ON THE VERTICES OF INDECOMPOSABLE SUMMANDS OF CERTAIN LEFSCHETZ MODULES

JOHN MAGINNIS AND SILVIA ONOFREI

Abstract. We study the reduced Lefschetz module of the complex of p-radical and p-centric subgroups. We assume that the underlying group G has parabolic characteristic p and the centralizer of a certain noncentral p-element has a component with central quotient H a finite group of Lie type in characteristic p. A non-projective indecomposable summand of the associated Lefschetz module lies in a non-principal block of G and it is a Green correspondent of an inflated, extended Steinberg module for a Lie subgroup of H. The vertex of this summand is the defect group of the block in which it lies.

1. Introduction

The best known example of a reduced Lefschetz module is the Steinberg module for a finite simple group of Lie type, with coefficients having the defining characteristic. This modular representation is both irreducible and projective. The Steinberg module equals the top homology module of the associated Tits building, but can also be described using an alternating sum of the modules of the chain complex of the building (and it is this type of virtual module which we will refer to as a Lefschetz module). Another important example is the Brown complex of an arbitrary finite group, with simplices the chains of inclusions of nontrivial p-subgroups for a fixed prime p. The reduced Lefschetz module of the Brown complex, with coefficients in a field of characteristic p, although not necessarily indecomposable is always projective both on the level of characters in the Grothendieck ring [Qui78, Corollary 4.3] and as a virtual module in the Green ring [Web87]. More generally, Webb [Web87, Theorem A′] proved that if the fixed point sets of elements of order p are contractible, for an admissible action of a finite group on a finite simplicial complex, then the reduced Lefschetz module is a projective virtual module (he worked with p-adic integers as coefficients). Ryba, Smith and Yoshiara [RSY90] applied Webb’s Theorem to various geometries admitting a flag transitive action by a sporadic or alternating group, or such geometries other than the building for Lie type groups. Their Table 1 summarizes information on 18 cases where the reduced Lefschetz module is projective, including descriptions of many indecomposable summands as projective covers of irreducible modules. Later work of Smith and Yoshiara [SY97] used a different approach,

Date: 2 September 2011.

1991 Mathematics Subject Classification. 20J05, 20C05, 20C20, 20C34, 20D08, 51E24, 51D20.

The research of the second author was partially supported by the National Security Agency under grant number H98230-10-1-0174.

2 JOHN MAGINNIS AND SILVIA ONOFREI

verifying equivariant homotopy equivalences between such sporadic geometries and the simplicial complex for the Quillen poset of nontrivial elementary abelian p-subgroups. Other subgroup complexes have been constructed with applications in representation theory and to homology decompositions of classifying spaces; see for example [DH01]. The collection of nontrivial p-centric and p-radical subgroups D_{p}(G) is often regarded as the best candidate for a p-local geometry and a generalization of the notion of building to general finite groups. A comprehensive study of 2-local sporadic geometries (in which all the vertex stabilizers are 2-local subgroups) with applications to mod-2 cohomology can be found in Benson and Smith [BS08]. In many cases, notably when the group G has parabolic characteristic 2 (see Section 2 for an explanation of this term), the 2-local geometries studied by Benson and Smith are G-homotopy equivalent with the complex of 2-centric and 2-radical subgroups. One of the first to study nonprojective reduced Lefschetz modules, Sawabe [Saw06a, Prop. 5] determined an upper bound for the orders of vertices of their indecomposable summands. Further results regarding the Lefschetz characters associated to several 2-local sporadic geometries were obtained by Grizzard [Gri09]. He studied the distribution into blocks of the indecomposable summands, for several sporadic groups whose 2-local geometries have nonprojective Lefschetz characters. Since the properties of the reduced Lefschetz modules are strongly influenced by the group structure and, in particular by the centralizers of p-elements, it seems natural to study these properties under appropriately chosen conditions imposed on the p-local structure of G. A p-central element is a nontrivial element of G that lies in the center of a Sylow p-subgroup of G. In [MO08] we introduced distinguished collections of p-subgroups, those p-subgroups of some standard collection that contain p-central elements in their centers. They have similar properties as their standard counterparts. If G has parabolic characteristic p, then the distinguished collection of p-radical subgroups B̂_{p}(G) is equal to the collection of p-centric and p-radical subgroups D_{p}(G). In the present paper we study the reduced Lefschetz module L̃ = L̃_{G}(B̂_{p}(G);k) = L̃_{G}(D_{p}(G);k) when G has parabolic characteristic p. Our approach is to use a theorem of Robinson, quoted in Section 5 of this paper as Theorem 5.1, which relates the indecomposable summands with vertex V in L̃_{G}(Δ;k) to the indecomposable summands with vertex V in L̃_{N_{G}(V)}(Δ^{V};k), for any simplicial complex Δ. We assume for most of this paper that G has parabolic characteristic p. We first determine the fixed point sets of p-subgroups T acting on Δ = B̂_{p}(G). If the largest normal p-subgroup O_{p}(T · C_{G}(T)) of T times its centralizer contains p-central elements then the fixed point set Δ^{T} is contractible; this is Theorem 3.3. If O_{p}(T · C_{G}(T)) does not contain p-central elements and if T · C_{G}(T) has a single component H that also has parabolic characteristic p then the fixed point set is equivariantly homotopy equivalent to the complex of distinguished p-radical subgroups in H. This result is proved in Theorem 3.6. Next we study the vertices of the indecomposable summands of L̃. We focus on the case when the centralizer of certain noncentral p-elements has a component H which is a finite group of Lie

VERTICES OF LEFSCHETZ MODULES 3

type in characteristic p. We find in Theorem 5.4, that a nonprojective indecomposable summand of L̃ is the Green correspondent of an inflated, extended Steinberg module which corresponds to H. This summand lies in a non-principal block of the group ring kG, and has vertex equal to the defect group of the block in which it lies. We discuss applications to the sporadic simple groups of parabolic characteristic p for p = 2 and 3. Our results verifies for these groups a conjecture of Grizzard [Gri09] and of Benson and Smith [BS08, p.164] who noticed: “...for all 15 sporadic groups G ... for which the 2-local geometry Δ is not homotopy equivalent to S_{2}(G), it seems that the reduced Lefschetz module involves an indecomposable in a suitable non-principal 2-block of G of small positive defect.” Specifically, we give an explanation for 12 of the sporadic simple groups that have parabolic characteristic 2.

We use the standard notation for finite groups as in the Atlas [CCN^{+} 85]. If p is a prime, then p^{n} denotes an elementary abelian p-group of rank n and p a cyclic group of order p. Also H.K denotes an extension of the group H by a group K (it does make no statement if the extension splits or not) and H : K denotes a split extension, while H^{.}K indicates a nonsplit extension. Finally H ·K denotes the product of two subgroups of a group G, with at least one normal in G. Regarding the conjugacy classes of elements of order p we follow the Atlas [CCN^{+} 85] notation. The simplified notation of the form 5A^{2} stands for an elementary abelian group 5^{2} whose 24 nontrivial elements are all of type 5A. The notation of the form 2A_{2}B_{1} stands for a group of order 4 which contains two elements from class 2A and one element from class 2B.

The structure of our paper is as follows. In Section 2 we review properties of groups of local characteristic p and parabolic characteristic p, as well as standard facts on collections of p-subgroups. We also recollect a few useful results on distinguished collection of p-subgroups. In Section 3 we determine the nature of the fixed point sets under the action of p-subgroups. Section 4 is dedicated to standard facts from modular representation theory. In Section 5 we prove a theorem that characterizes the Lefschetz module of the collection of p-centric and p-radical subgroups. Applications to the sporadic simple groups are discussed in Section 6.

Acknowledgements. We would like to thank Ron Solomon for providing us with an elegant argument for the existence of central elements in the 3-core of the centralizer of an element of type 3D in the groups F i_{22} and F i_{23} respectively.

2. Collections of p-subgroups

In this section we introduce the necessary notation, review the basic terminology regarding subgroup complexes and list a few results from the literature which will be used later in the proofs. We assume that G is a finite group and p is a prime dividing its order.

2.1. Groups of characteristic p and local characteristic p. The finite groups we are working with in the present paper have properties known as characteristic p, local characteristic p and parabolic

4 JOHN MAGINNIS AND SILVIA ONOFREI

characteristic p. For completeness of our presentation, in this section we define these notions and list some of their features. Most of the results mentioned here are well known and can be found elsewhere in the literature. If G is a finite group, we denote by O_{p}(G) the largest normal p-subgroup in G. A p-local subgroup is the normalizer of a nontrivial p-subgroup in G.

Definition 2.1. 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.

We remark here that the notion of “local characteristic p” is what group theorists usually call “characteristic p-type”.

Proposition 2.2. Let G be a finite group and let p be a prime dividing its order.

(i). Assume G has characteristic p. If P is a p-subgroup of G and H a subgroup of G with PC_{G}(P) ≤ H ≤ N_{G}(P), then H has characteristic p, see [Sol74, Lem. 1]. In particular, G has local characteristic p, see [GLS96, 12.6].

(ii). Let P be a p-subgroup of G. The subgroup C_{G}(P) has characteristic p if and only if N_{G}(P) has characteristic p, see [GLS96, 5.12].

Definition 2.3. A parabolic subgroup of G is an overgroup of a Sylow p-subgroup of G. The group G has parabolic characteristic p if all p-local, parabolic subgroups of G have characteristic p.

We would like to mention that our usage of the term “parabolic” differs from the classical notion of parabolics, in the special case of a Lie type group over a field of characteristic p, for in the latter, the parabolics are actually the overgroups not of just a Sylow p-subgroup S, but of a full Borel subgroup, i.e. of the normalizer N_{G}(S). Our notion of “parabolic characteristic p” appears elsewhere in group theory: notably for p = 2 it appears as “even characteristic” in Aschbacher and Smith [AS04, p.3]; this provides further evidence for the naturalness of this concept.

Collections of p-subgroups. Notation and known results. 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 order complex or the nerve Δ(C) is the simplicial complex which has as simplices proper inclusion chains from C; for a more detailed discussion see [BS08, 2.7]. If we let |Δ(C)| to denote the geometric realization of the nerve Δ(C), then the correspondence C → |Δ(C)| allows assignment of topological concepts to posets [Qui78, Sect. 1]. A collection C is contractible if |Δ(C)| is contractible.

