of A acts quadratically on D, Lemma 3.1(a) shows that (3.3) is satisfied. Hence, C_{D}(Γ) = C_{D}(N_{Γ}(J_{A}(S))) by Theorem 3.2 and Lemma 3.8 (where the latter applies by Definition 3.4). Since J_{A}(S) ∈ Q by assumption, we have N_{Γ}(J_{A}(S)) ⩽ Γ^{∗}. Hence

C_{D}(Γ) = C_{D}(N_{Γ}(J_{A}(S))) ⩾ C_{D}(Γ^{∗}) ⩾ C_{D}(Γ)

and (3.10) complete the proof. □

When compared with [Oli13, Proposition 3.2], the increased generality of Proposition 3.9 allows some simplifications in obtaining [Oli13, Proposition 3.3] when p is odd. We point out those simplifications now.

Proposition 3.11. Let (Γ, S, Y) be a general setup for the prime p. Set F = F_{S}(Γ), D = Z(Y), and G = Γ/C_{Γ}(D). Let R ⊆ 𝒮 (S)_{⩾Y} be an F-invariant interval such that for all Q ∈ 𝒮 (S)_{⩾Y}, Q ∈ R if and only if J_{A_{D}(G)}(Q) ∈ R. If p is odd, then L^{k}(F; R) = 0 for all k ⩾ 1.

Proof. Let (Γ, S, Y, R, k) be a counterexample for which the four-tuple (k, |Γ|, |Γ/Y |, |R|) is minimal in the lexicographic ordering. Steps 1 and 2 in the proof of [Oli13, Proposition 3.3] show that R = {P ⩽ S | J_{A_{D}(G)}(P) = Y } and k = 1 (since p is odd).

Let A be the set of nontrivial best offenders in G on D that are minimal under inclusion. Each best offender contains a member of A as a subgroup, and so J_{A_{D}(G)}(P) ∈ R if and only if J_{A}(P) ∈ R. By Lemma 3.6, each member of A acts quadratically on D.

If S ∈ R, then R = 𝒮 (S)_{⩾Y} since R is an interval, and L^{k}(F; R) = 0 for all k ⩾ 1 by Lemma 2.5(b). Hence S ∉ R and so J_{A}(S) ∉ R. Now Proposition 3.9 shows that (Γ, S, Y, R, 1) is not a counterexample. □

4. Norm arguments for p = 2

In this section, we define the notion of a solitary offender and prove the lemmas necessary to obtain Theorem 1.3. These results are used in §5 to give a proof of [Oli13, Proposition 3.2] except in the case where every minimal offender under inclusion is solitary.

The following version of Theorem 3.2 is easier to apply in applications for p = 2. The proof is very similar to that of Theorem 3.2 and is given in Appendix A.

Theorem 4.1. Suppose G is a finite group, S is a Sylow 2-subgroup of G, and D is an abelian p-group on which G acts. Let A be a nonempty set of subgroups of S, and set J = ⟨A⟩. Let H be a subgroup of G containing N_{G}(J). Set V = Ω_{1}(D). Assume that J is weakly closed in S with respect to G, and

(4.2) whenever A ∈ A, g ∈ G, and A ≰ H^{g}, then 𝔑_{A∩H^{g}}^{A} = 1 on V,

or more generally,

(4.3) whenever g ∈ G and J ≰ H^{g}, then 𝔑_{J∩H^{g}}^{J} = 1 on V.

Then C_{D}(H) = C_{D}(G).

Throughout the remainder of this section, we fix a finite group G, a Sylow 2-subgroup S of G, an abelian 2-group D on which G acts faithfully, and we set V = Ω_{1}(D).

Definition 4.4. Set

Â_{D}(G) = {A ∈ A_{D}(G) | |A||C_{D}(A)| > |D|},

the members of which are sometimes called over-offenders.

Denote by A_{D}(G)^{∘} the set of those members of A_{D}(G) that are minimal under inclusion, and denote by Â_{D}(G)^{∘} those members of Â_{D}(G) that are minimal under inclusion.

For a positive integer k, write

A_{D}(G)_{2} = {A ∈ A_{D}(G) | |A| = 2}; and
A_{D}(G)_{⩾4}^{∘} = {A ∈ A_{D}(G)^{∘} | |A| ⩾ 4}.

It may help to reiterate that a member of Â_{D}(G)^{∘}, while minimal under inclusion in the collection Â_{D}(G), may not be minimal under inclusion in A_{D}(G). By Lemma 3.6, each member of A_{D}(G)^{∘} acts quadratically on D.

Remark 4.5. Assume G is faithful on D and p = 2. If A ∈ A_{D}(G)_{2}, then |D/C_{D}(A)| = 2. In particular, every member of Â_{D}(G) is of order at least 4.

Lemma 4.6. Let A = A_{D}(G)_{⩾4}^{∘} ∪ Â_{D}(G)^{∘}. Assume that A is not empty and H is a subgroup of G containing N_{G}(J_{A}(S)). Then A ∩ S satisfies (4.2).

Proof. Fix A ∈ A ∩ S and g ∈ G with A ≰ H^{g}, and let A_{0} be a subgroup of A of index 2 that contains A ∩ H^{g}. Suppose 𝔑_{A∩H^{g}}^{A} is not 1 on V. Since 𝔑_{A∩H^{g}}^{A} = 𝔑_{A_{0}}^{A}𝔑_{A∩H^{g}}^{A_{0}}, we have C_{V} (A) < C_{V} (A_{0}) by Lemma 3.1. It follows that

|A_{0}||C_{D}(A_{0})| ⩾ \frac{1}{2}|A| · 2|C_{D}(A)| = |A||C_{D}(A)|. (4.7)

If A ∈ A_{D}(G)_{⩾4}^{∘}, then A is a best offender minimal under inclusion subject to A ≠ 1, so we have A_{0} = 1 and |A| = 2. This contradicts |A| ⩾ 4. Hence A ∈ Â_{D}(G)^{∘} and in particular |A| ⩾ 4. But then A_{0} ∈ Â_{D}(G) by (4.7), contradicting the minimality of A. □

Lemma 4.8. Let A = A_{D}(G)_{2}. Assume that Â_{D}(G) = ∅ and that A, B ∈ A. Then (a) if [A, B] = 1 and A ≠ B, then C_{D}(A) ≠ C_{D}(B) and AB is quadratic on D;

(b) if ⟨A, B⟩ is a 2-group, then [A, B] = 1;

(c) J_{A}(S) is elementary abelian; and

(d) if [A, B] ≠ 1, then L := ⟨A, B⟩ ≅ S_{3}, [D, L] is elementary abelian of order 4, and D = [D, L] × C_{D}(L).

Proof. Since A and B lie in A_{D}(G)_{2},

[D, A, A] = 1 = [D, B, B] and |D/C_{D}(A)| = 2. (4.9)

Suppose that [A, B] = 1 and A ≠ B, but that C_{D}(A) = C_{D}(B). Then AB is of order 4 and so C_{D}(C) = C_{D}(A) for every nontrivial subgroup C of AB. Since

|AB||C_{D}(AB)| = |A||B||C_{D}(A)| = 2|D| > |D|,

we have AB ∈ Â_{D}(G) contrary to assumption. Thus, the first statement of (a) holds. Since A ∈ A, [D, A] is B-invariant and of order 2. Thus [D, A, B] = 1. By symmetry [D, B, A] = 1. By a commutator identity and (4.9), [D, AB, AB] = 1, which establishes the second conclusion of (a).

Suppose that X := ⟨A, B⟩ is a 2-group and A ≠ B. Then |D/C_{D}(X)| = 4, and X acts on D/C_{D}(X). Since X is a 2-group, there is an X-invariant D_{1} with C_{D}(X) < D_{1} < D.

Set C = [A, B] and suppose that C ≠ 1. Provided it is shown that [D, A, B] = 1 = [D, B, A], then the Three Subgroups Lemma gives [C, D] = [A, B, D] = 1 and consequently the contradiction that C = 1 by faithfulness of the action. Thus, we may assume that

[D, A, B][D, B, A] ≠ 1. (4.10)

Since D_{1} is X-invariant and D/D_{1} is of order 2, we have [D, A] ⩽ D_{1} and [D, B] ⩽ D_{1}. Also, C_{D}(X) ⩽ C_{D}(A). If C_{D}(A) ≠ D_{1}, then [D, A] ⩽ C_{D}(A) ∩ D_{1} ⩽ C_{D}(X), and hence [D, A, B] = 1. Similarly, if C_{D}(B) ≠ D_{1}, then [D, B, A] = 1. Without loss of generality we may assume C_{D}(A) = D_{1} by (4.10). Then [C_{D}(A), A, B] = 1, and [C_{D}(A), B, A] = 1 as C_{D}(A)/C_{D}(X) is of order 2 and X-invariant. By the Three Subgroups Lemma, [C_{D}(A), C] = 1 and so C_{D}(A) ⩽ C_{D}(C). But then

C_{D}(A) = C_{D}(C) (4.11)

because C ≠ 1 and the action is faithful. Hence C ∈ A since Â_{D}(G) = ∅. In particular |C| = 2 and [A, C] = 1. Now (4.11) contradicts part (a) since A ≠ C, so (b) is established. Part (c) then follows immediately from (b).

To prove (d), let L = ⟨A, B⟩. Then L is a dihedral group, and L is not a 2-group by (b). Let K be the largest odd order subgroup of the cyclic maximal subgroup of L. Then

|D/C_{D}(K)| ⩽ |D/C_{D}(L)| ⩽ |D/C_{D}(A)||D/C_{D}(B)| = 4.

Since K has odd order and D is a 2-group, K acts faithfully on D/C_{D}(K) and D = [D, K]× C_{D}(K). So D/C_{D}(K) is elementary abelian of order 4, |K| = 3, and C_{D}(K) = C_{D}(L). Hence L acts faithfully on [D, K]. As Aut([D, K]) ≅ S_{3}, we see that L ≅ S_{3}. □

Definition 4.12. Let A = A_{D}(G)_{2}, J = J_{A}(S), and T ∈ A ∩ S. We say that T is solitary in G relative to S if there is a subgroup L of G containing T such that

(S1) L ≅ S_{3};

(S2) J = T × C_{J}(L) and C_{J}(L) = ⟨(A ∩ S) − {T }⟩; and (S3) D = [D, L] × C_{D}(L) and [D, L, C_{J}(L)] = 1.

