ON FIXED POINT SETS OF DISTINGUISHED COLLECTIONS FOR GROUPS OF PARABOLIC CHARACTERISTIC

JOHN MAGINNIS AND SILVIA ONOFREI

Abstract. We determine the nature of the fixed point sets of groups of order p, acting on complexes of distinguished p-subgroups (those p-subgroups containing pcentral elements in their centers). The case when G has parabolic characteristic p is analyzed in detail.

1. Introduction

The subgroup complexes associated to suitably chosen collections of p-subgroups are relevant to the understanding of the p-local structure of the underlying group G, and also provide valuable tools for investigating the modular representation theory and the mod-p cohomology of the group G.

This paper continues the systematic study, started in [MO], of certain collections of p-subgroups, which we call distinguished. These are subcollections of the standard collections of p-subgroups and which consist of those p-subgroups which contain p-central elements in their centers.

A group G has parabolic characteristic p if all the p-local subgroups which contain a Sylow p-subgroup of G have characteristic p. We study the homotopy type of the fixed point sets of subgroups of order p acting on the complex Δ of distinguished p-radical subgroups of G, for groups of parabolic characteristic p. Let P = ⟨x⟩ be such a psubgroup. If x is p-central then the fixed point set is contractible. If x is not of central type, then the homotopy type of the corresponding fixed point set is determined by the group structure of C_{G}(x). If C_{G}(x) has characteristic p, then Δ^{P} is again contractible. If we assume that C_{G}(x) does not have characteristic p, and that \overline{C} = C_{G}(x)/O_{p}(C_{G}(x)) has parabolic characteristic p, then the fixed point set Δ^{P} is equivariantly homotopy equivalent to the complex of distinguished p-radical subgroups of \overline{C}.

In Section 2, notation is introduced and a few basic results are reviewed. In Section 3, two varieties of collections of p-subgroups are defined and various homotopy properties are described; the fixed point sets of p-central elements acting on the corresponding complexes are shown to be contractible. In Section 4, under certain hypotheses, the fixed point sets of subgroups of order p of noncentral type are shown to be equivariantly homotopy equivalent to the complex for a quotient of the centralizer. In Section 5 we consider a few examples where G is a sporadic finite simple group, applications to modular representation theory are also given.

Date: August 2007.

1991 Mathematics Subject Classification. 20J05, 20D08, 20C34, 51E25, 55P91.

2. Notation, terminology and standard results

Throughout this paper G is a finite group and p a prime dividing its order. A p-subgroup R of G is called p-radical if R = O_{p}(N_{G}(R)), where O_{p}(H) is the largest normal p-subgroup of H. Every p-subgroup Q of G is contained in a p-radical subgroup of G uniquely determined by Q and G. This is called the radical closure of Q in G and it is the last term R_{Q} of the chain P_{i+1} = O_{p}(N_{G}(P_{i})) starting with Q = P_{0}. It is easy to see that N_{G}(Q) ≤ N_{G}(R_{Q}). A p-subgroup R is called p-centric if Z(R) is a Sylow p-subgroup of C_{G}(R), in which case C_{G}(R) = Z(R) × H, with H a subgroup of order relatively prime to p.

A collection C of p-subgroups of G is a set of p-subgroups which is closed under conjugation; a collection is a G-poset under the inclusion relation with G acting by conjugation. The nerve |C| is the simplicial complex which has as simplices proper inclusion chains in C; the correspondence C → |C| allows assignment of topological concepts to posets [Qui78, Sect.1]. A collection C is contractible if |C| is contractible. A poset map is a G-homotopy equivalence if and only if the induced map on H-fixed points is a homotopy equivalence for all H ≤ G; see [TW91, 1.3].

Notation 2.1. For Q ≤ G a subgroup let C^{P} = {Q ∈ C | P ≤ N_{G}(Q)} denote the subcollection of C fixed under the action of P. Next C_{>P} = {Q ∈ C | P < Q}. Similarly define C_{≥P} and also C_{<P} and C_{≤P}. We will also use the notation C_{>P}^{≤H} for the set C_{>P}^{≤H} = {Q ∈ C | P < Q ≤ H}.

Theorem 2.2. Let G be a finite group and C ⊆ D two collections of subgroups.

(1) [TW91, Prop. 1.7] Assume that D_{>P} is N_{G}(P)-contractible for all P ∈ D \ C. Then the inclusion C ↪ D is a G-homotopy equivalence.

(2) [TW91, Thm.1] Assume either that C_{≥P} is N_{G}(P)-contractible for all P ∈ D, or that C_{≤P} is N_{G}(P)-contractible for all P ∈ D. Then the inclusion C ↪ D is a G-homotopy equivalence.

(3) [GS06, 2.2(3)] Suppose that F is a G-equivariant poset endomorphism of C satisfying either F ≥ Id_{C} or F ≤ Id_{C}. Then, for any collection C′ containing the image of F, the inclusions F (C) ⊆ C′ ⊆ C are G-homotopy equivalences.

(4) [GS06, in proof Lemma 2.7(1)] Let C be a collection of p-subgroups that is closed under passage to p-overgroups. Let H be an arbitrary p-subgroup in G. Then the inclusion C_{≥H} ↪ C^{H} is a homotopy equivalence.

In what follows A_{p}(G) will denote the Quillen collection of nontrivial elementary abelian p-subgroups, S_{p}(G) the Brown collection of nontrivial p-subgroups and B_{p}(G) the Bouc collection of nontrivial p-radical subgroups. The inclusions A_{p}(G) ⊆ S_{p}(G) and B_{p}(G) ⊆ S_{p}(G) are G-homotopy equivalences [TW91, Thm.2].