Notation 2.4. Given a subgroup P of G, 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 use the notation C_{>P}^{≤H} for the set C_{>P}^{≤H} = {Q ∈ C | P < Q ≤ H}.

The following statement is a well known result, initially proved in a more general context, see for example [tD79, 8.2.4] and [JS78]:

2.5. A poset map f : C → C′ is a G-homotopy equivalence if and only if the induced map f^{H} : C^{H} → C^{′H}on H-fixed points is a homotopy equivalence for all subgroups H of G.

Theorem 2.6. Let C and C′be two collections of subgroups of G.

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

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

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

Theorem 2.7. [TW91, Thm. 1] Let G be a group and let φ : C → C′ be a poset map between two collections of subgroups of G. Suppose that for all P ∈ C′, the subcollection φ^{−1} (C′_{≥P}) is N_{G}(P)-contractible. Then φ is a G-homotopy equivalence.

We end this succinct review with a result due to Grodal [Gro02, pp. 420-421], see also [Saw06a, Thm. 1] or for an easy proof [MO10, Prop. 4.3].

Proposition 2.8. Let C be a collection of nontrivial p-subgroups of G, which is closed under passage to p-overgroups. Let Q be a p-subgroup of G. If C′⊆ C is a subcollection which contains every p-radical subgroup in C then C_{>Q} is N_{G}(Q)-homotopy equivalent to C′_{>Q}.

Standard collections of p-subgroups. A p-subgroup R of G is called p-radical if R = O_{p}(N_{G}(R)). 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 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. If R is p-centric and Q is a p-subgroup of G which contains R then Q is also p-centric and Z(Q) ≤ Z(R). In what follows S(G) will denote the Brown collection of nontrivial p-subgroups and B(G) will denote the Bouc collection of nontrivial p-radical subgroups^{1}. The inclusion B(G) ⊆ S(G) is a G-homotopy equivalence [TW91, Thm. 2]. Let D(G) denote the subcollection of S(G) consisting of the nontrivial p-centric and p-radical subgroups of G. This collection is not always homotopy equivalent with S(G).

1It is customary to denote these collections by S_{p}(G) and B_{p}(G), but since the prime p is fixed throughout the first sections of the paper, in order to simplify the notation we will drop the subscript p.

6 JOHN MAGINNIS AND SILVIA ONOFREI

Distinguished collections of p-subgroups. We recollect in this section the definitions and some facts on certain subfamilies of S(G), called distinguished collections of p-subgroups. These collections were introduced in [MO07] and further analyzed in [MO08] and in [MO10].

2.9. 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 Γ̂_{p}(G) denote the family of p-central elements of G.

Definition 2.10. For C(G) a collection of p-subgroups of G denote:

Ĉ(G) = {P ∈ C(G) | Z(P) ∩ Γ̂_{p}(G) ≠ ∅ }

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

C̃(G) = {P ∈ C(G) | P ∩ Γ̂_{p}(G) ≠ ∅ }

the collection of subgroups in C(G) which contain p-central elements. Obviously, the following inclusions hold Ĉ(G) ⊆ C̃(G) ⊆ C(G).

2.11. For later reference, we collect a few useful facts. The proofs are short and elementary, details can be found in [MO10, Lem. 3.3, Lem. 3.5 and Prop. 3.7].

(i). The collection S̃(G) is closed under passage to p-overgroups.

(ii). Let P ∈ S(G). If Q ∈ Ŝ(G)_{>P} then N_{Q}(P) ∈ Ŝ(G).

(iii). If N_{G}(P) has characteristic p, then O_{p}(N_{G}(P)) is p-centric and distinguished.

(iv). If G has parabolic characteristic p and if P ∈ S̃(G) then N_{G}(P) has characteristic p.

The distinguished collections of p-subgroups exhibit similar behavior as their standard counterparts. Some of these properties are collected in the following:

Theorem 2.12. Let G be a finite group.

(i). [MO10, Prop. 3.4] All p-centric subgroups of G are distinguished, hence D(G) ⊆ B̂(G). (ii). [MO10, Prop. 3.6] If G has local characteristic p, then D(G) = B̂(G) = B(G). (iii). [MO10, Prop. 3.7] If G has parabolic characteristic p, then D(G) = B̂(G) = B̃(G). (iv). [MO10, Prop. 3.9, 3.10] If G has parabolic characteristic p, then B̂(G), Ŝ(G) and S̃(G) are

G-homotopy equivalent.

2.13. In the case that G has parabolic characteristic p, the better known collection of p-centric pradical subgroups is equal to the collection of distinguished p-radical subgroups. The reason for using the “distinguished” point of view for this collection resides in the fact that our approach is far more useful and easier to work with in proofs.

3. The structure of the fixed point sets

We investigate the fixed point sets of p-elements of G acting on the order complex of the collection D(G) of p-centric and p-radical subgroups. We are interested in groups of parabolic characteristic p, in which case, as seen in Theorem 2.12, the p-centric and p-radical subgroups are the distinguished p-radical subgroups of G and the collections B̂(G), Ŝ(G) and S̃(G) are G-homotopy equivalent.

Notation 3.1. Throughout this section, G is a finite group of parabolic characteristic p, where p is a prime dividing its order. Let T be a p-subgroup of G; we use the shorthand notation C = T · C_{G}(T) and O_{C} = O_{p}(T · C_{G}(T)).

Lemma 3.2. Let G be a finite group of parabolic characteristic p. The inclusion X := Ŝ(G)_{>M}^{≤C} ↪ Y := S̃(G)_{>M}^{≤C} is a N_{G}(T)-homotopy equivalence, for every p-subgroup M that satisfies T ≤ M ≤ C and N_{G}(T) ≤ N_{G}(M).

Proof. Let P ∈ Y; we will show that X_{≥P} is equivariantly contractible and apply Theorem 2.6(i). Let R_{P} be the radical closure of P and let Q ∈ X_{≥P}. Consider the string of poset maps given by:

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

Note that T ≤ M < P ≤ R_{P}, so that Z(R_{P}) ≤ C. By 2.11(iii) and (iv), R_{P} is p-centric and distinguished. This implies that Z(R_{P}) is distinguished, and also PZ(R_{P}) is distinguished, since Z(R_{P}) ≤ Z(PZ(R_{P})). Next 2.11(ii) implies 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 it 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}) = Z(R_{P}), since R_{P} is p-centric and p-radical. This implies that Z(S) ≤ Z(N_{Q}(P)Z(R_{P})), so that N_{Q}(P)Z(R_{P}) is distinguished. Thus the above is a string of equivariant poset maps X_{≥P} → X_{≥P}, which proves the equivariant contractibility of the subcollection X_{≥P}. Note the second condition on M is needed so that N_{G}(T) acts on X and Y. □

The next result was first obtained for T a cyclic group of order p in [MO10], and the arguments from Propositions 4.4, 4.5 and 4.7 of that paper also apply to any p-group T. For completeness we include these proofs below. The notations from 3.1 are maintained.

Theorem 3.3. Let G be a group of parabolic characteristic p and let T be a p-subgroup of G. If O_{C} ∈ S̃(G) then the fixed point set D(G)^{T} is N_{G}(T)-contractible.

Proof. The result is obtained via a string of homotopy equivalences. Step 1: D(G)^{T} = B̂(G)^{T} ↪ Ŝ(G)^{T} ↪ S̃(G)^{T} are N_{G}(T)-homotopy equivalences, since D(G) = B̂(G) ↪ Ŝ(G) ↪ S̃(G) are G-homotopy equivalences by Theorem 2.12(iii) and (iv). Step 2: S̃(G)_{≥T} ↪ S̃(G)^{T} is an N_{G}(T)-homotopy equivalence by Theorem 2.6(iii) and 2.11(i). Step 3: If T ∈ S̃(G) then S̃(G)_{≥T} is contractible, a cone on T, and we are done. Thus we assume T ∉ S̃(G), so that S̃(G)_{≥T} = S̃(G)_{>T}.

Step 4: Ŝ(G)_{>T} ↪ S̃(G)_{>T} is an N_{G}(T)-homotopy equivalence by Proposition 2.8. We are also using 2.11(i) and Theorem 2.12(iii). The latter implies S̃(G) ∩ B(G) ⊆ Ŝ(G). Step 5: Ŝ(G)_{>T}^{≤C} ↪ Ŝ(G)_{>T} is an N_{G}(T)-homotopy equivalence by Theorem 2.6(ii), where the poset map used is F (P) = P ∩ C, for P ∈ Ŝ(G)_{>T}. Since T < P, we have Z(P) ≤ C_{G}(T) ≤ C, so that Z(P) ≤ P ∩ C. Step 6: The inclusion Ŝ(G)_{>T}^{≤C} ↪ S̃(G)_{>T}^{≤C} is an N_{G}(T)-homotopy equivalence by Lemma 3.2, applied with M = T. Step 7: S̃(G)_{≥O_{C}}^{≤C} ↪ S̃(G)_{>T}^{≤C} is an N_{G}(T)-homotopy equivalence by Theorem 2.6(ii), with poset map F (P) = P · O_{C} for P ∈ S̃(G)_{>T}^{≤C}. Since we have assumed that O_{C} ∈ S̃(G), we have S̃(G)_{≥O_{C}}^{≤C} is contractible, a cone on O_{C}. Thus we have shown that D^{T}(G) is N_{G}(T)-contractible. □

We arrive at the main result of this section. It asserts, that under certain assumptions on the shape of the centralizer of a purely noncentral p-subgroup T of G, the structure of the fixed point set D^{T}(G) is determined by the collection of p-centric and p-radical subgroups in a subquotient (or section) of this centralizer. Our Theorem 3.6 generalizes the result of [MO10, Thm. 4.14]. Before stating and proving the theorem, we need to introduce some more notation.

Notation 3.4. Let T be any p-subgroup of G, and denote C = T · C_{G}(T) and O_{C} = O_{p}(T · C_{G}(T)), maintaining the notation introduced in 3.1. Let L ⊴ C be a normal subgroup of C with O_{C} ≤ L. Denote the quotient groups by K = C/L and H = L/O_{C}, so we can write C = O_{C}.H.K = L.K. Let S_{C} ∈ Syl_{p}(C) and extend it to S ∈ Syl_{p}(G). Since T ≤ S_{C} ≤ S it follows that Z(S) ≤ Z(S_{C}). Next, define S_{L} = S_{C}∩ L, a Sylow p-subgroup of L. As S_{L} ⊴ S_{C}, Z(S_{C}) ∩ S_{L} ≠ 1. However, this does not guarantee that Z(S) ∩ S_{L} ≠ 1. The sequence of subgroups S_{L} ⊴ S_{C} ≤ S chosen as above will be denoted (S_{L},S_{C},S) and will be referred to as a triple of Sylow p-subgroups.