For a subgroup S_{0} of S, we say that T ⩽ S_{0} is semisolitary relative to S_{0}, if there are subgroups W and X of D that are normalized by ⟨A ∩ S_{0}⟩, such that (SS2) ⟨A ∩ S_{0}⟩ = T × ⟨(A ∩ S_{0}) − {T }⟩; and (SS3) W is elementary abelian of order 4, D = W ×X, T centralizes X, and ⟨A∩S_{0}−{T }⟩

centralizes W. Denote by T_{D}(G) the collection of subgroups of G that are solitary relative to some

Sylow 2-subgroup of G. Likewise, a member of A_{D}(G)_{2} is said to be semisolitary if it is semisolitary relative to some Sylow 2-subgroup of G.

Remark 4.13. If Â_{D}(G) = ∅, then given a subgroup T which is solitary in G relative to S, and given L as in Definition 4.12, we may take W = [D, L] and X = C_{D}(L) and see from Lemma 4.8(d) that T is semisolitary relative to S.

A solitary offender is of order 2 (by definition), and thus is generated by a transvection when D is elementary abelian. If G = S_{3} and D = C_{2}×C_{2}, then one may take L = G to see that each subgroup of order 2 in G is solitary. More generally, if G a symmetric group of odd degree and D is a natural module for G, then each subgroup generated by a transposition is solitary. Indeed, given a Sylow 2-subgroup S of G containing the transposition, L may be taken in this case to move only three points, namely the two points moved by the transposition and the unique point fixed by S. On the other hand, (S2) implies that SL_{n}(2) (n ⩾ 3) and even degree symmetric groups, for example, have no solitary offenders on their respective natural modules, despite being generated by transvections. We refer Lemma 6.1 for more details.

There is no “(SS1)” in Definition 4.12 because we view (SS2) and (SS3) as weakenings of (S2) and (S3). The reader should view the introduction of semisolitary offenders as auxilliary. They are used in relative sitations when the connection between solitary offenders in G and those in a subgroup H is difficult to ascertain. The following elementary lemma addresses a similar uncertainty.

Lemma 4.14. Assume that Â_{D}(G) = ∅. Then T_{D}(G) ∩ S is the set of subgroups that are solitary in G relative to S.

Proof. One containment follows from the definition. For the other containment, set A = A_{D}(G)_{2} and assume T ∈ T_{D}(G)∩S. Suppose T is solitary relative to the Sylow 2-subgroup S_{1} of G, and let g ∈ G with S_{1}^{g} = S. Set J_{1} = ⟨A ∩ S_{1}⟩ and J = ⟨A ∩ S⟩. Then J_{1} is elementary abelian by Lemma 4.8, and T^{g} ⩽ J_{1}^{g} = J. Fix a subgroup L_{1} ≅ S_{3} containing T so that (S1)-(S3) holds with S_{1} and J_{1} in the roles of S and J, respectively. Since J is abelian and weakly closed in S with respect to G, there is h ∈ N_{G}(J) with T^{gh} = T (by Lemma B.1). Setting L = L_{1}^{gh}, one establishes the validity of (S1)-(S3) in Definition 4.12 for L, S, and J from their validity for L_{1}, S_{1}, and J_{1}. □

The following lemma will be applied later with J^{∗} equal to the subgroup generated by members of A_{D}(G)_{2} ∩ S that are not semisolitary relative to S.

Lemma 4.15. Set A = A_{D}(G)_{2}, T = T_{D}(G), and B = A − T . Assume (a) Â_{D}(G) = ∅; and

(b) B ≠ ∅. Then for each subgroup J^{∗} ⩽ J_{B}(S) that is weakly closed in S with respect to G, A ∩ S satisfies (4.3) with H = N_{G}(J^{∗}).

Proof. Set J = J_{A}(S) for short, and let J^{∗} be a subgroup of J_{B}(S) that is weakly closed in S with respect to G. Then J^{∗} ⩽ J_{B}(S) ⩽ J since B ⊆ A, and all of these are weakly closed in S with respect to G, since A and B are G-invariant. From (a) and Lemma 4.8(c)

(4.16) J is elementary abelian.

Since J^{∗} is weakly closed,

S ⩽ N_{G}(J) ⩽ N_{G}(J^{∗}) = H. (4.17)

Assume (4.3) is false. That is, let g ∈ G with J ≰ H^{g}, set I = J ∩ H^{g}, and suppose 𝔑_{I}^{J} ≠ 1 on V.

Suppose first that |J : I| ⩾ 4. Let J_{0} be a subgroup of J with I ⩽ J_{0} and |J : J_{0}| = 4. By (4.16), and since J is generated by members of A_{D}(G)_{2}, we may write J = J_{0} ×A×B with A, B ∈ A. Then 𝔑_{I}^{J} = 𝔑_{J_{0}}^{J}𝔑_{I}^{J_{0}} and so 𝔑_{J_{0}}^{J} ≠ 1 on V. However, AB acts quadratically on C_{V} (J_{0})/C_{V} (J) by Lemma 4.8(a). Thus, 𝔑_{J_{0}}^{J} = 1 on V by Lemma 3.1(b)(ii), a contradiction. We conclude that

|J: I| = 2. (4.18)

Fix T ∈ A with J = IT = I × T. Let A ∈ A ∩ S and suppose that A ≠ T. Let B = AT ∩ I. Then B is of order 2 and |D/C_{D}(B)| ⩽ |D/C_{D}(A)||D/C_{D}(T)| = 4. If |D/C_{D}(B)| = 4, then

C_{D}(I) ⩽ C_{D}(B) = C_{D}(A) ∩ C_{D}(T) ⩽ C_{D}(T).

Consequently, J = IT centralizes C_{D}(I), and C_{D}(I) = C_{D}(J). Hence, also C_{V} (I) = C_{V} (J). So 𝔑_{I}^{J} = 1 on V in this case by Lemma 3.1(b)(i), a contradiction. Thus, |D/C_{D}(B)| = 2. That is, B ∈ A. This shows

(4.19) if A ∈ A ∩ S, then A = T or there exists B ∈ A ∩ I such that AT = BT.

In particular, J = ⟨A ∩ S⟩ = ⟨A ∩ I⟩T, and so

I = ⟨A ∩ I⟩. (4.20)

Recall that I = J ∩ H^{g}, so that I^{g^{−1}} ⩽ H. Let h ∈ H such that I^{g^{−1}h} ⩽ S. Then we have

I^{g^{−1}h} = ⟨A ∩ I⟩^{g^{−1}h} ⩽ ⟨A ∩ S⟩ = J

by (4.20). As H^{h^{−1}g} = H^{g}, we may replace g by h^{−1}g for convenience so that

I = J ∩ H^{g} ⩽ J ∩ J^{g}. (4.21)

We now show that T is solitary in G relative to S. Since J^{g} ≠ J, we may choose U ∈ (A ∩ S)^{g} such that U ≰ J. Then U ⩽ J^{g}. Let L = ⟨T, U⟩. Since T ⩽ J and J is abelian,

[I, L] = 1 (4.22)

by (4.21).

We first show that T and U do not commute. Suppose on the contrary that [T, U] = 1. Then IL is a 2-group generated by members of A, and hence is conjugate to a subgroup of J. Then since J = IT ⩽ IL, we see that J = IL. Thus U ⩽ L ⩽ J, contrary to the choice of U.

Thus,

L ≅ S_{3}, |[D, L]| = 4, and D = [D, L] × C_{D}(L) (4.23)

by (a) and Lemma 4.8(d). Further, C_{J}(L) = I by the structure of L, and so J = T ×C_{J}(L) by (4.22).

Since [D, T] is of order 2 and I-invariant, [D, T, I] = 1. Since I^{x} = I and [D, T^{x}] = [D, T]^{x} ⩽ C_{D}(I^{x}) = C_{D}(I) for every x ∈ L,

[D, L] ⩽ C_{D}(I) = C_{D}(C_{J}(L)). (4.24)

We have shown that (S1), (S3), and half of (S2) hold in Definition 4.12; it remains to prove that for each A ∈ A ∩ S, either A = T or A ⩽ I. Fix A ∈ A ∩ S and suppose that A ≠ T and A is not contained in I. By (4.19), there is B ∈ A ∩ I with AT = BT. By (4.24), B centralizes [D, L]. From C_{D}(L) ⩽ C_{D}(T) and (4.23) it follows that

C_{D}(A) = C_{[D,L]}(T) × (C_{D}(L) ∩ C_{D}(B)).

Therefore,

|D/C_{D}(A)| = |C_{D}(L)/(C_{D}(L) ∩ C_{D}(B))| · |[D, L]/C_{[D,L]}(T)| = 2 · 2 = 4

contrary to A ∈ A. This contradiction shows that A ∩ S − {T } = A ∩ I and together with (4.20) completes the proof of (S2). Thus, T ∈ T .

Recall that J^{∗} ⩽ J_{B}(S) where B consists of the members of A that are not solitary in G relative to S. We have just shown that J_{B}(S) ⩽ I ⩽ J ∩J^{g}, so that (J^{∗})^{g^{−1}} ⩽ J_{B}(S)^{g^{−1}} ⩽ J. Since J^{∗} is weakly closed in S with respect to G, it follows that g ∈ N_{G}(J^{∗}) = H and consequently that J ⩽ H = H^{g}. This contradicts the choice of g and completes the proof of the lemma. □

Lemma 4.25. Let T be a subset of A_{D}(G)_{2}∩S and Y = {[D, T] | T ∈ T }. Fix A ∈ A_{D}(G) and set B = C_{A}(⟨Y⟩). Assume that every member of T is semisolitary relative to S. Then

(a) ⟨Y⟩ is the direct product of [D, T] for T ∈ T ; and

(b) if A acts transitively on T by conjugation, then either A = B, or |Y| = 2, B has index 2 in A, C_{D}(A) has index 2 in C_{D}(B), and each element of A − B induces a transposition on T .

Proof. Whenever T ∈ T , set Z_{T} = [D, T] for short. To prove (a),

(4.26) it suffices to show that Z_{T} ∩ ∏_{R≠T} Z_{R} = 1 for each T ∈ T .