Let Ce_{p}(G) denote the subcollection of S_{p}(G) consisting of nontrivial p-centric subgroups and let B_{p}^{cen}(G) = Ce_{p}(G) ∩ B_{p}(G) be the collection of nontrivial p-radical and p-centric subgroups. These two collections are not in general homotopy equivalent with S_{p}(G); however the inclusion map B_{p}^{cen}(G) ⊆ Ce_{p}(G) is a G-homotopy equivalence; see [GS06, Thm.1.1].

Definition 2.3. The group G has characteristic p if C_{G}(O_{p}(G)) ≤ O_{p}(G). If all p-local subgroups of G have characteristic p then G has local characteristic p.

Proposition 2.4. Assume G has characteristic p. Then the following hold: (1) [GLS96, 12.6] G has local characteristic p.

(2) [Sol74, Lemma 1] If P is a p-subgroup of G and H a subgroup of G with P C_{G}(P) ≤ H ≤ N_{G}(P), then H has characteristic p.

Proposition 2.5. Let G be a finite group.

(1) Let P ≤ G be a p-subgroup. Then C_{G}(P) has characteristic p if and only if N_{G}(P) has characteristic p.

(2) [Sol74] Let Q be a p-subgroup of a finite group G, with C_{G}(Q) of characteristic p. Let P be a p-subgroup of G containing Q. Then C_{G}(P) has characteristic p.

Definition 2.6. A parabolic subgroup of G is defined to be a subgroup which contains a Sylow p-subgroup of G. The group G has parabolic characteristic p if all p-local, parabolic subgroups of G have characteristic p.

Remark 2.7. Examples of groups of local characteristic p are the groups of Lie type defined over fields of characteristic p, some of the sporadic groups (such as M_{22}, M_{24}, Co_{2} for p = 2, M_{11}, McL for p = 3, McL, Ly for p = 5), and any groups with self-centralizing Sylow p-subgroup of order p, such as Alt(p).

Remark 2.8. Any group of local characteristic p has parabolic characteristic p. Some examples of sporadic groups of parabolic characteristic p are: M_{12}, J_{2}, Co_{1} for p = 2, M_{12}, J_{3}, Co_{1} for p = 3, J_{2}, Co_{1}, Co_{2} for p = 5.

3. Distinguished collections of p-subgroups

An element x of order p in G is p-central if x is in the center of a Sylow p-subgroup of G. Let Γ_{p}(G) denote the family of p-central elements of G. For a p-subgroup P of G define:

P̂ = ⟨x | x ∈ Ω_{1}Z(P) ∩ Γ_{p}(G)⟩

Further, for C_{p}(G) a collection of p-subgroups of G denote:

Ĉ_{p}(G) = {P ∈ C_{p}(G) | P̂ ≠ 1 }

the collection of subgroups in C_{p}(G) which contain p-central elements in their centers. We call Ĉ_{p}(G) the distinguished C_{p}(G) collection. We shall refer to the subgroups in Ĉ_{p}(G) as distinguished subgroups. Also, denote

C̃_{p}(G) = {P ∈ C_{p}(G) | P ∩ Γ_{p}(G) ≠ 1 }

the collection of subgroups in C_{p}(G) which contain a p-central element. Obviously Ĉ_{p}(G) ⊆ C̃_{p}(G) ⊆ C_{p}(G).

Proposition 3.1. All p-centric subgroups of G are distinguished, that is Ce_{p}(G) ⊆ Ŝ_{p}(G). Consequently the collection of distinguished p-radical subgroups contains the collection of p-centric and p-radical subgroups: B_{p}^{cen}(G) ⊆ B̂_{p}(G).

Proof. Let P be a centric p-subgroup of G and let S be any Sylow p-subgroup of G which contains P. Then Z(S) ≤ C_{G}(P) and Z(S) ≤ Z(P). □

Remark 3.2. If R is a p-subgroup of G and if N_{G}(R) contains a Sylow p-subgroup of G, then R is distinguished. This is easy to see since in this case R ◁ S ∈ Syl_{p}(N_{G}(R)) ⊆ Syl_{p}(G). Thus R ∩ Z(S) ≠ 1.

Lemma 3.3. Let P ∈ S_{p}(G) and assume that N_{G}(P) has characteristic p. Then C_{G}(O_{p}(N_{G}(P))) = Z(O_{p}(N_{G}(P))), and thus O_{p}(N_{G}(P)) is p-centric and distinguished.

Proof. Since P ≤ O_{p}(N_{G}(P)), we have C_{G}(O_{p}(N_{G}(P))) ≤ C_{G}(P) ≤ N_{G}(P). Thus C_{G}(O_{p}(N_{G}(P))) = C_{N_{G}(P)}(O_{p}(N_{G}(P))) ≤ O_{p}(N_{G}(P)), and so C_{G}(O_{p}(N_{G}(P))) = Z(O_{p}(N_{G}(P))). Clearly O_{p}(N_{G}(P)) is p-centric, and Proposition 3.1 implies this group is distinguished. □

Proposition 3.4. Let G have local characteristic p. Then B_{p}^{cen}(G) = B_{p}(G).

Proof. Let R ∈ B_{p}(G), so that R = O_{p}(N_{G}(R)). Since N_{G}(R) has characteristic p, C_{G}(R) = Z(R) and R ∈ B_{p}^{cen}(G). □

Proposition 3.5. Let G have parabolic characteristic p. Then:

(a) If P ∈ S̃_{p}(G) then N_{G}(P) has characteristic p.

(b) B_{p}^{cen}(G) = B̂_{p}(G) = B̃_{p}(G).