3.5. We warn the reader that the “hat” symbol in Ŝ(G) and Ŝ(H) has different meanings; in the former it refers to those groups that are distinguished in G, while in the latter it refers to subgroups that are distinguished in H.

Theorem 3.6. Let G be a finite group of parabolic characteristic p, and let T be a p-subgroup of G. Assume the following hold:

(N1). The group O_{C} is purely noncentral in G. (N2). The group T ·C_{G}(T) has the form C = O_{C}.H.K, the group H has parabolic characteristic p,

and L = O_{C}.H is a normal subgroup of N_{G}(T). (N3). There exists a triple of Sylow p-subgroups (S_{L},S_{C},S) with the property that Z(S) ∩ S_{L} ≠ 1.

Then there is a N_{G}(T)-equivariant homotopy equivalence between D(G)^{T} and D(H).

3.7. The condition (N3) implies that if S′_{C} ∈ Syl_{p}(C), then there exists S′∈ Syl_{p}(G) such that Z(S′) ∩ (S′_{C}∩ L) ≠ 1. There exists an element g ∈ C with S′_{C} = gS_{C}g^{−1} . Define S′= gSg^{−1} . Since L ⊴ C, S′_{L} = S′_{C}∩ L = g(S_{C}∩ L)g^{−1} . Thus Z(S′) ∩ S′_{L} = g(Z(S) ∩ S_{L})g^{−1} .

VERTICES OF LEFSCHETZ MODULES 9

There are several examples involving sporadic finite simple groups in which a condition stronger than (N3) is satisfied: all p-central involutions of G which lie in C actually lie in L. This condition was used to prove a similar theorem; see [MO09, Thm. 2.1 and Lem. 2.3].

3.8. In some examples we will consider in Section 6, we have L = C_{G}(O_{C}). Then whenever we have a triple (S_{L},S_{C},S) of Sylow p-subgroups, we will have Z(S) ≤ L since O_{C} ≤ S. Thus (N3) is satisfied. Also, (N3) will be satisfied if we know that Z(S_{C}) ≤ L, which will be true in most cases.

There is no direct relationship between the elements which are p-central in G and those which are p-central in C, or in L. In order to deal with this difficulty, we introduce the following subcollection of Ŝ(G)_{>O_{C}}^{≤C}.

3.9. Use the notations from 2.4, 3.1 and 3.4 and set:

T = {P ∈ Ŝ(G)_{>O_{C}}^{≤C} | Z(P) ∩ Z(S) ∩ Z(S_{L}) ≠ 1, for some triple (S_{L},S_{C},S) withP ≤ S_{C} ≤ S}.

This collection is contained in Ŝ(C), and it contains all the p-centric subgroups P in C which properly contain O_{C}, since if P is p-centric and P ≤ S_{C} ≤ S, then Z(S) ≤ C_{G}(P), implying Z(S) ≤ Z(P). Another collection of interest will be T ^{≤L}; the p-subgroups it contains are distinguished as subgroups of G and also are distinguished as subgroups of L. The condition N_{G}(T) ≤ N_{G}(L) in (N2) is needed so that N_{G}(T) acts on T ^{≤L}. This condition will be satisfied if L is the generalized Fitting subgroup of C (there exists one component E of C with E/Z(E) = H).

3.10. Let P ∈ T and assume that O_{C} < Q ≤ P. Then N_{P}(Q) ∈ T . It is easy to see that if Q ≤ N_{P}(Q) ≤ P then Z(P) ≤ Z(N_{P}(Q)) and this suffices to show N_{P}(Q) ∈ T .

Proof of Theorem 3.6. The proof will consist of five steps and a number of homotopy equivalences between various intermediary collections. For the rest of the proof we assume that G and T satisfy all the conditions from the hypotheses of Theorem 3.6.

Step 1: D(G)^{T} and Ŝ(G)_{>O_{C}}^{≤C} are N_{G}(T)-homotopy equivalent. The proof of Theorem 3.3 shows that there is an N_{G}(T)-homotopy equivalence between D(G)^{T} and S̃(G)_{≥O_{C}}^{≤C} = S̃(G)_{>O_{C}}^{≤C}, using that O_{C} ∉ S̃(G). Then an application of Lemma 3.2 with M = O_{C} gives an equivalence with Ŝ(G)_{>O_{C}}^{≤C}.

Step 2: The inclusion T ↪ Ŝ(G)_{>O_{C}}^{≤C} is a N_{G}(T)-homotopy equivalence.

We will apply Theorem 2.6(i) once again. We need to show that T_{≥Q} is equivariantly contractible for every Q ∈ Ŝ(G)_{>O_{C}}^{≤C}. The equivariance here is with respect to the action of N_{G}(T) ∩ N_{G}(Q). Set O_{CQ}:= O_{p}(N_{C}(Q)). Assume now that P ∈ T_{≥Q} and consider the contracting homotopy given by the following string of equivariant poset maps:

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

In order to complete the proof we need to prove that each of the subgroups involved in the above string lies in T_{≥Q}. Since Q is a distinguished p-subgroup of G, N_{G}(Q) has characteristic p, by

10 JOHN MAGINNIS AND SILVIA ONOFREI

2.11(iv). Next, because T < Q ≤ C it follows that C_{G}(Q) ≤ C_{G}(T) ≤ C and thus QC_{G}(Q) ≤ N_{C}(Q) ≤ N_{G}(Q), and by Proposition 2.2(i), N_{C}(Q) has characteristic p. Hence C_{N_{C}(Q)}(O_{CQ}) ≤ O_{CQ}. But Q ≤ O_{CQ}so C_{G}(O_{CQ}) ≤ C_{G}(Q) ≤ N_{C}(Q) and therefore C_{G}(O_{CQ}) = Z(O_{CQ}) which implies Z(S_{C}) ≤ Z(O_{CQ}) for any S_{C} ∈ Syl_{p}(C) which contains O_{CQ}. Extend S_{C} to a Sylow p-subgroup S of G satisfying Z(S) ∩ S_{L} ≠ 1, using (N3) and 3.7. Recall that Z(S) ≤ Z(S_{C}). Since Z(S) ∩ Z(S_{L}) ≤ Z(S_{C}) ≤ O_{CQ}, it follows that O_{CQ}∈ T . The fact that N_{P}(Q) ∈ T_{≥Q} follows from 3.10. Next consider N_{P}(Q)O_{CQ}which is a p-subgroup of N_{C}(Q) and which contains Z(S_{C}) for any S_{C}∈ Syl_{p}(C) which contains it. Thus all the groups in the string are in T_{≥Q}.

Step 3: The inclusion T ^{≤L} ↪ T is a N_{G}(T)-homotopy equivalence. Apply 2.6(ii) with poset map F (P) = P ∩ L for P ∈ T . Note that Z(P) ∩ Z(S) ∩ Z(S_{L}) ≤ Z(P ∩ L). Step 4: Recall that L ≃ O_{C}.H and the quotient group H ≃ L/O_{C} has parabolic characteristic p. The quotient map will be denoted q : L → H. For M ≤ L, let \overline{M} = q(M). For Q ≤ L, denote O_{LQ} = O_{p}(N_{L}(Q)); for \overline{Q} ≤ H, denote O_{HQ} = O_{p}(N_{H}(\overline{Q})). If (S_{L},S_{C},S) is a triple of Sylow psubgroups defined as in 3.4, we let \overline{S}_{L} = q(S_{L}) ≃ S_{H} ∈ Syl_{p}(H). We prove the following:

The poset map q_{∗} : T ^{≤L} → Ŝ(H) induced by the quotient map q: L → H is an equivariant homotopy equivalence.

We first show that q_{∗}(T ^{≤L}) ⊆ Ŝ(H). Since O_{C} is purely noncentral in G, the map q : L → H is injective on the elements of Z(S) as they are p-central in G. Therefore, if P ∈ T ^{≤L} then Z(P) ∩ Z(S) ∩ Z(S_{L}) ≠ 1 for some triple of Sylow p-subgroups (S_{L},S_{C},S) with P ≤ S_{L}, and this implies Z(\overline{P}) ∩ Z(\overline{S}_{L}) ≠ 1, and we have q_{∗}(P) = \overline{P}∈ Ŝ(H). According to Theorem 2.7, the poset map q_{∗}: T ^{≤L} → Ŝ(H) is an equivariant homotopy equivalence if q_{∗}^{−1} (Ŝ(H)_{≥\overline{Q}}) is equivariantly contractible for any \overline{Q}∈ Ŝ(H). Define Q = q^{−1} (\overline{Q}). Recall that O_{C} ≤ P, thus we have:

q_{∗}^{−1} (Ŝ(H)_{≥\overline{Q}}) = \{P ∈ T ^{≤L} | \overline{Q}≤ \overline{P}\} = \{P ∈ T ^{≤L} | Q ≤ P\} = T_{≥Q}^{≤L} .

For P ∈ T_{≥Q}^{≤L}, consider the string of equivariant poset maps given by:

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

We need to show that all of these terms lie in T_{≥Q}^{≤L}. The fact that N_{P}(Q) ∈ T_{≥Q}^{≤L} follows from 3.10. Next, we have N_{L}(Q) = q^{−1} (N_{H}(\overline{Q})), using O_{C} ≤ Q. Thus O_{LQ} is equal to q^{−1} (O_{HQ}) by the correspondence theorem for normal subgroups applied to N_{L}(Q) → N_{H}(\overline{Q}). Since \overline{Q}∈ Ŝ(H), N_{H}(\overline{Q}) has characteristic p, by 2.11(iv) and using our assumption that H has parabolic characteristic p. Thus C_{H}(O_{HQ}) ≤ O_{HQ} and N_{H}(O_{HQ}) also has characteristic p, which follows by an application of Proposition 2.2(ii). Next, we have C_{L}(O_{LQ}) ≤ q^{−1} (C_{H}(O_{HQ})) ≤ q^{−1} (O_{HQ}) = O_{LQ}, which shows that the group O_{LQ} is p-centric in L. It follows that Z(S_{L}) ≤ Z(O_{LQ}) for every Sylow p-subgroup of L which contains O_{LQ}. But by our assumption (N3), and using 3.7, the subgroup Z(S_{L}) contains p-central elements of G; therefore it follows that O_{LQ}is distinguished in G. Consequently O_{LQ}lies in T_{≥Q}^{≤L}. Now consider N_{P}(Q)O_{LQ} which is a subgroup of N_{L}(Q). Since O_{LQ} lies in every Sylow p-subgroup of N_{L}(Q) and P ∈ T_{≥Q}^{≤L} it follows that the nontrivial elements in Z(P) ∩ Z(S) ∩ Z(S_{L}) also lie in Z(N_{P}(Q)O_{LQ}) and the subgroup N_{P}(Q)O_{LQ} is indeed in T_{≥Q}^{≤L} (that is O_{LQ}≤ S_{L}, with S_{L} the Sylow p-subgroup which contains P and satisfies the required condition from 3.9).