For each R in T , let W_{R} and X_{R} be as in Definition 4.12 in the roles of W and X. For each R ∈ T ,

Z_{R} = [D, R] = [W_{R}X_{R}, R] = [W_{R}, R] ⩽ W_{R},

so for each T ∈ T different from R,

Z_{R} = [D, R] = [W_{T} X_{T} , R] = [X_{T} , R] ⩽ X_{T} , (4.27)

since R centralizes W_{T} and normalizes X_{T} by Definition 4.12. Therefore,

Z_{T} ∩ ∏_{R≠T} Z_{R} ⩽ W_{T} ∩ X_{T} = 1,

by (SS3) and part (a) now follows from (4.26).

Assume A acts transitively on T by conjugation. Then A acts on Y in the same way. Now ⟨Y⟩ is a transitive permutation module for A by part (a), so |C_{⟨Y⟩}(A)| = 2. Suppose A > B and set m = |A/B|. Then |Y| = m > 1 and |A||C_{D}(A)| ⩾ |B||C_{D}(B)|, so that

m = |A/B| ⩾ |C_{D}(B)/C_{D}(A)| ⩾ |C_{⟨Y⟩}(B)/C_{⟨Y⟩}(A)| = 2^{m}/2. (4.28)

Hence m = 2 and equality holds in (4.28), so |C_{D}(B)/C_{D}(A)| = 2. Since A ∈ A_{D}(G), an element of A − B must act as a transposition on Y and on T . □

Lemma 4.29. Let P ⩽ S and let T ⊆ A_{D}(G)_{2} ∩ P be the collection of semisolitary subgroups relative to P. If A ∈ A_{D}(G) normalizes P, then A normalizes every element of T .

Proof. Let Y = {[D, T] | T ∈ T }, let B = C_{A}(⟨Y⟩), and set Z_{T} = [D, T] whenever T ∈ T , for short. By assumption A acts on T . By Lemma 4.25(b), each orbit of A on T has size at most 2. We assume that {T, R} is a nontrivial orbit and aim for a contradiction. Since T is semisolitary relative to P, we may choose subgroups W and X of D such that D = W ×X, |W | = 4, T centralizes X, [W, T] = Z_{T} = C_{W} (T), and R centralizes W.

We claim that C_{W} (A) = 1; assume otherwise. Since A transposes Z_{T} and Z_{R} and T does not centralize W, it follows that that T does not centralize C_{W} (A). Therefore, [C_{W} (A), T] = Z_{T} and so for a ∈ A − N_{A}(T),

Z_{R} = (Z_{T})^{a} = [C_{W} (A), T^{a}] = [C_{W} (A), R] ⩽ [W, R] = 1,

a contradiction. Thus C_{W} (A) = 1. Since |W | = 4, we have

|C_{D}(B)/C_{D}(A)| ⩾ 4

contrary to Lemma 4.25. □

5. Reduction to the transvection case

For this section, we fix a finite group Γ with Sylow p-subgroup S, we set F = F_{S}(Γ), and we let Q ⊆ F^{c} be an F-invariant interval such that S ∈ Q. In this situation, define Γ^{∗} to be the set of elements of Γ that conjugate some member of Q into Q.

The objective of this section is to give a proof of [Oli13, Proposition 3.2] in the case where some minimal offender under inclusion is not solitary. This result is obtained in Theorem 5.4.

We say that a 1-cocycle for the functor Z_{F}^{Q} is inclusion-normalized if it sends the class [ι_{P}^{Q}] ∈ Mor_{O(F^{c})}(P, Q) of any inclusion ι_{P}^{Q} to the identity element of Z(P) for each P, Q ∈ Q.

In what follows, we only specify 0- and 1-cochains for the functor Z_{F}^{Q} on subgroups in Q, and it is to be understood that they are the identity on F-centric subgroups outside Q. Alternatively, apply the isomorphism of cochain complexes in Lemma 2.5(a) to view these cochains as restrictions to the full subcategory of O(F^{c}) with objects in Q.

The reader may wish to recall the coboundary maps for 0- and 1-cochains in our righthanded notation from (2.2) and (2.3).

Lemma 5.1. Each 1-cocycle for Z_{F}^{Q} is cohomologous to an inclusion-normalized 1-cocycle. If t is an inclusion-normalized 1-cocycle, then

(a) t([φ_{1}]) = t([φ_{2}]) for each commutative diagram

\xymatrix{P_{2} \ar[r]^{φ_{2}} & Q_{2} \\ P_{1} \ar[u]^{ι_{P_{1}}^{P_{2}}} \ar[r]^{φ_{1}} & Q_{1} \ar[u]_{ι_{Q_{1}}^{Q_{2}}}}

in F among subgroups of Q;

(b) the function τ: Γ^{∗} → Γ^{∗} defined by the rule

g^{τ} = t([c_{g}])g,

is a bijection that restricts to the identity map on S, and

(g_{1}g_{2} ⋯ g_{n})^{τ} = g_{1}^{τ}g_{2}^{τ} ⋯ g_{n}^{τ}

for each collection of elements g_{i} ∈ Γ^{∗} with the property that there is Q ∈ Q such that Q^{g_{1}···g_{i}} ∈ Q for all 1 ⩽ i ⩽ n; and

(c) t = 0 if and only if τ is the identity on Γ^{∗}.

Proof. Given a 1-cocycle for Z_{F}^{Q}, define a 0-cochain u by u(P) = t([ι_{P}^{S}]) for each P ∈ Q. Then for any inclusion ι_{P}^{Q} in F with P, Q ∈ Q, we see that

du(P ⟶^{[ι_{P}^{Q})]} Q) = u(Q)u(P)^{−1} = t([ι_{Q}^{S}])t([ι_{P}^{S}])^{−1},

so that

(t du)(P ⟶^{[ι_{P}^{Q}]} Q) = t([ι_{P}^{Q}])t([ι_{Q}^{S}])t([ι_{P}^{S}])^{−1} = t([ι_{P}^{S}])t([ι_{P}^{S}])^{−1} = 1

by the 1-cocycle identity. Hence, t du is inclusion-normalized.

Assume now that t is inclusion-normalized, and let P_{i}, Q_{i}, and φ_{i} be as in (a). Since t sends inclusions to the identity, the 1-cocycle identity yields

t([φ_{1}ι_{Q_{1}}^{Q_{2}}]) = t([ι_{Q_{1}}^{Q_{2}}])^{φ_{1}^{−1}}t([φ_{1}]) = t([φ_{1}]), and
t([ι_{P_{1}}^{P_{2}}φ_{2}]) = t([φ_{2}])^{(ι_{P_{1}}^{P_{2}})^{−1}}t([ι_{P_{1}}^{P_{2}}]) = t([φ_{2}])

and so (a) follows by commutativity of the diagram.

Let τ be given by (b). Since g ∈ Γ^{∗}, the conjugation map c_{g}: ^{g}S ∩ S → S ∩ S^{g} is a map between subgroups in Q. Part (a) shows that t([c_{g}]) agrees with the value of t on the class of each restriction of c_{g} provided that the source and target of such a restriction lie in Q. This shows that τ is well defined. Then τ is a bijection since its inverse is induced by t^{−1} (which is inclusion-normalized) in the same way. Further, for s ∈ S, [c_{s}] = [id_{S}] is the identity in the orbit category, and so τ is the identity map on S, since t is normalized.

Let g_{1}, g_{2} ∈ Γ^{∗} and Q ∈ Q with Q^{g_{1}} ⩽ S and Q^{g_{1}g_{2}} ⩽ S. Then by the 1-cocycle identity,

g_{1}^{τ}g_{2}^{τ} = t([c_{g_{1}}])g_{1} t([c_{g_{2}}])g_{2} = t([c_{g_{1}}])t([c_{g_{2}}])^{g_{1}^{−1}} g_{1}g_{2} = t([c_{g_{1}}c_{g_{2}}]) g_{1}g_{2} = (g_{1}g_{2})^{τ}.

Now (b) follows by induction on n. Part (c) is clear. □

The function τ of Lemma 5.1(b) will be called the rigid map associated with the inclusionnormalized 1-cocycle t.

Lemma 5.2. Let t be an inclusion-normalized 1-cocycle for the functor Z_{F}^{Q} and let τ be the rigid map associated with t.

(a) For each Q ∈ Q ∩ F^{f}, there is z ∈ Z(N_{S}(Q)) such that τ is conjugation by z on N_{Γ}(Q);

(b) if z ∈ Z(S) and u is the constant 0-cochain defined by u(Q) = z for each Q ∈ Q, then du is inclusion-normalized and the rigid map υ associated with du is conjugation by z on Γ^{∗}; and

(c) if C ⊆ F^{f} is a conjugation family for F and τ is the identity on N_{Γ}(Q) for each Q ∈ C ∩ Q, then τ is the identity on Γ^{∗}.

Proof. We give two proofs for part (a). The first one uses elementary group-theoretic arguments and the norm map, and is given by Lemma B.2 in the appendix. The second is modeled on part of the proof of [AOV12, Lemma 4.2] and given now. From Lemma 5.1(b), τ induces an automorphism of N_{Γ}(Q) that is the identity on N_{S}(Q). By [OV09, Lemma 1.2] and its proof, there is a commutative diagram

\xymatrix{1 \ar[r] & Z^{1}(Out_{Γ}(Q); Z(Q)) \ar[r]^{η̃} \ar[d] & Aut(N_{Γ}(Q), Q) \ar[r] \ar[d] & Aut(Q) \ar[d] \\ 1 \ar[r] & H^{1}(Out_{Γ}(Q); Z(Q)) \ar[r]^{η} & Out(N_{Γ}(Q), Q) \ar[r] & Out(Q)}

(5.3)

with exact rows, where Aut(N_{Γ}(Q), Q) is the subgroup of automorphisms of N_{Γ}(Q) that leave Q invariant (and similarly for Out(N_{Γ}(Q), Q)). Also, η̃ maps the restriction of t (to Out_{Γ}(Q)) to the restriction of τ to N_{Γ}(Q). The restriction map H^{1}(Out_{F}(Q); Z(Q)) → H^{1}(Out_{S}(P); Z(P)) is injective since Out_{S}(Q) is a Sylow p-subgroup of Out_{F}(Q) by assumption on Q. Hence t represents the zero class in H^{1}(Out_{F}(P); Z(P)) since t is zero on Out_{S}(P). It then follows from (5.3) that τ induces an inner automorphism of N_{Γ}(Q). Hence τ is conjugation by an element in Z(N_{S}(Q)), since Q ∈ F^{c} and τ is the identity on N_{S}(Q).