Proof. (a) Let z ∈ P be a p-central element, so that N_{G}(⟨z⟩) contains a Sylow p-subgroup of G. Thus N_{G}(⟨z⟩) has characteristic p. By Proposition 2.5(1), C_{G}(z) has characteristic p. By Proposition 2.5(2), C_{G}(P) has characteristic p, and another application of Proposition 2.5(1) shows N_{G}(P) has characteristic p.

(b) Note that B_{p}^{cen}(G) ⊆ B̂_{p}(G) ⊆ B̃_{p}(G). Let R ∈ B̃_{p}(G). Then N_{G}(R) has characteristic p, which implies that C_{G}(R) = Z(R). Thus R ∈ B_{p}^{cen}(G). □

Remark 3.6. It follows from the above Proposition that if G has parabolic characteristic p and V ∈ B_{p}(G) \ B̂_{p}(G), then V does not contain any p-central elements.

Proposition 3.7. If G has parabolic characteristic p, then the collections B̂_{p}(G), Â_{p}(G) and Ŝ_{p}(G) are G-homotopy equivalent.

Proof. We first show that the inclusion map Â_{p}(G) ↪ Ŝ_{p}(G) is a G-homotopy equivalence. We attain this result by showing that Â_{p}(G)_{≤P} is N_{G}(P)-contractible for any P ∈ Ŝ_{p}(G) and then applying Theorem 2.2(2). If Q ∈ Â_{p}(G)_{≤P} then QP̂ ∈ Â_{p}(G)_{≤P}. We obtain the following contracting homotopy: Q ≤ QP̂ ≥ P̂. The N_{G}(P)-contractibility follows from the fact that the two inequalities correspond to poset maps which are N_{G}(P)-equivariant.

To show that B̂_{p}(G) is G-homotopy equivalent to Ŝ_{p}(G), we will use Theorem 2.2(1). Thus we have to prove that for each P ∈ Ŝ_{p}(G) \ B̂_{p}(G), the subcollection Ŝ_{p}(G)_{>P} is N_{G}(P)-contractible.

Let Q ∈ Ŝ_{p}(G)_{>P}, so P < N_{Q}(P) ≤ N_{G}(P). The subgroup N_{Q}(P) is distinguished since it contains Z(Q). As P is a distinguished p-subgroup, N_{G}(P) has characteristic p. Denote O_{NP} = O_{p}(N_{G}(P)) and observe that P < O_{NP}. By Lemma 3.3, O_{NP} is p-centric and distinguished. Since the collection of p-centric subgroups is closed under passage to p-overgroups, N_{Q}(P)O_{NP} is also p-centric and thus it is distinguished. Now consider the string of poset maps Ŝ_{p}(G)_{>P} → Ŝ_{p}(G)_{>P} given by:

Q ≥ N_{Q}(P) ≤ N_{Q}(P)O_{NP} ≥ O_{NP}

which proves the N_{G}(P)-contractibility of Ŝ_{p}(G)_{>P}. □

Proposition 3.8. Assume that G has parabolic characteristic p; then the inclusion Ŝ_{p}(G) ↪ S̃_{p}(G) is a G-homotopy equivalence.

Proof. We will show that the subcollection Ŝ_{p}(G)_{≥P} is N_{G}(P)-contractible for every P ∈ S̃_{P} (G) and then apply Theorem 2.2(2). Let Q ∈ Ŝ_{p}(G)_{≥P} so P ≤ N_{Q}(P) ≤ Q and N_{Q}(P) is a distinguished p-subgroup with Z(Q) ≤ Z(N_{Q}(P)). Next consider R_{P}, the radical closure of P defined in Section 2. Note that N_{Q}(P)R_{P} is a subgroup in N_{G}(R_{P}) since N_{Q}(P) ≤ N_{G}(P) ≤ N_{G}(R_{P}). By Proposition 3.5, N_{G}(R_{P}) has characteristic p and R_{P} is p-centric. Thus N_{Q}(P)R_{P} is also p-centric, and both R_{P} and N_{Q}(P)R_{P} are distinguished. Now consider the string of N_{G}(P)-equivariant poset maps Ŝ_{p}(G)_{≥P} → Ŝ_{p}(G)_{≥P} given by:

Q ≥ N_{Q}(P) ≤ N_{Q}(P)R_{P} ≥ R_{P}

which proves the contractibility of the subcollection Ŝ_{p}(G)_{≥P}. □

Proposition 3.9. The fixed point set S̃_{p}^{Z}(G) is N_{G}(Z)-contractible whenever Z = ⟨z⟩ with z a p-central element in G.

Proof. First note that Z ∈ S̃_{p}^{Z}(G). If P ∈ S̃_{p}^{Z}(G) then P Z ∈ S̃_{p}^{Z}(G) since Z normalizes P Z. There is a contracting homotopy P ≤ P Z ≥ Z via N_{G}(Z)-equivariant maps. □

Remark 3.10. Since Ŝ_{p}(G), Â_{p}(G), B̂_{p}(G) and S̃_{p}(G) are G-homotopy equivalent, for G of parabolic characteristic p, Ŝ_{p}^{Z}(G), Â_{p}^{Z}(G) and B̂_{p}^{Z}(G) are N_{G}(Z)-contractible as well.

4. Fixed point sets for noncentral elements

We shall investigate the fixed point set of an element of order p of noncentral type; these are elements of order p in G which are not conjugate to any element in the center of a Sylow p-subgroup of G. Under certain hypotheses, we will prove that the fixed point set is equivariantly homotopy equivalent to the complex for a quotient of the centralizer. This will require a combination of nine homotopy equivalences.