Step 5: There is an equivariant homotopy equivalence between D(G)^{T} and D(H). Combine the first four steps of the proof to obtain the chain of N_{G}(T)-homotopy equivalences:

D(G)^{T} ≃ Ŝ(G)_{>O_{C}}^{≤C} ≃ T ≃ T ^{≤L} ≃ Ŝ(H).

Recall that H has parabolic characteristic p and therefore Ŝ(H) and D(H) are H-homotopy equivalent, by Proposition 2.12(iii)-(iv). The proof in [MO10] of this homotopy equivalence can actually be seen to yield equivariance under any automorphism of H, so that here we have equivariance under N_{G}(T)/O_{C}, or under N_{G}(T) with O_{C} acting trivially. □

4. Some modular representation theory

On the group algebra of a group of parabolic characteristic p. Let G be a finite group and let k be a field of characteristic p, which is a splitting field for all the subgroups of G. Denote by kG the group algebra of G over the field of coefficients k. We are interested in the block structure of kG and in particular its defect groups. For details on the background material used in this section the interested reader is referred to [Alp86], or for a more succinct exposition to [Ben98]. Although not all the results from this Section are needed in the later proofs, the information given here provides valuable insight into the properties of the group algebras of those classes of groups considered in the next two Sections.

4.1. If a finite group G has characteristic p, then C_{G}(O_{p}(G)) ≤ O_{p}(G) and kG has only one block [Ben98, Prop. 6.2.2].

It is now easy to prove that:

Lemma 4.2. Let G be a finite group of local characteristic p. Then every nonprincipal block of kG has defect zero.

Proof. If B is a block of kG with a nontrivial defect group D, then by Brauer’s First Main Theorem there is a corresponding block b of kN_{G}(D). Since G has local characteristic p, the subgroup N_{G}(D) has characteristic p and thus the only block is the principal block, by 4.1. Then Brauer’s Third Main Theorem implies that B is the principal block of kG. □

Lemma 4.3. Assume that a finite group G has parabolic characteristic p. If D is a defect group of a nonprincipal block of kG, then D is purely noncentral.

Proof. Let B be a block of kG with a nontrivial defect group D, and let b denote its Brauer correspondent in kN_{G}(D). If D contains a p-central element, then according to 2.11(iv), N_{G}(D) has characteristic p. Thus b is the principal block of kN_{G}(D), and B is the principal block of kG. □

12 JOHN MAGINNIS AND SILVIA ONOFREI

4.4. If G has parabolic characteristic p, the only distinguished p-radical subgroup which is a defect group of kG is the Sylow p-subgroup of G, and the only block with this defect group is the principal block of kG.

Next, we relate the defect groups of kG to the defect groups of the group algebra kC_{G}(t) of the centralizer of an element t of order p.

Lemma 4.5. Let G be a finite group and let D be a defect group of a block of kG. If T ≤ Z(D) then D is a defect group of kC_{G}(T).

Proof. Let C denote T C_{G}(T) = C_{G}(T). Since T ≤ Z(D) it follows that DC_{G}(D) ≤ N_{G}(D) ∩ C ≤ N_{G}(D). By Brauer’s Extended First Main Theorem, kN_{G}(D) has a block B with defect group D, and kDC_{G}(D) has a block b with defect group D, with b^{N_{G}(D)}= B. The Brauer correspondents b^{N_{C}(D)} and b^{N_{G}(D)} are defined, and b^{N_{G}(D)}= (b^{N_{C}(D)})^{N_{G}(D)}. There is a defect group D′of b^{N_{C}(D)}containing the defect group D of b, and up to conjugacy D′is contained in the defect group D of b^{N_{G}(D)}; see [Alp86, Lem. 14.1(1)]. Thus D′= D and so kN_{C}(D) has a block b^{N_{C}(D)} with defect group D. Another application of Brauer’s First Main Theorem to N_{C}(D) ≤ C gives that kC has a block b^{C} with defect group D. □

4.6. If t is an element of order p and T := ⟨t⟩, the index of C_{G}(t) in N_{G}(T) is relatively prime to p. Since DC_{G}(D) ≤ C_{G}(t) ⊴ N_{G}(T), an application of [Alp86, Thm. 15.1(2)(4)(5)] tells us that the defect groups of kN_{G}(T) are the same as the defect groups of kC_{G}(t).

Lemma 4.5 does not give any result regarding a defect group for kG, when a defect group for kC_{G}(t) is given. However, the following consequence of [NT89, Thm. 5.5.21] is useful.

4.7. Let t be an element of order p in G and set C = C_{G}(t). Let b be a block of kC with defect group Q. The group T = ⟨t⟩ is normal in C, and since Q is a p-radical subgroup of C, it follows that T ⊴ Q and C_{G}(Q) ≤ C. Thus b^{G} is defined. Then for a suitable defect group D of b^{G}, Z(D) ≤ Z(Q) ≤ Q ≤ D and Q = D ∩ C.

We record the following direct consequence of the above discussion:

Proposition 4.8. Let G be a finite group of parabolic characteristic p, and let D be a defect group of a nonprincipal block of kG. For any t ∈ Z(D), the subgroup O_{p}(C_{G}(t)) is purely noncentral.

Proof. The subgroup D is purely noncentral by Lemma 4.3, and according to Lemma 4.5 the subgroup D is also a defect group of a block of kC_{G}(t). Thus O_{p}(C_{G}(t)) ≤ D. □

The group ring of O_{C}.H.K. Assume that C is a finite group of the form C = O_{C}.H.K, with O_{C} = O_{p}(C). Set H̃ = C/O_{C} = H.K. We also assume that K = H̃/H is either a p-group or a p′-group, and that H is either a group of Lie type in characteristic p or a group of local characteristic p. These cases will be significant to the study of those groups we are interested in.

VERTICES OF LEFSCHETZ MODULES 13

Lemma 4.9. Let C = O_{C}.H.K with O_{C} = O_{p}(C), and assume that K is a p′-group and that H has local characteristic p. The defect groups of blocks of kC are either equal to O_{C} or are Sylow p-subgroups of C.

Proof. Let B be a block of kC with defect group D. Since O_{C} is a normal p-subgroup, O_{C}≤ D, and there exists a block b̃ of H̃ dominated by B (or contained in B) with a defect group equal to D/O_{C} [Nav98, Thm 9.9(b)]. Next, there exists a block b of kH which is covered by the block b̃. If D′ is a defect group of b, then there exists a defect group D̃ of b̃ such that D′= D̃∩ H [Alp86, Thm. 15.1(2)]. Since K is a p′-group, we must have D′= D̃, which up to conjugacy equals D/O_{C}. We have assumed that H has local characteristic p, and all of the nonprincipal blocks of kH have defect zero. So either D = O_{C} or D/O_{C} ∈ Syl_{p}(H) and D ∈ Syl_{p}(C). □

Lemma 4.10. Let C = O_{C}.H.K with O_{C} = O_{p}(C). Assume K is a p-group and that H is a finite simple group of Lie type in characteristic p. Then kC has at most two blocks, and if a nonprincipal block exists, it has defect group of the form D = O_{C}.K.

Proof. Note that kH has exactly two blocks, the principal block b_{0} and a block b_{1} of defect zero (containing the Steinberg module). These are both invariant under the action of H̃. Since K is a p-group, each block of kH is covered by a unique block of H̃ [Nav98, Cor. 9.6]. So kH̃ has exactly two blocks, the principal block b̃_{0} (covering b_{0}) and one nonprincipal block b̃_{1} (covering b_{1}). The defect group D̃ of the nonprincipal block b̃_{1} satisfies D̃∩ H = 1 and has order equal to the order of K [Alp86, Thm. 15.1(2)(4)]. Thus D̃ is isomorphic to K. (Note this shows H̃ = H : K is a split extension.) Finally, since O_{C} is a normal p-subgroup of C, each block B of kC dominates some block of kH̃ and each block of kH̃ is dominated by exactly one block of kC. Thus either kC has one block only (dominating both blocks of kH̃), or kC has two blocks, the principal block B_{0} and one nonprincipal block B_{1} with defect group D. We have O_{C} ≤ D and D/O_{C} = D̃≃ K. □

Lemma 4.11. Let C = O_{C}.H.K with O_{C} = O_{p}(C). Assume K is a p-group and that H is a finite simple group of Lie type in characteristic p. Denote L = O_{C}.H, and assume that L is the generalized Fitting subgroup of C. Then kC has exactly two blocks.

Proof. Note that C has one component (a quasisimple group) E with center Z(E) a p-group and E/Z(E) = H. Elements of the component commute with elements of O_{C} [Asc00, 31.6(2)], so that C/C_{C}(O_{C}) is a p-group. This implies that there is a bijection between the blocks of kC and the blocks of kH̃ [Nav98, Thm. 9.10]. □

5. The vertices of the indecomposable summands of reduced Lefschetz module

In this section we determine the vertices of the indecomposable summands of the reduced Lefschetz module associated to the complex of p-centric and p-radical subgroups, denoted D(G), in a group with parabolic characteristic p which also has certain p-local properties. We start by reviewing a few basic facts on Lefschetz modules. All modules are assumed to be finitely generated.