With u as in (b), we see that du(P ⟶^{[ι_{P}^{Q}]} Q) = u(Q)u(P)^{−1} = zz^{−1} = 1, for any inclusion among subgroups when P ∈ Q (and when P ∉ Q by (2.2)). Also, for g ∈ Γ^{∗},

g^{υ} = du([c_{g}])g = z^{−1}z^{g^{−1}}g = g^{z}

so (b) holds.

Part (c) follows directly from Lemma 5.1(b); we give the details. Fix a conjugation family C ⊆ F^{f}, and suppose that τ is the identity on N_{Γ}(T) for each T ∈ C. Fix g ∈ Γ^{∗}, and choose Q ∈ Q with Q^{g} ⩽ S. Then there are a positive integer n, subgroups T_{1}, …, T_{n} ∈ C, and elements g_{i} ∈ N_{Γ}(T_{i}) such that g = g_{1} ⋯ g_{n}, Q ⩽ T_{1}, and Q^{g_{1}···g_{i}} ⩽ T_{i+1} for each i = 1, …, n − 1. As Q is F-invariant and closed under passing to overgroups, T_{i} ∈ Q for each i. Since τ fixes g_{i} for each i by assumption, τ fixes g by Lemma 5.1(b). □

Theorem 5.4. Let (Γ, S, Y) be a reduced setup for the prime 2. Set D = Z(Y), F = F_{S}(Γ), G = Γ/C_{Γ}(D), A = A_{D}(G)^{∘} ∪ Â_{D}(G)^{∘}, T = T_{D}(G), B = A − T , and

R = {P ∈ 𝒮 (S)_{⩾Y} | J_{A}(P) = Y }.

Assume B is not empty. Then L^{2}(F; R) = 0.

Proof. If R = 𝒮 (S)_{⩾Y}, then L^{2}(F; R) = 0 by Lemma 2.5(b) so we may assume Q := 𝒮 (S)_{⩾Y} − R is not empty. That is, A is not empty. Since Q is closed under passing to overgroups, S ∈ Q and J_{A}(Q) ∈ Q for each Q ∈ Q.

We will show L^{1}(F; Q) = 0. Since Q and R are F-invariant intervals that together satisfy the hypotheses of Lemma 2.6, the result then follows from part (a) of that lemma. Fix a 1-cocycle t for the functor Z_{F}^{Q}. To show that t is cohomologous to 0, we may assume by Lemma 5.1 that t is inclusion-normalized. Let τ: Γ^{∗} → Γ^{∗} be the rigid map associated with t.

The proof splits into two cases. In Case 1, some member of A has order at least 4. In Case 2, every member of A has order 2. We now fix notation for each case. Use bars to denote images modulo C_{Γ}(D). If P is a Sylow 2-subgroup of a subgroup H ⩽ Γ such that every member of A∩P̄ has order 2, then define B(P, H) to be the collection of subgroups in A ∩ P̄ that are not solitary in H̄ relative to P̄. In any situation, let B(P) denote the set of subgroups in A∩P̄ that are not semisolitary relative to P̄, and set A_{⩾4} = {A ∈ A | |A| ⩾ 4}.

Define

J_{1}(P) = J_{A_{⩾4}}(P); and
J_{2}(P) = J_{A_{⩾4}∪B(P)}(P),

so that Y ⩽ J_{1}(P) ⩽ J_{2}(P) ⩽ J_{A}(P) whenever P ⩾ Y. We define two subgroup mappings W_{1} and W_{2} on 𝒮 (S), to be employed in the respective cases. In all cases, set W_{i}(P) = P if P does not contain Y. For P ⩾ Y, set

W_{1}(P) = \begin{cases}J_{1}(P) & if A_{⩾4} ∩ P̄ ≠ ∅; \\ J_{2}(P) & if A_{⩾4} ∩ P̄ = ∅ and B(P) ≠ ∅; and \\ J_{A}(P) & otherwise, \end{cases}

and

W_{2}(P) = \begin{cases}J_{B(S,Γ)}(S) & whenever J_{B(S,Γ)}(S) ⩽ P; and \\ J_{A}(P) & otherwise. \end{cases}

In any case, W_{i}(P) ∈ Q and W_{i}(W_{i}(P)) = W_{i}(P) whenever P ∈ Q.

Set W = W_{i} for i = 1 or 2. Then W (S) is normal in S, so the restriction of τ to N_{Γ}(W (S)) is conjugation by an element z ∈ Z(S) by Lemma 5.2(a). Upon replacing t by t du where u is the constant 0-cochain defined by u(Q) = z^{−1} for each Q ∈ Q, and upon replacing τ by the rigid map associated with t du, we may assume by Lemma 5.2(b) that

5.4.1. τ is the identity on N_{Γ}(W (S)).

Now W_{1} is a Γ-conjugacy functor on 𝒮 (S) in the sense of Appendix A, since W_{1}(P) ≠ 1 whenever P ≠ 1 and since all subcollections used in defining W_{1} are G-invariant. In case W = W_{2} and A = A_{D}(G)_{2}, W_{2} is a Γ-conjugacy functor on 𝒮 (S) as then B(S, Γ) is weakly closed in S with respect to Γ by Lemma 4.14. By Theorem B.6, the collection C of subgroups of S that are well-placed with respect to W forms a conjugation family for F. If a subgroup P is well-placed with respect to W, then so is W (P) because W (W (P)) = W (P). Set

W = {Q ∈ C ∩ Q | W (Q) = Q}.

If Q ∈ C ∩ Q, then W (Q) ∈ W, and N_{Γ}(Q) ⩽ N_{Γ}(W (Q)). Therefore, it follows from Lemma 5.2(c) that

5.4.2. if τ is the identity on N_{Γ}(Q) for each Q ∈ W, then τ is the identity on Γ^{∗}.

For each Q ∈ W, the restriction of τ to H = N_{Γ}(Q) is conjugation by an element z_{H} ⩽ Z(N_{S}(Q)) ⩽ Z(Y) = D by Lemma 5.2(a). Hence, if N_{H}(W (N_{S}(Q))) fixes z_{H}, and and this normalizer controls fixed points in H on D, then τ is the identity on H. We record this important observation as follows.

5.4.3. Assume Q ∈ W and H = N_{Γ}(Q). If τ is the identity on N_{Γ}(W (N_{S}(Q))) and C_{D}(H) = C_{D}(N_{H}(W (N_{S}(Q))), then τ is the identity on H.

We now distinguish between the two cases. In each case, we prove that τ is the identity on N_{Γ}(Q) for each Q ∈ W by induction on the index of N_{S}(Q) in S. Fix Q ∈ Q and set S^{∗} = N_{S}(Q) and H = N_{Γ}(Q) for short.

The norm arguments in §4 are statements about control of fixed points in G. Each member of Q contains Y = C_{S}(D) since (Γ, S, Y) is a reduced setup, and so Lemma 3.8 provides the transition from control of fixed points by normalizers within G and those within Γ. We apply Lemma 3.8 implicitly for this transition in the arguments that follow.

Case 1: Some member of A has order at least 4.

Put W = W_{1}. Assume first that S^{∗} = S. Since Case 1 holds, the collection A_{⩾4} = A_{D}(G)_{⩾4}^{∘}∪Â_{D}(G)^{∘} is not empty. Hence W (S^{∗}) = J_{1}(S) by definition of W. By Lemma 4.6, (4.2) is satisfied with (H̄, S̄, D, A_{⩾4}∩H̄, NH̄(\overline{J_{1}(S)})) in the role of the five-tuple (G, S, D, A, H) of that lemma. Hence C_{D}(H) = C_{D}(N_{H}(J_{1}(S))) by Theorem 4.1. Further, τ is the identity on N_{Γ}(W (S)) = N_{Γ}(J_{1}(S)) by 5.4.1, so that τ is the identity on H by 5.4.3.

Assume now that S^{∗} < S. If A_{⩾4} ∩ S̄^{∗} ≠ ∅ (that is, if J_{1}(S^{∗}) > Y), then W (S^{∗}) = J_{1}(S^{∗}), and C_{D}(H) = C_{D}(N_{H}(W (S^{∗}))) by Lemma 4.6 and Theorem 4.1 as before. Since S^{∗} < N_{S}(W (S^{∗})) (by Lemma B.5(c)), τ is the identity on N_{Γ}(W (S^{∗})) by induction, so that τ is the identity on H by 5.4.3.

Assume for the remainder of Case 1 that A_{⩾4} ∩ S̄^{∗} is empty. In particular, Â_{D}(G) ∩ S̄^{∗} is empty. However, A ∩ S̄^{∗} is not empty since S^{∗} ⩾ Q ∈ Q, and every member of A ∩ S̄^{∗} is of order 2. Moreover,

(5.5) Q = W (Q) ⩽ J_{A}(Q) ⩽ J_{A}(S^{∗}) and \overline{J_{A}(S^{∗})} is elementary abelian

from Lemma 4.8(c).

Assume next that B(S^{∗}) is not empty. Then W (S^{∗}) = J_{2}(S^{∗}) > Y. Since no member of B(S^{∗}) is solitary in H̄ relative to S̄^{∗} by Remark 4.13, we see that B(S^{∗}) ⊆ B(S^{∗}, H), and that

W (S^{∗}) = J_{2}(S^{∗}) = J_{B(S^{∗})}(S^{∗}) ⩽ J_{B(S^{∗},H)}(S^{∗}).

Since \overline{J_{A}(S^{∗})} is elementary abelian and semisolitary subgroups relative to S^{∗} are invariant under conjugation in NH̄(\overline{J_{A}(S^{∗})}), Lemma B.1 shows that W (S^{∗}) is weakly closed in S^{∗} with respect to H. Apply Lemma 4.15 with (H̄, S̄^{∗}, D, A ∩ S̄^{∗}, \overline{W (S^{∗})}) in the role of the five-tuple (G, S, D, A, J^{∗}) of that lemma to obtain the hypotheses of Theorem 4.1, which gives C_{D}(H) = C_{D}(N_{H}(W (S^{∗}))) as before. However, S^{∗} < N_{S}(W (S^{∗})) since S^{∗} < S (by Lemma B.5(c)). Since τ is the identity on N_{Γ}(W (S^{∗})) by induction, we have that τ is the identity on H by 5.4.3.