Notation 4.1. Throughout this section, T will be a subgroup of order p of noncentral type in G. We will use the shorthand notation C = C_{G}(T) and O_{C} = O_{p}(C). The quotient group will be denoted \overline{C} = C/O_{C}; the quotient map is q : C → \overline{C}. For H ≤ C, let \overline{H} = q(H). For Q ≤ C, denote O_{Q} = O_{p}(N_{C}(Q)); for \overline{Q} ≤ \overline{C}, denote O\overline{Q} = O_{p}(N\overline{C}(\overline{Q})). Let S_{T} ∈ Syl_{p}(C), and extend it to S ∈ Syl_{p}(G). Since T ≤ S_{T} ≤ S, we have Z(S) ≤ C; thus Z(S) ≤ S_{T} and in fact Z(S) ≤ Z(S_{T}). Note that \overline{S_{T}} = q(S_{T}) ∈ Syl_{p}(\overline{C}).

Remark 4.2. The proof of our main result (Theorem 4.12) will require the following hypotheses:

(1) G is a finite group of parabolic characteristic p;

(2) C = C_{G}(T) does not have characteristic p;

(3) The quotient group \overline{C} = C_{G}(T)/O_{p}(C_{G}(T)) has parabolic characteristic p.

We first recall a result which is due to Grodal [Gro02, pp. 420-421], see also Sawabe [Saw06, Thm.1]. For completeness we provide a proof.

Proposition 4.3. (in the proof of [Gro02, Thm.1.1]) Let C be a collection of nontrivial p-subgroups of G, which is closed under passage to p-overgroups. Let Q ∈ S_{p}(G). If C′ is a collection satisfying: C ∩ B_{p}(G) ⊆ C′ ⊆ C then C_{>Q} is N_{G}(Q)-homotopy equivalent to C′_{>Q}.

Proof. To simplify the notation, we shall denote by X = C_{>Q} and by Y = C′_{>Q}. We will prove that X_{>P} is N_{G}(P)-contractible for all P ∈ X \ Y. Then, an application of Theorem 2.2(1) will give the result. Note that X_{>P} = {R ∈ C | P < R}. Denote O_{NP} = O_{p}(N_{G}(P)); since P is not p-radical, P < O_{NP}. By elementary group theory, for P < R, we have P < N_{R}(P) ≤ R. Since C is closed under passage to p-overgroups, the subgroups N_{R}(P), N_{R}(P)O_{NP} and O_{NP} are also in X_{>P}. Thus we obtain a contracting homotopy given by the string of N_{G}(P)-equivariant poset maps X_{>P} → X_{>P} :

R ≥ N_{R}(P) ≤ N_{R}(P)O_{NP} ≥ O_{NP} .

Hence X_{>P} is N_{G}(P)-contractible. □

Proposition 4.4. Let G be a finite group of parabolic characteristic p. The inclusion Ŝ_{p}(G)_{>T}^{≤C} ↪ Ŝ_{p}^{T} (G) is a N_{G}(T)-homotopy equivalence.

Proof. We verify the following chain of N_{G}(T)-homotopy equivalences:

Ŝ_{p}^{T} (G) ≃ S̃_{p}^{T} (G) ≃ S̃_{p}(G)_{≥T} = S̃_{p}(G)_{>T} ≃ Ŝ_{p}(G)_{>T} ≃ Ŝ_{p}(G)_{>T}^{≤N_{G}(T)} = Ŝ_{p}(G)_{>T}^{≤C}.

The subcollections Ŝ_{p}^{T} (G) and S̃_{p}^{T} (G) are N_{G}(T)-homotopy equivalent because S̃_{p}(G) and Ŝ_{p}(G) are G-homotopy equivalent. The next step follows by an application of Theorem 2.2(4). The collection S̃_{p}(G) is closed under passage to p-overgroups; thus the inclusion S̃_{p}(G)_{≥T} ↪ S̃_{p}^{T} (G) is a homotopy equivalence. Next, since T ∉ S̃_{p}(G) it follows that S̃_{p}(G)_{≥T} = S̃_{p}(G)_{>T}. The homotopy equivalence between S̃_{p}(G)_{>T} and Ŝ_{p}(G)_{>T} follows from an application of Proposition 4.3 with C = S̃_{p}(G) and C′ = Ŝ_{p}(G). Note that Proposition 3.5(b) provides the necessary hypothesis C ∩ B_{p}(G) ⊆ C′. To see that Ŝ_{p}(G)_{>T}^{≤N_{G}(T)} ↪ Ŝ_{p}(G)_{>T} is a homotopy equivalence, consider the poset map Ŝ_{p}(G)_{>T} → Ŝ_{p}(G)_{>T} given by Q ↦ N_{Q}(T) whose image lies in Ŝ_{p}(G)_{>T}^{≤N_{G}(T)}, and apply Theorem 2.2(3). The final equality follows from T ≤ P ≤ N_{G}(T) if and only if T ≤ P ≤ C, since T ⊴ P implies that T ≤ Z(P). □

Lemma 4.5. Let G be a finite group of parabolic characteristic p. Then O_{C} ∈ S̃_{p}(G) if and only if C has characteristic p.

Proof. If O_{C} ∈ S̃_{p}(G) then N_{G}(O_{C}) has characteristic p, by Proposition 3.5(a). As T ≤ O_{C} ⊴ C ≤ N_{G}(O_{C}), it follows that C_{N_{G}(O_{C})}(T) = C_{G}(T) = C and so C = T · C_{N_{G}(O_{C})}(T). Thus C has characteristic p, by Proposition 2.4(2). Conversely, assume that C has characteristic p. Note T ≤ O_{C}, so C_{G}(O_{C}) ≤ C. Thus C_{G}(O_{C}) = C_{C}(O_{C}) ≤ O_{C}, and so O_{C} is p-centric. Thus O_{C} ∈ Ŝ_{p}(G) ⊆ S̃_{p}(G). □

Proposition 4.6. Let G be a finite group of parabolic characteristic p, and assume that O_{C} ∈ S̃_{p}(G). Then the fixed point set Ŝ_{p}(G)^{T} is contractible.