14 JOHN MAGINNIS AND SILVIA ONOFREI

The representation ring or the Green ring a(G) has generators the isomorphism classes [M] of kG-modules and relations given by direct sums and k-tensor products. Since the Krull-Schmidt theorem applies, each element of the Green ring can be written as [M] = ∑ n_{i}[M_{i}], where [M_{i}] are isomorphism classes of indecomposable kG-modules. The additive structure of a(G) is of a free abelian group with one generator for each class [M_{i}]. The Grothendieck ring R(G) = a(G)/a_{0}(G) is a quotient ring of the Green ring a(G) by the ideal spanned by difference elements of the form M_{2}−M_{1}−M_{3} where 0 → M_{1} → M_{2} → M_{3} → 0 is a short exact sequence of kG-modules. Each module is identified with its simple factors from the composition series, so two modules with the same composition factors are identified in the Grothendieck ring. The Grothendieck ring of G over k is a free abelian group on the set of isomorphism classes of simple kG-modules. The simple modules are determined by their (complex) characters and the Grothendieck ring is isomorphic to the ring of characters. When a group G acts admissibly on a simplicial complex Δ, we can construct the Lefschetz module by taking the alternating sum of the k-vector spaces spanned by the chain groups. To obtain the reduced Lefschetz module, subtract the trivial one dimensional representation. The n^{th}chain group C_{n}(Δ;k) is a permutation module and a k-module with oriented simplices as generators. Thus the reduced Lefschetz module is a virtual module, an element of the Green ring:

L̃_{G}(Δ;k) = ∑_{σ∈Δ/G} (−1)^{|σ|}Ind_{G_{σ}}^{G}(k) − k

The reduced Lefschetz module associated to a subgroup complex Δ is not always projective; however, it was shown by Thévenaz [Thé87, Thm. 2.1] that this virtual module turns out to be, in many cases, projective relative to a collection of very small order p-groups. Specifically, L̃_{G}(Δ;k) is X - relatively projective, where X is a collection of p-subgroups such that Δ^{Q} is acyclic (for example contractible) for every p-subgroup Q which is not in X. Information about fixed point sets leads to details about the vertices of indecomposable summands of this virtual module. The following theorem is due to Robinson [Rob88, in proof of Cor. 3.2]; also see [Saw06a, Lem. 1]. This result is our main tool for finding vertices for reduced Lefschetz modules.

Theorem 5.1 (Robinson). The number of indecomposable summands of L̃_{G}(Δ;k) with vertex Q is equal to the number of indecomposable summands of L̃_{N_{G}(Q)}(Δ^{Q};k) with vertex Q.

We give below a few applications of Robinson’s Theorem 5.1.

Lemma 5.2. Assume G has parabolic characteristic p. Suppose that t is an element of order p in G such that O_{p}(C_{G}(t)) contains central elements. Then no vertex of the reduced Lefschetz module L̃_{G}(D(G);k) contains a conjugate of t.

Proof. Recall that according to Theorem 3.3, the fixed point set D(G)^{t}is contractible. Hence D(G)^{t} is mod-p acyclic, and an application of Smith theory [Bro94, Thm. VII.10.5(b)] gives that D(G)^{Q} is mod-p acyclic for any p-group Q containing t. Therefore L̃_{N_{G}(Q)}(D(G)^{Q};k) = 0. Then Theorem 5.1 implies that the reduced Lefschetz module L̃_{G}(D(G);k) has no indecomposable summand with vertex Q containing a conjugate of t. □

Proposition 5.3. Assume G has parabolic characteristic p, and let Q be a vertex of an indecomposable summand of the reduced Lefschetz module L̃_{G}(D(G);k). Assume that t is an element of order p in the center Z(Q) such that conditions (N1), (N2) and (N3) hold for C = C_{G}(t). Then Q is a vertex of a summand of L̃_{C}(D(H);k).

Proof. By Robinson’s Theorem 5.1, Q is a vertex of a summand of L̃_{N_{G}(Q)}(D(G)^{Q};k). Consider the restriction of this Lefschetz module to the subgroup C ∩N_{G}(Q), obtaining the Lefschetz module L̃_{N_{C}(Q)}(D(G)^{Q};k). There exists an indecomposable summand of L̃_{N_{C}(Q)}(D(G)^{Q};k) with vertex Q [Alp86, Lem. 9.5]. Since D(G)^{Q} = (D(G)^{t})^{Q}, another application of Theorem 5.1 yields a summand of L̃_{C}(D(G)^{t};k). Theorem 3.6 says that D(G)^{t} is equivariantly homotopy equivalent to D(H), so L̃_{C}(D(G)^{t};k) and L̃_{C}(D(H);k) are isomorphic virtual kC-modules. □

Note that t can be replaced in the above proof by any subgroup T ≤ Z(Q), where C = T C_{G}(T). Also C can be replaced by N_{G}(T).

Theorem 5.4. Let G be a finite group of parabolic characteristic p. Let T be a p-subgroup of G and set C = T C_{G}(T). Assume that the following conditions hold:

(i). C = O_{C}.H.K where O_{C} = O_{p}(C) and L = O_{C}.H is the generalized Fitting subgroup of C; (ii). The group H = L/O_{C} is a finite simple group of Lie type in characteristic p; (iii). There exists a triple of Sylow p-subgroups (S_{L},S_{C},S) with Z(S) ∩ S_{L} ≠ 1.

Then T is a vertex of an indecomposable summand of the reduced Lefschetz module L̃_{G}(D(G);k) if and only if:

(a). T = O_{C} and T is purely noncentral, (b). K = C/L is a p′-group, (c). the index of C in N_{G}(T) is relatively prime to p.

Under these conditions, there will exist a unique summand of L̃_{G}(D(G);k) with vertex T, which will lie in a block with defect group T.

Proof. Theorem 5.1 tells us that T is a vertex of a summand of L̃_{G}(D(G);k) if and only if T is a vertex of a summand of L̃_{N_{G}(T)}(D(G)^{T};k). By Lemma 5.2, if O_{C} contains central elements T is not a vertex; thus O_{C} is purely noncentral. The generalized Fitting subgroup L is characteristic in C and so is normal in N_{G}(T). Denote the quotient group by K′= N_{G}(T)/L. Note that K′will be a p′-group if and only if K is a p′-group and the index of C in N_{G}(T) is relatively prime to p. We have N_{G}(T) = O_{C}.H.K′. The group C has one component E with center Z(E) a p-group and E/Z(E) = H. Elements of O_{C} commute with elements of E.

16 JOHN MAGINNIS AND SILVIA ONOFREI

Theorem 3.6 tells us that the fixed point set D(G)^{T} is N_{G}(T)-homotopy equivalent to the Tits building Δ for the Lie group H. Thus their reduced Lefschetz modules are isomorphic virtual kN_{G}(T)-modules. Denote by M = L̃_{N_{G}(T)}(Δ;k). The subgroup O_{C} acts trivially on M (by definition of the action of N_{G}(T) on Δ) and so M is the inflation of a module M̃ over the group H̃′= N_{G}(T)/O_{C} = H.K′. The restriction of M̃ to the Lie group H is up to a sign the Steinberg module St_{H}, which is irreducible. This implies that M and M̃ are also irreducible modules. The restriction M′of M to L = O_{C}.H is also irreducible, being the inflation of ±St_{H}. The module M̃ is referred to as an extended Steinberg module; see [Sch92] and [MS95]. Next, kH has only two blocks, the principal block b_{0} and a defect zero block b_{1} containing the Steinberg module. Then kL has two blocks also, the principal block B_{0} and a block B_{1} which contains the module M′, using that E centralizes O_{C} and [Nav98, Thm. 9.10]. The block B_{1} dominates (or contains) the block b_{1}. The normal p-subgroup O_{C} is contained in the vertex Q_{1} of the irreducible module M′[NT89, Thm. 4.7.8]. The defect group D_{1} of the block B_{1} contains O_{C} (and also Q_{1}), and the quotient D_{1}/O_{C} equals 1, the defect group of the block b_{1} [Nav98, Thm. 9.9(b)]. Therefore Q_{1} = D_{1} = O_{C}. Let B be the block of kN_{G}(T) which contains the module M. Note that B covers the block B_{1}. The defect group D of B satisfies D ∩ L = D_{1} = O_{C} by [Alp86, Thm. 15.1(2)]. If Q is the vertex of M, then O_{C} ≤ Q and O_{C} ≤ Q ∩ L ≤ D ∩ L = O_{C}. Also the quotient S′= Q/(Q ∩ L) = Q/O_{C} is a Sylow p-subgroup of K′= N_{G}(T)/L by [Alp86, Lem. 9.8]. Since Q = O_{C}.S′is a subgroup of the p-group D, the quotient map from N_{G}(T) → K′applied to D has kernel O_{C} and image S′. Therefore Q = D = O_{C}.S′. Since T ≤ O_{C}, we have T = Q if and only if T = O_{C} and S′= 1. In this situation, K′ is a p′- group. Under these conditions, there will be a unique summand of the reduced Lefschetz module L̃_{G}(D(G);k) with vertex T, and this summand will lie in a block with defect group T because of the relationship between the Brauer correspondence and the Green correspondence. □

5.5. Work of Mühlherr and Schmid [MS95, Thm. C and Lem. 7] concerning the action of an automorphism of H on the associated building Δ can be applied to H ⊴ H̃′. If the value of the rational character χ of the extended Steinberg module M̃evaluated on an element g ∈ H̃′is nonzero, then the fixed point set Δ^{g} is a Moufang complex. Also, χ(g) ≠ 0 if and only if the p-part of g is conjugate to an element of the defect group S′of the block containing M̃. If g ∈ S′, then Δ^{g} is a building for the group O^{p′}(C_{H}(g)). Since O_{C} acts trivially on Δ, we can phrase this result in terms of the character of the inflated extended Steinberg module M, with fixed point set Δ^{g}, a building when g ∈ D = O_{C}.S′. It follows that D(G)^{D} is homotopy equivalent to the building Δ^{D} =Δ^{S′} for O_{C}.O^{p′}(C_{H}(S′)). As noted in the proof of Lemma 4.10 the extension H : S′ is split and S′ is a subgroup of H̃′. This homotopy is equivariant with respect to N_{G}(T) ∩ N_{G}(D), which contains the group DC_{G}(D). To obtain a homotopy equivariant with respect to N_{G}(D), we require more information about DC_{G}(D) so that we can apply Theorem 3.6 to DC_{G}(D).

VERTICES OF LEFSCHETZ MODULES 17

Proposition 5.6. Assume that H is a normal subgroup of N_{G}(T) with complement a group D containing O_{C} = O_{p}(T C_{G}(T)). Denote S′= D/O_{C}, and assume that D = O_{C}: S′is a split extension. Then DC_{G}(D) = DC_{H}(S′).