Finally, assume that B(S^{∗}) is empty. Then W (S^{∗}) = J_{A}(S^{∗}) by definition of W, and every element of A ∩ S̄^{∗} is semisolitary relative to S̄^{∗}. We will show this leads to a contradiction – this is a critical step in the proof. Since Case 1 holds, J_{A}(S^{∗}) < J_{A}(S). Now J_{A}(S^{∗}) < J_{A}(N_{S}(J_{A}(S^{∗}))) (by Lemma B.5(b)), and so there exists A ⩽ S with Ā ∈ A ∩ S̄ such that

A ⩽ J_{A}(N_{S}(J_{A}(S^{∗}))), but A ≰ J_{A}(S^{∗}). (5.6)

It follows from the definitions that each subgroup of S^{∗} that is semisolitary relative to S^{∗} is also semisolitary relative to J_{A}(S^{∗}). As A normalizes J_{A}(S^{∗}), we see that Ā permutes the elements of A ∩ S̄^{∗} = A ∩ \overline{J_{A}(S^{∗})} by conjugation. We are thus in the situation of Lemma 4.29 with \overline{J_{A}(S)} in the role of P, and A ∩ S̄^{∗} in the role of T there. By that lemma, Ā normalizes every member of A ∩ S̄^{∗}. Thus A normalizes each of their preimages in S. However Q is generated by the preimages of a subset of A ∩ S̄^{∗} by (5.5), where W (Q) = J_{A}(Q) in the current situation. Hence A normalizes Q. But then A ⩽ J_{A}(S^{∗}), contrary to (5.6). This contradiction completes the proof of Case 1.

Case 2: Each member of A is of order 2.

Put W = W_{2} and assume first that S^{∗} = S. By assumption, A = A_{D}(G)_{2}, and so Â_{D}(G) = ∅ by Remark 4.5. Set J = J_{B(S,H)}(S) and note that W (S) = J_{B(S,Γ)}(S) by definition of W_{2}. By Definition 4.12, every element of A that is solitary in H̄ relative to S̄ is also solitary in G relative to S̄. Thus B(S, Γ) ⊆ B(S, H), and

W (S) ⩽ J ⩽ J_{A}(S).

We saw earlier (after 5.4.1) that W (S) is weakly closed in S with respect to Γ whenever A = A_{D}(G)_{2}, which holds in the present case. So we may apply Lemma 4.15 with (H̄, S̄, D, A∩H̄, \overline{W (S)}) in the role of the five-tuple (G, S, D, A, J^{∗}) of that lemma to obtain the hypotheses of Theorem 4.1, which yields C_{D}(H) = C_{D}(N_{H}(W (S))). Further, τ is the identity on N_{Γ}(W (S)) by 5.4.1, so that τ is the identity on H by 5.4.3.

Assume now that S^{∗} < S. As \overline{J_{A}(S)} is elementary abelian and Q ∈ W,

Q = W (Q) ⩽ J_{A}(Q) ⩽ J_{A}(S),

and \overline{J_{A}(S)} centralizes Q̄. Hence J_{A}(S) ⩽ S^{∗} so that

J_{A}(S^{∗}) = J_{A}(S).

Since J_{B(S,Γ)}(S) ⩽ J_{A}(S), this shows that W (S^{∗}) = J_{B(S,Γ)}(S) = W (S). As in the situation where Q was normal in S, we have that B(S, Γ) ⊆ B(S^{∗}, H) and that W (S^{∗}) is weakly closed in S^{∗} with respect to H. Apply Lemma 4.15 with (H̄, S̄^{∗}, D, A ∩ H̄, \overline{W (S^{∗})}) in the role of the five-tuple (G, S, D, A, J^{∗}) of that lemma to obtain the hypotheses of Theorem 4.1, which yields C_{D}(H) = C_{D}(N_{H}(W (S^{∗}))) as before. Further, since W (S^{∗}) = W (S), τ is the identity on N_{Γ}(W (S^{∗})) by 5.4.1, so that τ is the identity on H by 5.4.3. This concludes the proof in Case 2.

The theorem now follows from 5.4.2 and Lemma 5.1(c). □

6. Transvections

The aim of this section is to give a proof of Proposition 3.3 of [Oli13] for p = 2 in Theorem 6.9. This result and the proof of [Oli13, Theorem 3.4] give Theorem 1.1 when p = 2.

Using McLaughlin’s classification of irreducible subgroups of SL_{n}(2) generated by transvections, we first classify in Theorem 6.2 those finite groups which have no nontrivial normal 2-subgroups and are generated by solitary offenders. Recall that by a natural S_{m}-module (m ⩾ 3), we mean the nontrivial composition factor of the standard permutation for S_{m} over the field with two elements.

Lemma 6.1. Let G be a finite group acting irreducbily on an elementary abelian 2-group W. Assume that G is generated by transvections. Then T_{W} (G) is not empty if and only if G is isomorphic to a symmetric group of odd degree and W is a natural module for G. Moreover, in this case, T_{W} (G) is the set of transpositions.

Proof. Assume first that G is generated by transvections on the irreducible module W. By a result of McLaughlin [McL69], G is isomorphic to SL(W), or the dimension n of W is even and at least 4 and G is isomorphic to Sp(W), O^{−}(W), O^{+}(W), S_{n+1}, or S_{n+2}. For the classical groups, A_{W} (G)_{2} is the set of transvections; for the symmetric groups, A_{W} (G)_{2} is the set of transpositions. In all cases, A_{W} (G)_{2} is a single G-conjugacy class.

Fix a Sylow 2-subgroup S of G, and assume first that G = SL(W) with n ⩾ 3. Since A_{W} (G)_{2} is a single conjugacy class, either T_{W} (G) is empty or T_{W} (G) = A_{W} (G)_{2}. Since S is itself generated by transvections, (S2) forces S to be abelian in the latter case, a contradiction.

Assume that G is a symmetric group of degree n + 2 ⩾ 6. We may assume S stabilizes the partition {{1, 2}, …, {n + 1, n + 2}}, and then ⟨A_{W} (G)_{2} ∩ S⟩ is the centralizer of this partition. Fix A = ⟨(2j − 1, 2j)⟩ ∈ A_{W} (G)_{2} ∩ S, and let L ≅ S_{3} be a subgroup of G containing A. Then all members of A_{W} (G)_{2} ∩ L are conjugate, and so the support of L is a three-element set, say {2j − 1, 2j, k}. Hence L does not centralize the element of A_{W} (G)_{2} ∩ S moving k, and thus A is not solitary in G relative to S.

Assume that G = Sp(W) preserves the symplectic form 𝔟. Fix a maximal isotropic subspace W_{0} stabilized by S, and let U be the unipotent radical of its stabilizer in G. Then all members of A_{W} (G)_{2} ∩ S are contained in U. Let A be one of them, having center ⟨e⟩ ⊆ W_{0}, and let L ≅ S_{3} be a subgroup of G containing A. Since two symplectic transvections commute if and only if their centers are orthogonal with respect to 𝔟, [W, L] is a hyperbolic line. Since n ⩾ 4, we may find e′ ∈ [W, L]^{⊥} ∩ W_{0} − ⟨e⟩, and then L does not centralize the member of A_{W} (G) ∩ S with center ⟨e + e′⟩. So A is not solitary.

Assume that G is an orthogonal group preserving the quadratic form 𝔮 with associated symplectic form 𝔟. If n = 4, then since G is generated by transvections, G ≅ S_{5}. Thus, we may assume that n ⩾ 6. Choose a maximal isotropic subspace W_{0} (with respect to 𝔟) stabilized by S and such that W_{0} contains a nonsingular vector. Let U be the unipotent radical of the stabilizer of the radical of 𝔮|_{W_{0}}. Fix a nonsingular vector e ∈ W_{0}, let A ⩽ U be generated by the transvection with center ⟨e⟩, and let L ≅ S_{3} be a subgroup of G such that L contains A. As before, the restriction of 𝔟 to [W, L] is nondegenerate, and as L acts on [W, L], 𝔮|_{[W,L]} is of minus type. As dim(W_{0}) ⩾ 3, there is a singular vector e′ in [W, L]^{⊥} ∩ W_{0}, and then L does not centralize the orthogonal transvection with center ⟨e + e′⟩. So A is not solitary.

Therefore, G is a symmetric group of odd degree and W is a natural module for G. For the converse, let G = S_{2n+1} (n ⩾ 1), Ω the standard G-set, and identify W with

the set of even order subsets of Ω. To show that each transposition generates a solitary subgroup, we may restrict our attention to T = ⟨(2n − 1, 2n)⟩. Consider the partition {{2i − 1, 2i} | 1 ⩽ i ⩽ n} of Ω − {2n + 1}, and let S be a Sylow 2-subgroup of G stabilizing this partition. Then A_{W} (G)_{2} ∩ S = {⟨(1, 2)⟩, …, ⟨(2n − 1, 2n)⟩}. Hence, taking L be the symmetric group induced on {2n − 1, 2n, 2n + 1}, it is easily checked that (S1)-(S3) hold in Definition 4.12, so that T is solitary in G relative to S. □

Theorem 6.2. Let G be a finite group, let D be an abelian 2-group on which G acts faithfully, and set T = T_{D}(G). Assume that O_{2}(G) = 1 and that G = ⟨T ⟩ is generated by its solitary offenders. Then there exist a positive integer r and subgroups E_{1}, …, E_{r} such that

(a) G = E_{1} × ⋯ × E_{r}, T = (T ∩ E_{1}) ∪ ⋯ ∪ (T ∩ E_{r}), and E_{i} ≅ S_{m_{i}} with m_{i} odd for each i; and (b) D = V_{1} × ⋯ × V_{r} × C_{D}(G) with V_{i} = [D, E_{i}] a natural S_{m_{i}}-module, and with

[V_{i}, E_{j}] = 1 for j ≠ i.

Proof. Set W = [D, G]C_{D}(G)/C_{D}(G), and let (G, D) be a counterexample with |G| + |D| minimal. We show in Step 1 that [D, G, G] = [D, G], in Step 2 that C_{D/C_{D}(G)}(G) = 1, in Step 3 that G is faithful on W with W = [W, G] and C_{W} (G) = 1, in Step 4 that the theorem for (G, W) implies the theorem for (G, D), and in Step 5 that W is irreducible. The theorem then follows from Lemma 6.1. We prefer to give essentially complete proofs from first principles.