Proof. Consider the poset map φ : S̃_{p}(G)_{>T}^{≤C} → S̃_{p}(G)_{>T}^{≤C} given by φ(P) = P · O_{C}. If O_{C} contains p-central elements, the poset map φ will have image equal to S̃_{p}(G)_{≥O_{C}}^{≤C}; this is contractible, a cone on O_{C}. Apply Theorem 2.2(3); combining this with Proposition 4.4 proves the result. □

Proposition 4.7. Let G be a finite group of parabolic characteristic p, and assume that C does not have characteristic p. The inclusion X = S̃_{p}(G)_{>O_{C}}^{≤C} ↪ Y = S̃_{p}(G)_{>T}^{≤C} is a N_{G}(T)-homotopy equivalence.

Proof. The poset map φ : S̃_{p}(G)_{>T}^{≤C} → S̃_{p}(G)_{>T}^{≤C} given by φ(P) = P · O_{C} now has image in S̃_{p}(G)_{>O_{C}}^{≤C} since O_{C} is purely noncentral, containing no p-central elements of G. The result follows by an application of Theorem 2.2(3). □

Proposition 4.8. Let G be a finite group of parabolic characteristic p. The inclusion X = Ŝ_{p}(G)_{>H}^{≤C} ↪ Y = S̃_{p}(G)_{>H}^{≤C} is a N_{G}(T)-homotopy equivalence, where H satisfies T ≤ H ≤ C and N_{G}(T) ≤ N_{G}(H).

Proof. Let P ∈ Y; we will show that X_{≥P} is equivariantly contractible and apply Theorem 2.2(2). Let R_{P} be the radical closure of P; note that T < P ≤ R_{P}, so that Z(R_{P}) ≤ C. By Proposition 3.5(b), R_{P} is p-centric and distinguished. This implies that Z(R_{P}) is distinguished, and also P ·Z(R_{P}) is distinguished, since Z(R_{P}) ≤ Z(P ·Z(R_{P})). Let Q ∈ X_{≥P}. P ≤ Q implies Z(Q) ≤ Z(N_{Q}(P)), so that N_{Q}(P) is distinguished. Observe that N_{Q}(P) ≤ N_{G}(P) ≤ N_{G}(R_{P}), implying N_{Q}(P)Z(R_{P}) is a group. Choose S_{R} ∈ Syl_{p}(N_{G}(R_{P})) satisfying N_{Q}(P) ≤ S_{R}, and extend to S ∈ Syl_{p}(G). Note R_{P} ≤ S_{R} ≤ S. Then Z(S) ≤ C_{G}(N_{Q}(P)) and Z(S) ≤ C_{G}(R_{P}), so that Z(S) ≤ Z(R_{P}). This implies that Z(S) ≤ Z(N_{Q}(P)Z(R_{P})), so that N_{Q}(P)Z(R_{P}) is distinguished. Now consider the string of equivariant poset maps X_{≥P} → X_{≥P} given by:

Q ≥ N_{Q}(P) ≤ N_{Q}(P)Z(R_{P}) ≥ P · Z(R_{P})

which proves the equivariant contractibility of the subcollection X_{≥P}. □

There is no obvious relationship among those elements which are p-central in G, or in C, or in \overline{C}. In order to overcome this difficulty, we define a subcollection 𝔖 of Ŝ_{p}(G)_{>O_{C}}^{≤C} as follows:

Definition 4.9. Assume the notation from 4.1 and set:

𝔖 = {P ∈ Ŝ_{p}(G)_{>O_{C}}^{≤C} | Z(P) ∩ Z(S) ≠ 1, for some S_{T} and S with P ≤ S_{T} ≤ S}

This subcollection is also contained in Ŝ_{p}(C), and it contains all p-centric subgroups P in C which properly contain O_{C}, since if P ≤ S_{T} ≤ S, then Z(S) ≤ C_{G}(P), implying Z(S) ≤ Z(P).

Proposition 4.10. Let G be a finite group of parabolic characteristic p. The inclusion 𝔖 ↪ Ŝ_{p}(G)_{>O_{C}}^{≤C} is an N_{G}(T)-homotopy equivalence.

Proof. We will apply Theorem 2.2(2) once again. We need to show that 𝔖_{≥Q} is equivariantly contractible whenever Q ∈ Ŝ_{p}(G)_{>O_{C}}^{≤C}. Since Q is a distinguished p-subgroup of G, N_{G}(Q) has characteristic p, by Proposition 3.5(a). Also QC_{G}(Q) ≤ N_{C}(Q) ≤ N_{G}(Q) since T < Q ≤ C; by Proposition 2.4(2), it follows that N_{C}(Q) has characteristic p. Set O_{Q} = O_{p}(N_{C}(Q)) and observe that, according to Lemma 3.3, O_{Q} is p-centric. For P ≤ C, N_{P} (Q)O_{Q} is a group which is also p-centric. Thus O_{Q} and N_{P} (Q)O_{Q} lie in 𝔖. Assume now that P ∈ 𝔖_{≥Q} and consider the contracting homotopy given by the following string of equivariant poset maps:

P ≥ N_{P} (Q) ≤ N_{P} (Q)O_{Q} ≥ O_{Q}.