Proof. Note that N_{G}(T) = H · D = (O_{C}× H) : S′. Since T ≤ O_{C} ≤ D, we have C_{G}(D) ≤ C_{G}(T) ≤ N_{G}(T), so that C_{G}(D) = C_{N}(D), denoting N = N_{G}(T). An element of C_{N}(D) can be written uniquely as a product xy with x ∈ H and y ∈ D, such that for all z ∈ D, xy = z(xy)z^{−1} = (zxz^{−1} )(zyz^{−1} ). Since zxz^{−1} ∈ H and zyz^{−1} ∈ D, we must have zxz^{−1} = x and so x ∈ C_{H}(D). Since elements of H commute with elements of O_{C}, C_{H}(D) = C_{H}(S′). It now follows that DC_{G}(D) = DC_{H}(D) = DC_{H}(S′). □

5.7. In the above situation, we can write DC_{G}(D) in the form D.H_{D}.K_{D}, with H_{D} = O^{p′}(C_{H}(S′)) and K_{D} = C_{H}(S′)/H_{D}. Usually, when H is a finite simple group of Lie type, the extension H : S′is split and H_{D} will also be a finite simple group of Lie type. Then H_{D} will be a component of DC_{G}(D) and will be normal in N_{G}(D). As noted in 5.5, under the conditions of Theorem 5.4, D(G)^{D} will be a building and thus not contractible, so D must be purely noncentral. Finally, K_{D} is a p′-group, and the conditions for Theorem 3.6 will be satisfied.

Proposition 5.8. Assume N = N_{G}(T) = O_{C}.(H : S′) and assume C_{H}(S′) is a simple group. Also assume that D = O_{C}.S′is purely noncentral in G. Then DC_{G}(D) = D.C_{H}(S′).

Proof. Note that DC_{G}(D) = DC_{N}(D) ⊴ N_{N}(D), and under the quotient map N → H : S′, the image of N_{N}(D) equals N_{H:S′}(S′) = S′N_{H}(S′). Next, take the quotient modulo S′to obtain N_{H}(S′). The image of DC_{G}(D) under the composition of these two quotient maps is a normal subgroup of the simple group C_{H}(S′). It is impossible for C_{G}(D) ≤ D to be purely noncentral, since if D ≤ S ∈ Syl_{p}(G) then Z(S) ≤ C_{G}(D). So we must have DC_{G}(D) = D.C_{H}(S′). □

Proposition 5.9. Let G be a finite group of parabolic characteristic p. Let T be a p-subgroup of G and set C = T C_{G}(T). Assume that:

(i). Conditions (N1), (N2) and (N3) hold for C. In particular C = O_{C}.H.K with O_{C}:= O_{p}(C). (ii). The subgroup L = O_{C}.H is the generalized Fitting subgroup of C.

Then O_{C}· C_{G}(O_{C}) is of the form O_{C}.H.\overline{K}for some subgroup \overline{K}≤ K, and conditions (N1), (N2) and (N3) are satisfied for O_{C}· C_{G}(O_{C}).

Proof. The group C has one component E with center Z(E) a p-group and E/Z(E) = H. We have E ≤ C_{G}(O_{C}), and thus L = O_{C}.H ≤ O_{C}·C_{G}(O_{C}). In order to prove that (N2) holds for O_{C}·C_{G}(O_{C}), we have to show that L is a normal subgroup of N := N_{G}(O_{C}). Observe that it suffices to show that E is the only component of N, since in this case L is the product of two normal subgroups of N, namely O_{C} and E. Let E(N) denote the layer of N, the central product of all the components of N. Since E(N) centralizes the Fitting subgroup of N, which contains O_{C}, it follows that E(N) ≤ O_{C}·C_{G}(O_{C}) ≤ C. Thus E(N) = E. Define \overline{K}to be the image of O_{C}·C_{G}(O_{C}) under the quotient map C → C/L = K. Thus conditions (N1) and (N2) are satisfied for O_{C}· C_{G}(O_{C}) = O_{C}.H.\overline{K}. Condition

18 JOHN MAGINNIS AND SILVIA ONOFREI

(N3) also holds since any Sylow p-subgroup S_{O_{C}} of O_{C}· C_{G}(O_{C}) can be extended to a Sylow psubgroup S_{C} of T C_{G}(T), and we have S_{L}= S_{C}∩L is a Sylow p-subgroup of L. Then using 3.7 there exists a Sylow p-subgroup S of G extending S_{C} such that Z(S) ∩ S_{L} ≠ 1. □

The reduced Lefschetz module L̃_{G}(C(G);k) was studied by Sawabe [Saw06a], in the more general context when G is any finite group and C(G) is a collection of subgroups of G which is closed under p-overgroups. Sawabe applied his results to the collection C(G) of p-centric subgroups of G, which is equivariantly homotopy equivalent to D(G). The collection S̃(G) also satisfies this condition by 2.11(i), and for G of parabolic characteristic p is equivariantly homotopy equivalent to D(G) by Theorem 2.12(iii-iv). We give below two of Sawabe’s results.

Proposition 5.10 (Sawabe). Assume that G is a finite group and let C(G) denote a collection of p-subgroups of G that is closed under taking p-overgroups. Take V in B(G) \ C(G) of maximal order. Then the following hold:

(i). [Saw06a, Prop. 4] For any p-subgroup Q of order greater than that of V, the fixed point set

C(G)^{Q} is N_{G}(Q)-contractible. In particular Q is not a vertex of L̃_{G}(C(G);k). (ii). [Saw06b] The group V is a vertex of an indecomposable summand of L̃_{G}(C(G);k) if and only

if L̃_{N_{G}(V)}(C(G)^{V};k) ≠ 0.

We end this Section with a result which asserts that under our working assumptions the defect groups of the nonprincipal blocks of kG are vertices for the nonprojective indecomposable summands of the reduced Lefschetz module associated to the collection of p-centric and p-radical subgroups of G.

Theorem 5.11. Let G be a finite group of parabolic characteristic p. Let T be a p-subgroup of G and set C = T C_{G}(T). Assume the following hold:

(i). Conditions (N1), (N2) and (N3) hold for C. Thus C = O_{C}.H.K and N_{G}(T) = O_{C}.H.K′. (ii). H is a finite group of Lie type in characteristic p. (ii). The defect group D of the block B for the group N_{G}(T) which contains the inflated extended

Steinberg module of H is a noncentric p-radical subgroup of G of maximal order.

Then D is a vertex of an indecomposable summand of the reduced Lefschetz module L̃_{G}(D(G);k), and if this summand lies in a nonprincipal block, the defect group of this block equals D.

Proof. Theorem 3.6 tells us that D(G)^{T} is equivariantly homotopy equivalent to the building Δ for the Lie group H. Thus D(G)^{D} is homotopy equivalent to Δ^{D}, a building by [MS95], see also 5.5. This latter homotopy is only equivariant with respect to N_{G}(T) ∩ N_{G}(D), but all we need is an ordinary homotopy to conclude that the reduced Lefschetz module L̃_{N_{G}(D)}(D(G)^{D};k) is nonzero. Thus there is at least one indecomposable summand of L̃_{G}(D(G);k) with vertex D by Sawabe’s Proposition 5.10(ii). The defect group D_{0} of the block containing this summand must contain D, and if this block is not the principal block, the defect group D_{0} is purely noncentral by Lemma 4.3.

VERTICES OF LEFSCHETZ MODULES 19

If D ⪇ D_{0} then D_{0} is centric (note D_{0} is p-radical, and D was assumed to be a maximal noncentric p-radical subgroup). But for any Sylow p-subgroup S of G with D_{0} ≤ S, we have Z(S) ≤ C_{G}(D_{0}) and thus Z(S) ≤ D_{0} since Z(D_{0}) is the unique Sylow subgroup of C_{G}(D_{0}). This contradicts D_{0} being purely noncentral. Thus D = D_{0}. □

With additional information, including DC_{G}(D) = D.C_{H}(S′), as in Propositions 5.6 and 5.8, we would be able to obtain a homotopy equivariance with respect to N_{G}(D), so that L̃_{N_{G}(D)}(D(G)^{D};k) would be the irreducible inflated extended Steinberg module for the Lie group O^{p′}(C_{H}(S′)). Then there would be exactly one indecomposable summand of L̃_{G}(D(G);k) with vertex D.

6. Applications: sporadic simple groups of parabolic characteristic p

In this final Section we consider a few examples. We discuss how our results apply to the sporadic simple groups of parabolic characteristic p. We start with even characteristic, after which we discuss examples in characteristic 3.

Sporadic simple groups of parabolic characteristic 2. Among the 26 sporadic groups nine have local characteristic 2; these are: M_{11}, M_{22}, M_{23}, M_{24}, J_{1}, J_{3}, J_{4}, Co_{2} and T h. The twelve groups given in Table 6.1 have parabolic characteristic 2 and contain noncentral involutions whose centralizers have component type. The remaining five groups have central involutions with centralizer of component type; these are Co_{3}, F i_{23}, McL, O′N and Ly. In Table 6.1 below we gather information regarding the sporadic simple groups of parabolic characteristic 2. The notation is as follows. We let t be a noncentral involution in G. In the second column, the centralizer C_{G}(t) is given. H denotes a group associated to the component (a central quotient or an extension of the component) of C_{G}(t), usually a group of Lie type described in the Atlas [CCN^{+} 85] notation. We are studying D_{2}(G), the collection of 2-centric and 2-radical subgroups, or equivalently the distinguished 2-radical subgroups, as defined in 2.10. In the fourth column we describe the fixed point set for T = ⟨t⟩; this is usually a building, and our notation follows [Ron89, p. 214]. For the last three groups, D_{2}(G)^{T} is not a building, but it is homotopy equivalent to D_{2}(H) and H also has parabolic characteristic 2. Columns 5, 6 and 7 contain information on a vertex V of a non-projective indecomposable summand of L̃_{C_{G}(t)}(D_{2}(G)^{t};𝔽_{2}). Finally, H_{V} denotes the Lie type component of V C_{G}(V) and the column D_{2}(G)^{V} gives the type of the corresponding building associated to this fixed point set under the action of V. The main sources of information are the Atlas [CCN^{+} 85], Chapter 5 in the third volume of the classification of the finite simple groups [GLS98] and Landrock’s paper [Lan78] on the non-principal 2-blocks of the sporadic simple groups. Additional references are provided in the foremost right column of the table. The following Proposition is a direct consequence of our results from Theorems 3.6 and 5.4 combined with the information from Table 6.1.