When D > D_{1} > ⋯ > D_{k} > 1 is a chain of G-invariant subgroups of D (or of any other abelian p-group with faithful action from G) we say for short that a subgroup K of G acts nilpotently on the chain if K acts trivially on successive quotients. If a subgroup K acts nilpotently on such a chain and k ⩾ 1, then K ⩽ O_{2}(G) by [Gor80, Theorem 5.3.3]. Step 1: For each T ∈ T , choose a subgroup L of G such that L ⩾ T and L ≅ S_{3} as in Definition 4.12. Then [D, T] ⩽ [D, L] ⩽ [D, G] and [D, L, L] = [D, L] by (SS3), so that [D, T] ⩽ [D, L, L] ⩽ [D, G, G]. Hence T centralizes G/[D, G, G]. It follows that G centralizes G/[D, G, G] since G = ⟨T ⟩ and the choice of T was arbitrary. That is, [D, G] ⩽ [D, G, G]. Since the reverse inclusion holds, we conclude that [D, G] = [D, G, G]. Step 2: Let D_{1} be the preimage of C_{D/C_{D}(G)}(G) in D. Suppose that D_{1} > C_{D}(G). Then C_{D}(G) > 1. As before fix T ∈ T and choose L as in Definition 4.12 for T. Then O^{2}(L) acts nilpotently on the chain D_{1} > C_{D}(G) > 1. Since O^{2}(L) is of odd order, it centralizes D_{1}. Hence, by (S3),

D_{1} ⩽ C_{D}(O^{2}(L)) = C_{D}(L) ⩽ C_{D}(T).

That is, T centralizes D_{1}. We conclude that G centralizes D_{1} since G = ⟨T ⟩, and since the choice of T was arbitrary. This contradicts D_{1} > C_{D}(G). Therefore, D_{1} = C_{D}(G). Step 3: Set W = [D, G]C_{D}(G)/C_{D}(G) as above.

Using faithfulness of G on D and the assumption O_{2}(G) = 1, we see that

W > 1 (6.3)

since otherwise G acts nilpotently on D > C_{D}(G) > 1. Let K be the kernel of the action of G on W. Then K acts nilpotently on the chain D ⩾ [D, G]C_{D}(G) > C_{D}(G) ⩾ 1, with the strict inclusion from (6.3), and hence K ⩽ O_{2}(G) = 1. We conclude that

(6.4) G is faithful on W .

By Steps 1 and 2,

W = [W, G] and C_{W} (G) = 1. (6.5)

Step 4: Assume now that G is a direct product of symmetric groups E_{i} of odd degree m_{i} and that W is a direct sum of natural modules W_{i} for E_{i} (1 ⩽ i ⩽ r) satisfying [W_{i}, E_{j}] = 1 whenever j ≠ i. Then part (a) holds for G and D, so we need only show part (b). Set V_{1} = [D, E_{1}]. Then under the projection from D onto D/C_{D}(G), V_{1} maps onto [W, E_{1}] = W_{1}. On the other hand, we may choose m_{1} − 1 elements T_{1}, …, T_{m_{1}−1} ∈ T ∩ E_{1} corresponding to adjacent transpositions that generate E_{1}, and see that

|V_{1}| = |⟨[D, T_{1}], …, [D, T_{m_{1}−1}]⟩| ⩽ ∏_{1⩽i⩽m_{1}−1} |[D, T_{i}]| = 2^{m_{1}−1} = |W_{1}|,

since |[D, T_{i}]| = 2 for each 1 ⩽ i ⩽ m_{1} − 1. Hence V_{1} ≅ W_{1} is a natural E_{1}-module. It follows that C_{D}(E_{1}) ∩ V_{1} = 1. Moreover,

|D/C_{D}(E_{1})| = |D/(∩_{i}C_{D}(T_{i}))| ⩽ ∏_{1⩽i⩽m_{1}−1} |D/C_{D}(T_{i})| = 2^{m_{1}−1} = |V_{1}|,

since |D/C_{D}(T_{i})| = 2 for each i. We conclude that D = V_{1}×C_{D}(E_{1}). In the case that r = 1, this shows that (b) holds for G and D. Otherwise, apply induction (on r) to E_{2} ⋯ E_{r}, T ∩ E_{2} ⋯ E_{r}, and C_{D}(E_{1}) to obtain

D = V_{1} × ⋯ × V_{r} × C_{D}(E_{1} ⋯ E_{r})

yielding part (b) in general for G and D. Step 5: We next show that W is irreducible for G. Assume on the contrary that W_{1} is a nontrivial proper G-invariant subgroup of W. Set

T_{1} = {T ∈ T | [W, T] ⩽ W_{1}},
T_{2} = {T ∈ T | [W_{1}, T] = 1},

G_{1} = ⟨T_{1}⟩, and G_{2} = ⟨T_{2}⟩. Then G_{1} and G_{2} are normal in G.

Let T ∈ T − T_{2}. Then

1 < [W_{1}, T] ⩽ [W, T] ⩽ [D, T]C_{D}(G)/C_{D}(G),

so as |[D, T]| = 2, all these inclusions are equalities. In particular, [W, T] = [W_{1}, T] ⩽ W_{1}, which yields T ∈ T_{1}. Hence

T = T_{1} ∪ T_{2} and G = G_{1}G_{2}. (6.6)

Set K = C_{G}(W/W_{1}) ∩ C_{G}(W_{1}). Then K acts nilpotently on the chain W > W_{1} > 1, and so K ⩽ O_{2}(G) = 1 by (6.4) and assumption on G. Since [G_{1}, G_{2}] ⩽ G_{1} ∩ G_{2} ⩽ K, we see that

T_{1} ∩ T_{2} = ∅ and G = G_{1} × G_{2} (6.7)

from (6.6).

Now [W, G_{1}, G_{2}] = 1 by construction and [G_{1}, G_{2}, W] = 1 from (6.7), so [W, G_{2}, G_{1}] = 1 by the Three Subgroups Lemma. Hence [W, G_{1}G_{2}] = [W, G_{1}][W, G_{2}]. Further, [W, G_{1}] ∩ [W, G_{2}] ⩽ C_{W} (G_{1}G_{2}), which is the identity by (6.5), and so

W = [W, G] = [W, G_{1}] × [W, G_{2}] (6.8) again by (6.5) and (6.6).

Finally, T_{1} is not empty since otherwise G = G_{2} centralizes W_{1} contrary to (6.5). Similarly, T_{2} is not empty. Hence 1 < |G_{1}| < |G| and 1 < |G_{2}| < |G|. One then checks that T ∈ T_{k} is solitary in G_{k} (on W_{k}, k = 1, 2), using (6.8) and the fact that an L ≅ S_{3} containing T in G is generated by G-conjugates of T.

Induction applied to (G_{1}, W_{1}) and (G_{2}, W_{2}) now yields the theorem for (G, W). By Step 4, (G, D) is not a counterexample. We conclude that W is irreducible for G as desired. In particular, W is elementary abelian, and each element of T induces a transvection on W.

Now Lemma 6.1 shows that G is a symmetric group of odd degree and W is a natural module for G, and T is the collection of subgroups generated by a transposition. Again by Step 4, (G, D) is not a counterexample. □

Proposition 6.9. Let (Γ, S, Y) be a general setup for the prime 2. Set F = F_{S}(Γ), D = Z(Y), and G = Γ/C_{Γ}(D). Let R ⊆ 𝒮 (S)_{⩾Y} be an F-invariant interval such that for each Q ∈ 𝒮 (S)_{⩾Y}, Q ∈ R if and only if J_{A_{D}(G)}(Q) ∈ R. Then L^{k}(F; R) = 0 for all k ⩾ 2.

Proof. Assume the hypotheses of the proposition, but assume that the conclusion is false. Let (Γ, S, Y, R, k) be counterexample for which the four-tuple (k, |Γ|, |Γ/Y |, |R|) is minimal in the lexicographic ordering. Steps 1–3 in the proof of [Oli13, Proposition 3.3] show that R = {P ⩽ S | J_{A_{D}(G)}(P) = Y }, k = 2, and (Γ, S, Y) is a reduced setup.

Let Q = 𝒮 (S)_{⩾Y} − R and A = A_{D}(G)^{∘} ∪ Â_{D}(G)^{∘}. Since (Γ, S, Y) is a counterexample, Theorem 5.4 shows that

6.9.1. A = T_{D}(G).

That is, Â_{D}(G) is empty and every best offender minimal under inclusion is solitary in G relative to S̄. Since L^{2}(F; R) ≠ 0, we see that

6.9.2. L^{1}(F, Q) ≠ 0.

from Lemma 2.6. We prove next that

6.9.3. G = ⟨A⟩.

Proof. Let G_{0} = ⟨A⟩. Let Γ_{0} be the preimage of G_{0} in Γ, set S_{0} = S ∩Γ_{0}, set F_{0} = F_{S_{0}}(G_{0}), and set Q_{0} = 𝒮 (S_{0})_{⩾Y} ∩ Q. Then Γ_{0} ⊴ Γ and by (6.9.1), we have Y ⩽ S_{0}. Further, (Γ_{0}, Y, S_{0}) is a reduced setup and Q_{0} is an F_{0}-invariant interval. Since each member of A is contained in G_{0}, we have Γ_{0} ∩ Q ∈ Q for each Q ∈ Q. By Lemma 2.7, the restriction map induces an injection L^{1}(F; Q) → L^{1}(F_{0}; Q_{0}) and so L^{1}(F_{0}; Q_{0}) ≠ 0 by 6.9.2. Hence Γ = Γ_{0} by minimality of |Γ|. □

Therefore, G and its action on D are described by Proposition 6.2. We adopt the notation in that proposition for the remainder of the proof. In the decomposition of part (b) there, each V_{i} is G-invariant and so each is S-invariant. Thus, the centralizer of S in D factors as

6.9.4. C_{D}(S) = C_{V_{1}}(S) × ⋯ × C_{V_{r}}(S) × C_{D}(G).

Fix an inclusion-normalized 1-cocycle t for Z_{F}^{Q} representing a nonzero class in L^{1}(F; Q) by 6.9.2, and let τ: Γ^{∗} → Γ^{∗} be the rigid map associated with t. We show next that

6.9.5. r = 1.