This concludes the proof of the Proposition. □

Proposition 4.11. Let G be a finite group of parabolic characteristic p, and assume that C does not have characteristic p. Also assume that \overline{C} has parabolic characteristic p. Then the map q_{∗} : 𝔖 → Ŝ_{p}(\overline{C}) induced by the quotient map q : C → \overline{C} is a homotopy equivalence.

Proof. To see that q_{∗}(𝔖) ⊆ Ŝ_{p}(\overline{C}), let P ∈ 𝔖. Recall that Z(S) ≤ Z(S_{T}) and \overline{S_{T}} ∈ Syl_{p}(\overline{C}). Thus \overline{Z(S)} ≤ \overline{Z(S_{T})} ≤ Z(\overline{S_{T}}). Since O_{C} is purely noncentral (Lemma 4.5), the map q : C → \overline{C} is injective on the p-central elements of Z(S). Therefore Z(S)∩Z(P) ≠ 1 implies Z(\overline{S_{T}}) ∩ Z(\overline{P}) ≠ 1, and we have q_{∗}(P) = \overline{P} ∈ Ŝ_{p}(\overline{C}).

According to a result of Thévenaz and Webb [TW91, Thm.1], the poset map q_{∗} : 𝔖 → Ŝ_{p}(\overline{C}) is an equivariant homotopy equivalence if q_{∗}^{−1}(Ŝ_{p}(\overline{C})_{≥\overline{Q}}) is equivariantly contractible for any \overline{Q} ∈ Ŝ_{p}(\overline{C}). Define Q = q^{−1}(\overline{Q}). Then q_{∗}^{−1}(Ŝ_{p}(\overline{C})_{≥\overline{Q}}) = \{P ∈ 𝔖|\overline{Q} ≤ \overline{P} \} = \{P ∈ 𝔖|Q ≤ P \}, since O_{C} ≤ P. This latter set is just 𝔖_{≥Q}. Given P ∈ 𝔖_{≥Q}, consider the string of equivariant poset maps 𝔖_{≥Q} → 𝔖_{≥Q} given by:

P ≥ N_{P} (Q) ≤ N_{P} (Q)O_{Q} ≥ O_{Q}.

We need to show that all of these terms lie in 𝔖_{≥Q}. Note that N_{P} (Q) ≤ P ≤ S_{T} ≤ S, and Q ≤ P implies that Z(P) ≤ Z(N_{P} (Q)). Thus N_{P} (Q) ∈ 𝔖. Next, we have N_{C}(Q) = q^{−1}(N\overline{C}(\overline{Q})), using O_{C} ≤ Q. Thus O_{Q} := O_{p}(N_{C}(Q)) is equal to q^{−1}(O\overline{Q}) by the correspondence theorem for normal subgroups applied to N_{C}(Q) → N\overline{C}(\overline{Q}). Since \overline{Q} ∈ Ŝ_{p}(\overline{C}), N\overline{C}(\overline{Q}) has characteristic p, by Proposition 3.5(a). Then C\overline{C}(O\overline{Q}) ≤ O\overline{Q}. Since T ≤ Q ≤ O_{Q}, C_{G}(O_{Q}) ≤ C_{G}(T) = C and so C_{G}(O_{Q}) = C_{C}(O_{Q}). Therefore C_{G}(O_{Q}) = C_{C}(O_{Q}) ≤ q^{−1}(C\overline{C}(O\overline{Q})) ≤ q^{−1}(O\overline{Q}) = O_{Q}. The group O_{Q} is p-centric in G. But 𝔖 contains all subgroups of C, properly containing O_{C}, which are p-centric in G. The p-overgroup N_{P} (Q)O_{Q} is also p-centric and lies in 𝔖. □

We can now state the main result of this section.

Theorem 4.12. Maintain the notation in 4.1 and the hypotheses in Remark 4.2. There is an N_{G}(T)-equivariant homotopy equivalence Ŝ_{p}^{T} (G) ≃ Ŝ_{p}(\overline{C}).

Proof. We have the chain of N_{G}(T)-homotopy equivalences:

Ŝ_{p}^{T} (G) ≃ Ŝ_{p}(G)_{>T}^{≤C} ≃ S̃_{p}(G)_{>T}^{≤C} ≃ S̃_{p}(G)_{>O_{C}}^{≤C} ≃ Ŝ_{p}(G)_{>O_{C}}^{≤C} ≃ 𝔖 ≃ Ŝ_{p}(\overline{C}).

The first step is Proposition 4.4; then apply Proposition 4.8 with H = T. Next, use Proposition 4.7, and then Proposition 4.8 again with H = O_{C}. Finally, a combination of Propositions 4.10 and 4.11 completes the proof of the theorem. □

5. Examples and Lefschetz modules

We will discuss three examples, and give an application to modular representation theory. Recall that if a group G acts on a simplicial complex Δ, we can construct the virtual Lefschetz module by taking the alternating sum of the vector spaces (over a field of characteristic p) spanned by the chains. To obtain the reduced Lefschetz module, subtract the trivial one dimensional representation. Information about fixed point sets leads to details about the vertices of irreducible summands of this module.

The Fischer group F i_{22} and p = 2 We begin with the sporadic simple Fischer group F i_{22}, which has parabolic characteristic 2 and has three conjugacy classes of involutions, denoted 2A, 2B and 2C in the Atlas [CCN^{+}85]. The class 2B is 2-central. Their centralizers are C_{F i_{22}}(2A) = 2.U_{6}(2), C_{F i_{22}}(2B) = (2 × 2_{+}^{1+8} : U_{4}(2)) : 2 and C_{F i_{22}}(2C) = 2^{5+8} : (S_{3} × 3^{2} : 4).