Proposition 6.1. Let G be a sporadic simple group of parabolic characteristic 2 and let L̃_{G}denote the reduced Lefschetz module for the complex of 2-radical and 2-centric subgroups in G, over 𝔽_{2}.

(i). Let G be one of the groups M_{12}, J_{2}, HS, Ru, Suz, F i_{22}, He, Co_{1} or BM. The virtual module L̃_{G} has precisely one indecomposable non-projective summand M, and this summand lies in a nonprincipal 2-block of 𝔽_{2}G and has vertex V equal to the defect group of this block. The module M is the Green correspondent of the inflated extended Steinberg module for the building given in column D_{2}(G)^{V} of Table 6.1.

(ii). If G is one of the groups F i′_{24},HN or M then the reduced Lefschetz module L̃_{G}has at least one indecomposable non-projective summand, which lies in a non-principal block of 𝔽_{2}G and has vertex V equal to the defect group of this block.

6.2. In every case, the vertex V referred to in the Proposition is the unique maximal noncentric 2radical subgroup of G, as suggested by the work of Sawabe, see Proposition 5.10(i). For example, when G is either HN or M, there are five conjugacy classes of noncentric 2-radical subgroups, with the semidihedral group SD_{16} maximal; see [Yos02, Table VI] for HN and [Yos05b, Table 2] for M. In both of these cases^{2}, there are inclusions 2A ≤ 2A^{2} ≤ D_{8} ≤ SD_{16} and also 2A ≤ Q_{8} ≤ SD_{16}. For G = BM, there are four conjugacy classes of noncentric 2-radical subgroups, with inclusions 2A ≤ 2A_{2}C_{1} ≤ D_{8} and 2C^{3} ≤ D_{8}; see [Yos05b, Table 1]. For G = F i′_{24}, there are four classes of noncentric 2-radical subgroups, including two classes of elementary abelian subgroups 2A^{2} of rank 2, with inclusions 2A ≤ 2A_{I}^{2} ≤ D_{8} and 2A ≤ 2A_{II}^{2} ≤ D_{8}; see [KY02, Table 2]. Regarding the remaining groups from Table 6.1: J_{2}, Ru and F i_{22} have one conjugacy class of noncentric 2-radical subgroups each; M_{12} and HS have two classes of such subgroups each and Co_{1}, Suz and He have three classes each. It is easy to check that the poset of noncentric 2-radical subgroups has a unique maximal member in each case. Although not all the details are given, the relevant references for each sporadic simple group can be found in Chapter 7 of the book by Benson and Smith [BS08].

6.3. The three cases in Table 6.1 in which the fixed point set is contractible are verified using Theorem 3.3 and Lemma 5.2 since O_{C} contains central elements. The following information is available in the Atlas [CCN^{+} 85]. For t = 2C in G = F i_{22}, there is a purely central elementary abelian 2-group 2B^{4} of rank 4 in the subgroup 2^{5} = 2A_{6}B_{15}C_{10} of 2^{5+8} . For t = 2C in G = Co_{1}, the elementary abelian 2^{11} = 2A_{759}C_{1288} contains central elements of class 2A. The normalizer is N(2^{11}) = 2^{11} : M_{24}, and the subgroup M_{12}.2 ≤ M_{24} has index 1288. And for t = 2D in G = BM, the elementary abelian subgroup 2^{9} of 2^{9}.2^{16} contains a purely central 2B^{8} of rank 8. The normalizer is N(2B^{8}) = 2^{9}.2^{16}.Sp_{8}(2). The lift of the group 2^{9} in 2.BM ≤ M is a group 2^{10} referred to as an “ark” in the work of Meierfrankenfeld and Shpectorov [MS02, Mei03]. The notation O_{8}^{+} (2) in C(2D) = 2^{9}.2^{16}.O_{8}^{+} (2).2 is the Atlas notation for the simple group, also denoted Ω_{8}^{+} (2), so that O_{8}^{+} (2).2 = SO_{8}^{+} (2).

6.4. The next step in the proof of the Proposition 6.1 is to verify that the hypotheses of Theorem 3.6 hold for C_{G}(t) and for V C_{G}(V). In particular, we need to check that condition (N3) holds in these

2Recall the notation described in Introduction.

VERTICES OF LEFSCHETZ MODULES 21

groups. We first recall this condition. Let T be a p-subgroup of G. If T C_{G}(T) = C = O_{C}.H.K = L.K, then (N3) asserts that there exists a triple of Sylow p-subgroups S_{L} ⊴ S_{C} ≤ S of L,C and G respectively, such that Z(S) ∩ S_{L} ≠ 1; in particular the group S_{L} is distinguished in G. Note this condition is trivially satisfied when K is a group of odd order, since then S_{L} = S_{C}. This happens in the following four cases: when G equals J_{2}, Ru or M, and for t = 2A in G = F i_{22}. It also occurs when C = V C_{G}(V) for every group V from the fifth column of Table 6.1. We now prove a Lemma which yields condition (N3) in five additional cases from Table 6.1.

Lemma 6.5. Let G be one of the following groups: M_{12}, HS, Suz, He, or BM. Let t be a noncentral involution in G (where for G = BM we assume t is of class 2A), and let L be the component of the centralizer C_{G}(t). Let \overline{L}= L except in the case G = HS, where \overline{L}= S_{6} ≤ Aut(A_{6}). Then the 2-central involutions of G which lie in C_{G}(t) actually lie in \overline{L}.

Proof. The first three groups (and also J_{2} and Ru) are discussed in [MO09, Lemma 2.3]. For G = HS, this result is Lemma 1.7 of Aschbacher [Asc03, p. 24]. For G = Suz, see the paragraph on p. 456 before Lemma 1 of Yoshiara [Yos01]. For G = M_{12}, the centralizer C_{G}(t) = 2 × S_{5} has five conjugacy classes of involutions, fusing to two classes in M_{12}. The involutions in A_{5} are squares of elements of order 4, which are 2-central in M_{12} by the Atlas [CCN^{+} 85]. The other four classes in C_{G}(t) are noncentral in M_{12}. There exists a purely noncentral elementary abelian 2^{2} in M_{12}, so the involutions in S_{5} \ A_{5} are noncentral. The 2-central involutions are closed under nontrivial commuting products; the class multiplication coefficient is zero [BS08, p. 107, 162]. Thus the diagonal involutions in 2 × A_{5} are noncentral in M_{12}. Next, for G = He, the Lemma follows from [Hel69, p.277–278]. Finally, for t = 2A in G = BM, the result follows from Segev [Seg91, 2.5.2 and 3.18]. □

Note that if L = C_{G}(O_{C}) then whenever S_{L} ≤ S, we have Z(S) ≤ L. This argument shows that condition (N3) is satisfied for G equal to Suz, He, also for t = 2A in F i_{22}, t = 2B in Co_{1} and for t = 2C in G = BM. Condition (N3) holds for G = HN by a description of the groups S_{L} and S_{C} given in [Har76, (L) on p. 128], where it is shown that Z(S_{C}) = 2^{2} ≤ S_{L}. Finally, let G = F i′_{24}, with C_{G}(t) = 2^{.}F i_{22} : 2 and L = 2^{.}F i_{22}. Let S denote a Sylow 2-subgroup of F i_{22}, with S_{L} = 2^{.}S and S_{C} = 2^{.}S : 2. Let z be an involution in the center Z(S_{C}) and assume that z is not in S_{L}. Consider the centralizer C_{G}(2^{2}) of the elementary abelian 2^{2} = ⟨t,z⟩, which must contain S_{C}. But by [Wil87, Table 10], this centralizer is divisible by only 2^{14} (it equals either 2^{2}× O_{8}^{+} (2) : 3 or 2^{2}× 2^{6} : U_{4}(2)). This contradiction implies that the subgroup Ω_{1}(Z(S_{C})) containing all involutions of the center Z(S_{C}) must be a subgroup of S_{L}. This implies that condition (N3) holds.

6.6. In what follows we determine those 2-subgroups which are candidates for the vertices of L̃_{G}. Note that Proposition 5.3 implies that we need only to consider those 2-subgroups Q which are vertices of the reduced Lefschetz module L̃_{C_{G}(t)}(D_{2}(G)^{t};𝔽_{2}) for the action of the centralizer C_{G}(t),

22 JOHN MAGINNIS AND SILVIA ONOFREI

with t an involution in Z(Q), on the complex D_{2}(H) for H described in the fourth column of Table 6.1. This yields those groups given in column five, denoted V. Let us consider the first nine groups in Table 6.1. Theorem 5.4 applies for both C = C_{G}(t) and for C = V C_{G}(V) in every case. Observe that for G = BM, the group V = 2^{2} = 2A_{1}C_{2} is not a vertex of a summand of the reduced Lefschetz module for G since the centralizer of V has index 2 in the normalizer of V. In the remaining three cases, Theorem 5.4 does not apply if we let C = C_{G}(t). The subgroup C_{G}(t) is of the form O_{C}.H.K but H is not a group of Lie type. For each of these groups, Theorem 3.6 tells us that the fixed point set D_{2}(G) is equivariantly homotopy equivalent to the complex of 2-centric and 2-radical subgroups in the component H of C_{G}(t). We can, however, apply Theorem 5.4 to the group C = V C_{G}(V) from the sixth column of the Table. Specifically, V = D_{8} in G = F i′_{24} and V = SD_{16} in G = HN and G = M, obtaining the existence of summands with vertex equal to the dihedral or semidihedral group. For G = M it is possible that T = 2A is a vertex of a summand of the reduced Lefschetz module (if BM has a projective summand in its reduced Lefschetz module). For G = HN, we do not know the vertices for the action of 2.HS: 2 on the complex D_{2}(HS). Finally, we remark that for G = F i′_{24} and for G = HN the group V = 2^{2} is not a vertex, since the centralizer of V has even index in the normalizer of V.