Proof. We assume r > 1 and aim for a contradiction. Let G_{1} = E_{1} and G_{2} = E_{2} ⋯ E_{r}. For i = 1, 2, let K_{i} be the preimage of G_{i} in Γ, and set Γ_{i} = K_{i}S and F_{i} = F_{S}(Γ_{i}). We have that Γ_{i} < Γ by assumption and that (Γ_{i}, S, Y) is a general setup. Hence L^{1}(F_{i}; Q) ≅ L^{2}(F_{i}; R) = 0 by minimality of |Γ|. As the restriction of t to O(F_{i}^{c}) represents the zero class, there are elements z_{i} ∈ C_{D}(S) by 6.9.4 such that z_{i} ∈ [D, G_{i}] ⩽ C_{D}(G_{3−i}) and such τ is conjugation by z_{i} on (Γ_{i})^{∗}. Set t′ = t du where u is the constant 0-cochain defined by u(P) = (z_{1}z_{2})^{−1} for each P ∈ Q. Then by Lemma 5.2(b), upon replacing t by t′ and τ by the rigid map τ′ associated with t′, we may assume that τ is the identity when restricted to (Γ_{i})^{∗} for each i = 1, 2.

The objective is now to show that τ is the identity on Γ^{∗}. Let W be the Γ-conjugacy functor defined by W (P) = J_{A}(P) for each P ⩾ Y, and by W (P) = P otherwise, and let C be the collection of subgroups P of S such that W (P) = P and P is well-placed with respect to W. Since N_{Γ}(P) ⩽ N_{Γ}(W (P)) and W (W (P)) = W (P) for each P ⩽ S, the collection C is a conjugation family for F. It thus suffices to show that τ is the identity on N_{Γ}(Q) for each Q ∈ C ∩ Q by Lemma 5.2(c).

Let Q ∈ C ∩ Q, so that the image of Q in G is generated by members of A. It then follows from Proposition 6.2(a) that Q = Q_{1}Q_{2} with Q_{1} ∩ Q_{2} = Y, where Q̄_{1} and Q̄_{2} are the projections in G_{1} and G_{2} of Q̄. If it happens that Q_{i} ∉ Q for i = 1 or 2, this means that Q_{i} = Y. Since Y ∉ Q but Q ∈ Q, we may assume without loss that Q_{2} ∈ Q. Now Q is well-placed and W (Q) = Q, so that Q is fully F-normalized. A straightforward argument shows that Q_{2} is also fully F-normalized.

Let g ∈ N_{Γ}(Q_{2}), and write g = g_{1}g_{2} with g_{i} ∈ K_{i}. Since N_{S}(Q_{2}) is a Sylow 2-subgroup of N_{Γ}(Q_{2}), we have by Lemma 3.8 that \overline{N_{Γ}(Q_{2})} = N_{G}(Q̄_{2}), and the latter is G_{1} × N_{G_{2}}(Q̄_{2}). Since ḡ_{1} ∈ G_{1} centralizes Q̄_{2}, we may write g_{1} = h_{1}c_{1} where h_{1} ∈ C_{Γ_{1}}(Q_{2}/Y) ⩽ N_{Γ_{1}}(Q_{2}) and c_{1} ∈ C_{Γ}(D). So as h_{1} ∈ (Γ_{1})^{∗} and c_{1}g_{2} ∈ (Γ_{2})^{∗} (both send Q_{2} to Q_{2}), we see that τ fixes g. We conclude that τ is the identity on N_{Γ}(Q_{2}). However, then τ is the identity on N_{Γ}(Q) since N_{Γ}(Q) ⩽ N_{Γ}(Q_{2}). We conclude that τ is the identity on Γ^{∗}, a contradiction. Thus, r = 1 as desired. □

By 6.9.5, we may fix m = 2n + 1 such that G = E_{1} ≅ S_{m} and write Ω for the set of even order subsets of {1, …, m}. Identify G with S_{m} and V := V_{1} with Ω. We may assume that S stabilizes the collection {{2i−1, 2i} | 1 ⩽ i ⩽ n}. For each 1 ⩽ i ⩽ n, set z_{i} = {1, …, 2i}, set z′_{i} = {2i + 1, …, 2n}, and let Q_{i} be the preimage in S of ⟨(1, 2), …, (2i − 1, 2i)⟩. Then C_{V} (N_{Γ}(Q_{n})) = ⟨z_{n}⟩, and

C_{V} (N_{Γ}(Q_{i})) = ⟨z_{i}⟩ < ⟨z_{i}, z′_{i}⟩ = C_{V} (N_{Γ}(Q_{i}) ∩ N_{Γ}(Q_{n})) and z_{n} = z_{i}z′_{i}. (6.10)

for all 1 ⩽ i ⩽ n − 1.

Set Γ_{n} = C_{Γ}(z_{n}) (the setwise stabilizer of z_{n}). Then S ⩽ N_{Γ}(Q_{n}) ⩽ Γ_{n}, C_{V} (S) = ⟨z_{n}⟩, and the image of Γ_{n} in G is isomorphic with S_{2n}. Since (Γ_{n}, S, Y) is a general setup and not a counterexample, we may adjust t by a coboundary and assume that

(6.11) τ is the identity on Γ_{n}^{∗}.

Arguing as before, it suffices to show that τ is the identity on N_{Γ}(Q) for each Q ∈ Q∩F^{f} with Q = J_{A}(Q). Each Q with these properties is F-conjugate to Q_{i} for some 1 ⩽ i ⩽ n, so it suffices to prove that τ is the identity on N_{Γ}(Q_{i}) for each such i.

Now for i = n, we have that N_{Γ}(Q_{n}) ⊆ Γ_{n}^{∗}, and so τ is the identity on this normalizer by (6.11). Hence n ⩾ 2 since Γ is a counterexample. Since z_{n−1} centralizes N_{Γ}(Q_{n−1}) and z_{n} = z_{n−1}z′_{n−1} by (6.10), if necessary we may replace t by t′ = t du where u(P) = z_{n} for each P ∈ Q, and obtain that τ is the identity on N_{Γ}(Q_{n−1}). Then τ remains the identity on Γ_{n}^{∗}, since z_{n} ∈ C_{V} (Γ_{n}^{∗}), and since the rigid map associated with du is conjugation by z_{n} on Γ_{n}^{∗}. Thus, n ⩾ 3 since Γ is a counterexample.

Finally, fix i with 1 ⩽ i ⩽ n − 2. Then \overline{N_{Γ}(Q_{i})} = N_{G}(Q̄_{i}) by Lemma 3.8. Since Q̄_{i} = ⟨(1, 2), …, (2i − 1, 2i)⟩, we see that N_{G}(Q̄_{i}) = G_{i} × G′_{i} where G_{i} ≅ C_{2} ≀ S_{i} moves the first 2i points, and where G′_{i} ≅ S_{m−2i} moves the remaining points in the natural action. Thus, we may write N_{Γ}(Q_{i}) = Γ_{i}Γ′_{i} with Γ_{i} ∩ Γ′_{i} = C_{Γ}(D) ∩ N_{Γ}(Q_{i}), Γ̄_{i} = G_{i}, and Γ̄′_{i} = G′_{i}. Fix g ∈ N_{Γ}(Q_{i}) and write g = g_{i}g′_{i} where g_{i} ∈ Γ_{i} and g′_{i} ∈ Γ′_{i}. Then g_{i} ∈ Γ_{n}^{∗}, and τ is the identity on Γ_{n}^{∗}, so τ fixes g_{i}. But Γ′_{i} is generated by Γ_{n}^{∗} ∩ Γ′_{i} and N_{Γ}(Q_{n−1}) ∩ Γ′_{i} by Lemma B.3, and so τ fixes g′_{i} as well. Hence τ is the identity on N_{Γ}(Q_{i}), and together with our reductions, this implies that τ is the identity on Γ^{∗}. Now Lemma 5.1(c) shows that this is contrary to our choice of t. □

Appendix A. Modified norm argument

Given a group G and two nonempty subsets X and Y of G, define the product set

X · Y = {xy | x ∈ X, y ∈ Y }.

Definition A.1. Let G be a finite group and V an abelian group on which G acts. Let X be a subset of G.

(a) A subset Y of G is a transversal to X in G if for each g ∈ G, there are unique x ∈ X and y ∈ Y such that g = xy.

(b) The norm from X to G relative to the transversal Y is the group homomorphism 𝔑_{X; Y}^{G} : C_{V} (X) → C_{V} (G) given by v ↦ ∏_{y∈Y} v^{y}.

Given a subset X, a transversal Y to X in G, and an element g ∈ G, one sees that the map y ↦ y_{g} is a bijection Y → Y, where y_{g} ∈ Y is the unique element such that yg = xy_{g} for some x ∈ X. Hence the image of 𝔑_{X; Y}^{G} does indeed lie in C_{V} (G).

Lemma A.2. Let P be a finite p-group, let V an abelian group on which P acts, and let Q and R be subgroups of P. Then there exists a transversal to Q · R in P, and 𝔑_{R}^{P} = 𝔑_{Q·R; Y}^{P} 𝔑_{Q∩R}^{Q}|_{C_{V} (R)} for any such transversal Y.

Proof. This is a combination of Lemmas A1.1 and A1.2 in [Gla71], with the statement on norms following from Lemma A1.1(a) there and Definition A.1(b) here. □

Theorem A.3. Suppose G is a finite group, S is a Sylow p-subgroup of G, and D is an abelian p-group on which G acts. Let A be a nonempty set of subgroups of S, and set J = ⟨A⟩. Let H be a subgroup of G containing N_{G}(J), and set V = Ω_{1}(D). Assume that J is weakly closed in S with respect to G, and that

(A.4) whenever A ∈ A, g ∈ G, and A ≰ H^{g}, then 𝔑_{A∩H^{g}}^{A} = 1 on V.

or more generally,

(A.5) whenever g ∈ G and J ≰ H^{g}, then 𝔑_{J∩H^{g}}^{J} = 1 on V.

Then C_{D}(H) = C_{D}(G).

Proof. We follow the argument from [Gla71, Theorem A1.4]. Let H be a subgroup of G containing N_{G}(J). Then S ⩽ H since J is weakly closed in S with respect to G.