We consider the simplicial complex Δ whose vertex stabilizers are the four 2-local subgroups of F i_{22}.

H_{1} = (2 × 2_{+}^{1+8} : U_{4}(2)) : 2  H_{2} = 2^{5+8} : (S_{3} × A_{6})
H_{3} = 2^{6} : Sp_{6}(2)  H_{4} = 2^{10} : M_{22}

The flag stabilizers are listed below.

H_{1,2} = 2^{5+8} : (S_{3} × S_{4})  H_{1,2,3} = 2^{6} : 2^{5} : (2 × S_{4})
H_{1,3} = 2^{6} : 2^{5} : S_{6}  H_{1,2,4} = 2^{5+8} : (2 × S_{4})
H_{1,4} = 2^{10} : 2^{4} : S_{5}  H_{1,3,4} = 2^{6} : 2^{5} : (2 × S_{4})
H_{2,3} = 2^{6} : 2^{1+6} : (S_{3} × S_{3})  H_{2,3,4} = 2^{6} : 2^{1+6} : (S_{3} × 2)
H_{2,4} = 2^{5+8} : (2 × A_{6})  H_{1,2,3,4} = 2^{6} : 2^{1+6} : 2^{2}
H_{3,4} = 2^{6} : 2^{6} : L_{3}(2)

The subgroup complex Δ is also known as the “standard” 2-local geometry for F i_{22}. The geometry Δ is G-homotopy equivalent to B_{2}^{cen}(F i_{22}), and since F i_{22} has parabolic characteristic 2, this is equal to the distinguished collection B̂_{2}(F i_{22}); for details we refer the reader to Benson and Smith [BS07, Sections 8.16 and 9.4].

We shall use the notation from the Modular Atlas homepage [MOC07], where φ_{i} denotes an irreducible module of F i_{22} and P_{F i_{22}}(φ_{i}) is its corresponding projective cover.

Proposition 5.1. Let Δ be the standard 2-local geometry for the Fischer group F i_{22}.

(a) The reduced Lefschetz module is

L̃_{F i_{22}}(Δ) = −P_{F i_{22}}(φ_{12}) − P_{F i_{22}}(φ_{13}) − 6φ_{15} − 12P_{F i_{22}}(φ_{16}) − φ_{16}.

(b) The fixed point sets Δ^{2B} and Δ^{2C} are contractible.

(c) The fixed point set Δ^{2A} is equivariantly homotopy equivalent to the building for the Lie group U_{6}(2).

(d) There is precisely one nonprojective summand of the reduced Lefschetz module, it has vertex ⟨2A⟩ and lies in a block with the same group as defect group.

Proof. (a) The first two terms of the reduced Lefschetz module, the projective covers of φ_{12} and φ_{13}, lie in the principal block. Also φ_{15} is projective and lies in block 2, with defect zero. Next, φ_{16} is not projective, and lies in block 3 with defect one. This formula will be shown to be valid at the level of the Green ring of virtual modules, and not just the Grothendieck ring of characters. The alternating sum of the induced characters was computed using GAP[GAP00], and the character corresponding to the formula for L̃_{F i_{22}}(Δ) given above was obtained. The eight terms lying in H_{3} = 2^{6} : Sp_{6}(2) can be combined to yield the character of the inflation of the Steinberg module for the symplectic group.

(b) Proposition 3.9 (also see Remark 3.10) tells us that the fixed point set Δ^{2B} is contractible. The contractibility of Δ^{2B} implies that Δ^{Q} is mod 2 acyclic for any 2-group Q containing an involution of type 2B (by Smith theory), and thus the reduced Lefschetz module L̃_{N_{G}(Q)}(Δ^{Q}) = 0. A theorem due to Burry and Carlson [BC82, Theorem 5] and to Puig (unpublished) was applied by Robinson [Rob88, in Corollary 3.2] to Lefschetz modules to obtain the following result; also see [Saw06, Lemma 1]. The number of indecomposable summands of L̃_{G}(Δ) with vertex Q is equal to the number of indecomposable summands of L̃_{N_{G}(Q)}(Δ^{Q}) with the same vertex Q. The proof of this result uses the Green correspondence, and the relationship to the Brauer correspondence permits a conclusion regarding the blocks in which the summands lie. This implies that the vertices of the indecomposable summands of L̃_{F i_{22}}(Δ) do not contain any 2-central involutions. Proposition 4.6 applies to the element 2C since O_{C} = O_{2}(C_{F i_{22}}(2C)) contains 2-central elements. The elementary abelian 2^{5} ≤ 2^{5+8} is of the type 2^{5} = 2A_{6}B_{15}C_{10}; there is a purely 2-central 2^{4} ≤ 2^{5}. Note that Δ^{2C} being contractible implies that no vertex of a summand of L̃_{F i_{22}}(Δ) contains an involution of type 2C.

(c) Theorem 4.12 applies to the fixed point set Δ^{2A}, since the quotient group of the centralizer C_{F i_{22}}(2A)/O_{2}(C_{F i_{22}}(2A)) = U_{6}(2) is a Lie group. Therefore Δ^{2A} is homotopy equivalent to the building for U_{6}(2). The reduced Lefschetz module associated to this action on the building is the irreducible Steinberg module. The vertex under the action of 2.U_{6}(2) is ⟨2A⟩. Thus there exists one indecomposable summand of the Lefschetz module with vertex 2 = ⟨2A⟩, lying in a block with the same group as defect group (block 3 with defect one). Note that since this block has cyclic defect group of order two, there is only one nonprojective indecomposable module lying in this block, namely the irreducible module φ_{16}. Also note that since Δ^{2A} is homotopy equivalent to a building, Δ^{Q} will be contractible for any 2-group Q of order at least four which contains an involution of type 2A. This implies that such a group Q cannot be a vertex of a summand of L̃_{F i_{22}}(Δ).