In the situation of (A.5), there is A ∈ A with A ≰ J ∩ H^{g}, since J = ⟨A⟩. Then A∩(J ∩H^{g}) = A∩H^{g}, and we see that (A.5) follows from (A.4) upon applying Lemma A.2 with J, A, and J ∩ H^{g} in the roles of P, Q, and R, respectively.

Thus, we assume (A.5) and prove C_{D}(H) = C_{D}(G) by induction on the order of D. We may assume D > 1. The p-th power homomorphism on D has kernel V and image ℧^{1}(D), and so D/V ≅ ℧^{1}(D). Since ℧^{1}(D) < D and Ω_{1}(℧^{1}(D)) ⩽ V, the pair (G, ℧^{1}(D)) satisfies the hypotheses of the theorem in place of (G, D). Thus

C_{D/V} (G) = C_{D/V} (H). (A.6)

by induction.

Let z ∈ C_{D}(H) and suppose first that ⟨V, z⟩ < D. The coset V z is fixed by H, and so it is fixed by G by (A.6). Thus, ⟨V, z⟩ is G-invariant. Apply induction with ⟨V, z⟩ in the role of D to obtain that z ∈ C_{D}(G) as required.

Next assume that ⟨V, z⟩ = D and V < D. Then C_{V} (H) = C_{V} (G) by induction. Set z′ = 𝔑_{H}^{G}(z). Then z′ ∈ C_{D}(G) ⩽ C_{D}(H). Since V z is G-invariant, z′ ≡ z^{|G:H|} modulo V. Then as |G : H| is prime to p, we see that ⟨V, z′⟩ = D and

z ∈ C_{D}(H) = C_{V} (H)⟨z′⟩ = C_{V} (G)⟨z′⟩ = C_{D}(G)

as required.

Finally assume that V = D. Given a set [H\G/J] of H-J double coset representatives in G containing the identity, and a transversal [J/J ∩H^{g}] to J ∩H^{g} in J for each g ∈ [H\G/J], then the disjoint union of g[J/J ∩ H^{g}] as g ranges over [H\G/J] is a transversal to H in G. Further, 𝔑_{J∩H}^{J}(z) = 𝔑_{J}^{J}(z) = z. Thus, the norm map decomposes as

𝔑_{H}^{G}(z) = ∏_{g∈[H\G/J]} 𝔑_{J∩H^{g}}^{J}(z^{g}) = z ∏_{g∈[H\G/J]−{1}} 𝔑_{J∩H^{g}}^{J}(z^{g}). (A.7)

If g ∈ [H\G/J] − {1} and J ⩽ H^{g}, then we may choose h ∈ H such that J^{g^{−1}h} ⩽ S. Then g^{−1}h ∈ N_{G}(J) ⩽ H, since J is weakly closed in S with respect to G, and so HgJ = Hh^{−1}gJ = H yields g = 1 by our choice, a contradiction. Thus, J ≰ H^{g} for each g ∈ [H\G/J] − {1}. We conclude that 𝔑_{J∩H^{g}}^{J}(z^{g}) = 1 for each such g from (A.5), and then z = 𝔑_{H}^{G}(z) ∈ C_{V} (G) from (A.7). □

Appendix B. Conjugacy and conjugacy functors

We give here some elementary lemmas from finite group theory that are needed at various places in the paper. We also discuss the notion of a Γ-conjugacy functor W, and describe how it gives rise to the Γ-conjugation family of subgroups well-placed with respect to W, which is also a conjugation family for the fusion system of Γ.

Lemma B.1 (Burnside). Let G be a finite group and S a Sylow p-subgroup of G. Assume that J is an abelian subgroup of S that is weakly closed in S with respect to G and that X and Y are subgroups of J. If X and Y are conjugate in G, then they are conjugate in N_{G}(J).

Proof. Assume X and Y are conjugate in G, and fix g ∈ G with X^{g} = Y. Since ⟨J, J^{g}⟩ ⩽ C_{G}(Y), we may choose h ∈ C_{G}(Y) such that ⟨J^{h}, J^{g}⟩ is a p-group. Choose g_{1} ∈ G with ⟨J^{hg_{1}}, J^{gg_{1}}⟩ ⩽ S. Then J^{hg_{1}} = J = J^{gg_{1}} since J is weakly closed in S with respect to G. Thus, gh^{−1} ∈ N_{G}(J), and X^{gh^{−1}} = Y^{h^{−1}} = Y since h^{−1} centralizes Y. □

The following lemma gives an alternative, more elementary, argument for Lemma 5.2(a) using the norm map. One should apply it there by taking (N_{Γ}(Q), N_{S}(Q), Q, τ) in the role of (Γ, S, Y, τ) below.

Lemma B.2. Let (Γ, S, Y) be a general setup for the prime p and τ an automorphism of Γ centralizing S and having order a power of p. Then τ is conjugation by an element of Z(S).

Proof. Set D = Z(Y) for short. Denote by Γ̂ = Γ⟨τ⟩ the semidirect product of Γ by ⟨τ⟩, and set Ŝ = S⟨τ⟩ = S × ⟨τ⟩ and D̂ = D⟨τ⟩ = D × ⟨τ⟩. Then Y and D are normal in Γ̂, and D̂ = CΓ̂(Y), so that D̂ is normal abelian in Γ̂.

Consider the norm 𝔑 := 𝔑Γ̂ ; Ŝ: CD̂(Ŝ) → CD̂(Γ̂), and set n = |Γ : S|. Since Γ̂ centralizes D̂/D and since n is prime to p, the restriction 𝔑|_{⟨τ⟩} is injective. Choose an integer m such that mn = 1 (mod |D|). Then 𝔑(τ) ≡ τ^{n} (mod D) and 𝔑(τ)^{m} ≡ τ (mod D). Let σ = 𝔑(τ)^{m}, and choose d ∈ D with σ = τd. Then τ ≡ d^{−1} modulo Z(Γ̂) since Γ̂ centralizes σ. Further, since Ŝ centralizes τ, we see that d ∈ CΓ̂(Ŝ) ∩ D = Z(S), as desired. □

Lemma B.3. Let n ⩾ 2 and let G be the symmetric group S_{2n+1}. Set

R_{1} = ⟨(1, 2), (3, 4), …, (2n − 1, 2n)⟩, and
R_{2} = ⟨(1, 2), (3, 4), …, (2n − 3, 2n − 2)⟩.

Then G is generated by N_{G}(R_{1}) and N_{G}(R_{2}).

Proof. Let H = ⟨N_{G}(R_{1}), N_{G}(R_{2})⟩ and Ω = {1, 2, …, 2n + 1}. Now N_{G}(R_{1}) is transitive on Ω − {2n + 1}. Similarly, N_{G}(R_{2}) is transitive on Ω − {2n − 1, 2n, 2n + 1} and contains a subgroup inducing the symmetric group on {2n − 1, 2n, 2n + 1}. Therefore, H is transitive on Ω and the stabilizer of 2n + 1 in H is transitive on Ω − {2n + 1}. Since H contains the transposition (2n, 2n + 1) and is 2-transitive on Ω, it contains all transpositions. Hence, H = G. □

We next give the background on conjugacy functors and well-placed subgroups, which are used in §5 and §6.

Definition B.4. Let Γ be a finite group with Sylow p-subgroup S. A Γ-conjugacy functor on 𝒮 (S) is a mapping W: 𝒮 (S) → 𝒮 (S) such that for all P ⩽ S,

(a) W (P) ⩽ P;

(b) W (P) ≠ 1 whenever P = 1; and

(c) W (P)^{g} = W (P^{g}) whenever g ∈ Γ with P^{g} ⩽ S.

Lemma B.5. Let Γ be a finite group, S a Sylow p-subgroup of Γ, and W a Γ-conjugacy functor on 𝒮 (S). Then for all P ⩽ S,

(a) N_{S}(P) ⩽ N_{S}(W (P));

(b) W (P) = W (N_{S}(W (P))) if and only if W (P) = W (S); and

(c) P = N_{S}(W (P)) if and only if P = S.

Proof. Let P ⩽ S and T = N_{S}(W (P)). Part (a) holds by Definition B.4(c). If W (T) = W (P), then by (a), N_{S}(T) ⩽ N_{S}(W (T)) = N_{S}(W (P)) = T, so that T = S and W (P) = W (T) = W (S). Now (b) holds since the converse is clear.

If P = T, then again N_{S}(P) ⩽ T = P, so that P = S. Now (c) holds since the converse is clear. □

A Γ-conjugacy functor W on 𝒮 (S) can be uniquely extended to a Γ-conjugacy functor Ŵ in the sense of [Gla71, §5]: given a p-subgroup P of Γ, choose g ∈ Γ with P^{g} ⩽ S and define Ŵ (P) = W (P^{g})^{g^{−1}}. Then Ŵ is a mapping on all p-subgroups of Γ which is uniquely determined by (b).

Each Γ-conjugacy functor W gives rise to a conjugation family via its well-placed subgroups. A conjugation family for the fusion system F over S is a collection C of subgroups of S such that every morphism in F is a composition of restrictions of F-automorphisms of the members of C. A conjugation family C for S in Γ in the sense of [Gla71, §3] is itself a conjugation family for F_{S}(Γ) in the above sense.

For a Γ-conjugacy functor W on 𝒮 (S) and a subgroup P ⩽ S, define W_{1}(P) = P and, for all i ⩾ 2, define inductively W_{i}(P) = W (N_{S}(W_{i−1}(P))). Then P is said to be well-placed (with respect to W) if W_{i}(P) is fully F_{S}(Γ)-normalized for all i ⩾ 1.

Theorem B.6. Let Γ be a finite group, S a Sylow p-subgroup of Γ, and W a Γ-conjugacy functor on 𝒮 (S). Then every subgroup of S is Γ-conjugate to a well-placed subgroup of S. The set of well-placed subgroups of S forms a conjugation family for F_{S}(Γ).

Proof. This is a combination of Lemma 5.2 and Theorem 5.3 of [Gla71], given the above remarks. □

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Department of Mathematics, University of Chicago, 5734 S. University Ave, Chicago, IL 60637

E-mail address: gg@math.uchicago.edu

Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd, Piscataway, NJ 08854

E-mail address: jlynd@math.rutgers.edu