(d) It follows from the previous steps that there is precisely one nonprojective summand of the reduced Lefschetz module; it has vertex ⟨2A⟩ and it lies in a block with the same group as defect group. The remaining summands are projective and are determined by their characters, and so the formula for L̃_{F i_{22}}(Δ) is valid at the level of the Green ring of virtual modules, and not just the Grothendieck ring of characters. □

The Conway group Co_{3} and p = 3 The group Co_{3} has parabolic characteristic 3, and it has three conjugacy classes of elements of order three, denoted 3A, 3B and 3C, with the 3-central elements being those of type 3A. The normalizers are N_{Co_{3}}(⟨3A⟩) = 3_{+}^{1+4} : 4S_{6}, N_{Co_{3}}(⟨3B⟩) = 3^{5}(2 × S_{5}) and N_{Co_{3}}(⟨3C⟩) = S_{3} × L_{2}(8) : 3. Note that L_{2}(8) : 3 = Ree(3) = ^{2}G_{2}(3) is a Chevalley group having local characteristic 3.

Proposition 5.2. Let Δ be the subgroup complex associated to the distinguished 3radical collection B̂_{3}(Co_{3}).

(a) The fixed point sets Δ^{3A} and Δ^{3B} are contractible, and Δ^{3C} is equivariantly homotopy equivalent to the building for ^{2}G_{2}(3).

(b) The reduced Lefschetz module L̃_{Co_{3}}(Δ) has precisely one nonprojective irreducible summand, which has vertex ⟨3C⟩ and lies in a block with the same group as defect group.

Proof. Proposition 3.9 (and Remark 3.10) implies that the fixed point set Δ^{3A} of the 3-central element is contractible. Proposition 4.6 implies that Δ^{3B} is contractible since O_{C} = O_{3}(C_{Co_{3}}(3B)) = 3^{5} = 3A_{55}B_{66} contains 3-central elements. The reduced Lefschetz module L̃_{Co_{3}}(Δ) has no summands with a vertex containing either an element of type 3A or an element of type 3B, by a similar argument to the one in part (b) of the previous proposition.

Theorem 4.12 applies to Δ^{3C}. The reduced Lefschetz module for the ^{2}G_{2}(3) building is the irreducible Steinberg module, and under the action of N_{Co_{3}}(⟨3C⟩), the vertex is ⟨3C⟩. Also, since Δ^{3C} is equivariantly homotopy equivalent to a building, Δ^{Q} will be contractible for any 3-group Q of order at least nine which contains an element of type 3C. This implies that such a group Q cannot be a vertex of a summand of L̃_{Co_{3}}(Δ). The aforementioned Burry-Carlson theorem in Robinson’s formulation implies now the result of part (b). □

The Harada-Norton group HN and p = 5 The group HN has parabolic characteristic 5 and has five conjugacy classes of elements of order 5, denoted 5A, 5B, 5C, 5D and 5E; see [CCN^{+}85]. The elements of type 5B are 5-central. An element of type 5D is the square of an element of type 5C, so they generate a group of order 5 denoted 5CD. The centralizers are C_{HN}(5A) = 5 × U_{3}(5), C_{HN}(5B) = 5^{1+4}.2^{1+4}.5, C_{HN}(5CD) = 5^{3}.SL_{2}(5) and C_{HN}(5E) = 5 × 5^{1+2} : 2^{2}.

Proposition 5.3. Let Δ be the subgroup complex associated to the distinguished 5radical collection B̂_{5}(HN).

(a) The fixed point sets Δ^{5B}, Δ^{5CD} and Δ^{5E} are contractible, and Δ^{5A} is equivariantly homotopy equivalent to the building for U_{3}(5).

(b) The reduced Lefschetz module L̃_{HN}(Δ) has precisely one nonprojective summand, which has vertex ⟨5A⟩ and lies in a block with the same group as defect group.

Proof. Proposition 3.9 (and Remark 3.10) applies to the 5-central elements of type 5B, so that the fixed point set Δ^{5B} is contractible. The two groups 5^{3} = B_{6}(CD)_{25} and 5 × 5^{1+2} contain 5-central elements (note that Z(5^{1+2}) = ⟨5B⟩), thus according to Proposition 4.6 the fixed point sets Δ^{5CD} and Δ^{5E} are both contractible. Thus the reduced Lefschetz module L̃_{HN}(Δ) has no summands with a vertex containing elements of type 5B, 5C, 5D or 5E.

Theorem 4.12 applies to the elements of type 5A, so that Δ^{5A} is equivariantly homotopy equivalent to the building for U_{3}(5). The reduced Lefschetz module for this building is the irreducible Steinberg module, and under the action of N_{HN}(⟨5A⟩), the vertex is ⟨5A⟩. Thus L̃_{HN}(Δ) has one summand with vertex ⟨5A⟩ lying in a block with the same group as defect group. Since Δ^{5A} is homotopy equivalent to a building, Δ^{Q} will be contractible for any 5-group Q of order at least 25 which contains an element of type 5A. This implies that such a group Q cannot be a vertex of a summand of L̃_{HN}(Δ). □

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Department of Mathematics, Kansas State University, 137 Cardwell Hall, Manhattan, Kansas 66506, Email: maginnis@math.ksu.edu

Department of Mathematics, Kansas State University, 137 Cardwell Hall, Manhattan, Kansas 66506, Email: onofrei@math.ksu.edu