Our results and the Benson-Grizzard-Smith conjecture. The book of Benson-Smith [BS08] describes a 2-local geometry for each of the 26 sporadic finite simple groups. On pages 105 − 108, they list those sporadic groups for which the 2-local geometry is equivariantly homotopy equivalent to the Brown complex^{3} S_{2}(G) [BS08, (5.9.2)], or to the Benson complex^{4}^{,}^{5} E_{2}(G) [BS08, (5.10.3)], or to the complex^{6} D_{2}(G) = B_{2}^{cen}(G) [BS08, (5.11.3)]. The eleven groups in list [BS08, (5.9.2)] have projective reduced Lefschetz modules; all but one case^{7} appear in [RSY90] and [SY97]. The six groups in list [BS08, (5.10.3)] are studied by Grizzard in his thesis [Gri07]. Grizzard computes reduced Lefschetz characters for these six groups, and also for the groups O′N and He. The characters were already known for three of these eight groups, namely M_{12} [BW95, p. 44], J_{2} [BS08, after Thm. 7.7.1] and HS (Klaus Lux, unpublished). Four of the groups in the list [BS08, (5.10.3)] also appear in [BS08, (5.11.3)], and it seems that HS should also appear in both lists. Note that although HS does not appear in [BS08, (5.11.3)], in [BS08, p. 193] a quotient map is described yielding a homotopy equivalence of the 2-local geometry of HS with D_{2}(HS). The sporadic groups of parabolic characteristic 2 given in Table 6.1 consist of eleven of the thirteen groups in list [BS08, (5.11.3)], plus HS. Thus Proposition 6.1 applies to the 2-local geometries for these twelve groups. The remaining two groups in [BS08, (5.11.3)] are O′N and F i_{23}, and in both groups all involutions are central. Then our distinguished 2-radical complex

3These are: M_{11}, M_{22}, M_{23}, M_{24}, J_{1}, J_{3}, J_{4}, McL, Co_{2}, T h, Ly.

4Corresponds to a certain collection of elementary abelian 2-groups, see [BS08, (5.10.1)] for a precise definition.

5These are: M_{12}, J_{2}, HS, Suz, Co_{3}, Ru.

6These are: M_{12}, J_{2}, Suz, Co_{1}, F i_{22}, F i_{23}, F i′_{24}, HN, B, M, He, Ru, O′N.

7The group Ly.

VERTICES OF LEFSCHETZ MODULES 23

equals the Bouc complex of all nontrivial 2-radical subgroups, which is homotopy equivalent to the Brown complex of all nontrivial 2-subgroups. Thus the results in this paper do not apply to the 2-local geometries for the two groups O′N and F i_{23}. In a footnote in [BS08, p. 164], the following comment appears: “For all 15 sporadic groups G in (5.10.3) and (5.11.3) for which the 2-local geometry is not homotopy equivalent to S_{2}(G), it seems that the reduced Lefschetz module involves an indecomposable in a suitable non-principal 2-block of G of small positive defect”. Grizzard refers to this as the “Nonprincipal Block Observation” [Gri09, Rem. 3.1] and conjectures that each sporadic group affording a nonprojective Lefschetz module will have a nonprojective part in a nonprincipal block. Grizzard verifies this conjecture for the eight groups in his thesis. Grizzard also shows that for seven of his groups, the defect group of this nonprincipal block has order equal to the ratio of the 2-part |G|_{2} of the order of the group and the 2-part of the reduced Euler characteristic of the 2-local geometry. This was also predicted by Steve Smith; see for example [Smi05]. Grizzard observed that this was not true for O′N, which also differed from the other examples by having a nonprojective part of the reduced Lefschetz module lying in the principal block. Sawabe approached this latter idea from the viewpoint of vertices of modules, and proves a theorem concerning a lower bound for the p-part of the reduced Euler characteristic of the complex D_{p}(G) [Saw06a, Prop. 7] in terms of the order of a noncentric p-radical subgroup V of maximal order. An unpublished result of Sawabe [Saw06b] also shows that such a maximal noncentric p-radical subgroup V is a vertex of a summand of the reduced Lefschetz module L̃_{G}(D(G);k) if and only if L̃_{W}(S(W);k) ≠ 0, for W = N_{G}(V)/V. This result can be applied to many groups, including all of those in Table 6.1, showing the existence of a summand of the reduced Lefschetz module with vertex V. This result also applies to the sporadic group F i_{23} and the maximal noncentric 2radical subgroup V = D_{8} (see [AO99] for the radical subgroups of F i_{23}, and for the normalizer N_{G}(V) = D_{8} × Sp_{6}(2)). It follows that the dihedral group D_{8} is a vertex of a summand of the reduced Lefschetz module for the 2-local geometry of F i_{23}. But the work of Sawabe does not give any information on the block in which this summand lies. Also, his results yield existence of nonprojective summands for all the groups in Table 6.1, but does not supply the uniqueness of the nonprojective summand that we obtain in Proposition 6.1(i). Indeed, it has been shown that there are examples where there are more than one nonprojective summand. Grizzard shows that the reduced Lefschetz module for the 2-local geometry of the sporadic group O′N has at least two nonprojective summands, one in the principal block and one in a block of defect three. And for Co_{3}, there are precisely three nonprojective summands, all lying in a nonprincipal block of defect three; see [MO09].

Sporadic simple groups of parabolic characteristic p = 3 . We now discuss the cases when G is a sporadic simple group of parabolic characteristic 3. The notations from Table 6.1 are maintained in Table 6.2 and below. In what follows t will denote an element of order 3 and O_{C}= O_{3}(C_{G}(⟨t⟩)).

Proposition 6.7. Let G be a sporadic simple group of parabolic characteristic 3 and let L̃_{G}denote the reduced Lefschetz module for the complex of 3-radical and 3-centric subgroups in G, over 𝔽_{3}.

(i). Let G be one of the groups M_{12}, Co_{3}, J_{3}, Co_{2}, F i_{22}, F i_{23}, F i′_{24} or T h. The virtual module L̃_{G} has precisely one indecomposable non-projective summand M, and this summand lies in a nonprincipal 3-block of 𝔽_{3}G and has vertex equal to the defect group of this block. The module M is the Green correspondent of the inflated extended Steinberg module for the building given in the eighth column of Table 6.2.

(ii). If G is one of the groups HN,BM or M then the reduced Lefschetz module L̃_{G}has at least one non-projective indecomposable summand which lies in a non-principal block of 𝔽_{3}G.

Proof. The eight cases in Table 6.2 in which the fixed point set is contractible are verified using Theorem 3.3 and Lemma 5.2 since O_{C} contains central elements. For two of these cases, the necessary information in is the Atlas [CCN^{+} 85]. Let t = 3B in G = Co_{3}. The elementary abelian 3^{5} is of type 3A_{55}B_{66}, containing central elements of type 3A. Next, let t = 3C in G = F i′_{24}. The elementary abelian 3^{7} is of type 3A_{378}B_{364}C_{351}, containing central elements of type 3B. The normalizer of 3^{7} is N(3^{7}) = 3^{7}.O_{7}(3) and the subgroup 2.U_{4}(3) : 2 ≤ O_{7}(3) has index 351. The group 3^{7} is the natural orthogonal module for O_{7}(3) with the 364 subgroups of type 3B corresponding to the totally isotropic points [KY02, Sect. 4.1]. We now quote information from three papers of R. A. Wilson. For t = 3C in G = T h the elementary abelian 3 × 3^{4} is of type 3B_{40}C_{81} with type 3B central; see [Wil88b, p.21]. For t = 3C in G = F i_{22} the elementary abelian 3^{5} is of type 3A_{36}B_{40}C_{45}, containing central elements of type 3B; see [Wil84, p. 201]. Note that N(3^{5}) = 3^{5}.U_{4}(2).2 and the subgroup 2.(A_{4}× A_{4}).2 ≤ U_{4}(2) has index 45. For t = 3C in G = F i_{23} the elementary abelian 3^{6}, whose normalizer is 3^{6}.L_{4}(3).2, contains elements of type 3A,3B and 3C, with 3B being 3-central. This follows from the fact that the group 3^{6} has an invariant quadratic form defined by putting the norms of elements of classes 3A, 3B and 3C equal to −1, 0 and 1 respectively; see [Wil87, Prop. 1.3.3]. This indirectly states that 3-central elements 3B occur in 3^{6}. For G = F i′_{24}, with type 3B central, see [Wil87, p. 87] for a discussion of the normalizer of a purely central elementary abelian 3B^{2} of rank 2. The normalizer is N(3B^{2}) = 3^{2}.3^{4}.3^{8} : (A_{5}× 2.A_{4}(2).2). The group 3^{2}.3^{4} = 3^{6} has an A_{5} action on the 3^{4}-factor with orbits of size 5, 10, 10, 15 on the 40 cyclic subgroups. The corresponding cosets of 3^{2} ≤ 3^{2}.3^{4} consist of elements of type 3D, 3A, 3B and 3C respectively. This yields a subgroup C(3D) = 3^{2}.3^{4}.3^{6}.(A_{4}× 2.A_{4}) ≤ N(3B^{2}). Thus the 3^{2} in C(3D) is the purely central 3B^{2}. Lastly, let G be either F i_{22} or F i_{23} and let t = 3D. The fact that O_{C} contains central elements of type 3B was not immediately evident from the literature, and we would like to thank Ronald Solomon for providing the following argument. Let N = N_{G}(⟨t⟩), and denote Q = O_{3}(N), which is equal to O_{C}. Then the quotient group N/Q is either 2.S_{4} = GL_{2}(3) if G = F i_{22}, or 2 × 2.S_{4} if

VERTICES OF LEFSCHETZ MODULES 25

G = F i_{23}. Let P ∈ Syl_{3}(N) be a Sylow 3-subgroup of the normalizer, and extend it to a Sylow 3-subgroup S of G, so that P ≤ S ∈ Syl_{3}(G). Note that Q ≤ P is a subgroup of index three. Since t ∈ Q ≤ P ≤ S, we have Z(S) ≤ C_{G}(t) ≤ N and also Z(S) ≤ C_{G}(Q), so that Z(S) ≤ C_{N}(Q). We want to show that Z(S) ≤ Q. Assume that this is not true, so that P = Q · Z(S) and also under the quotient map q: N → N/Q, the image q(Z(S)) is a Sylow 3-subgroup of GL_{2}(3). Note that for any g ∈ N, the conjugate gZ(S)g^{−1} ≤ C_{N}(Q), and the images of these conjugates in the quotient group N/Q are the Borel subgroups of GL_{2}(3) and thus generate 2.A_{4} = SL_{2}(3). Therefore there is an involution z ∈ C_{N}(Q) which maps to the central involution of the matrix group in N/Q. Clearly Q ≤ C_{G}(z), and it also follows that Z(S) ≤ C_{G}(z). Thus P ≤ C_{G}(z) and the order of P divides the order of C_{G}(z), with |P| = 3^{7} if G = F i_{22} and |P| = 3^{10} if G = F i_{23}. Each of these Fischer groups has three conjugacy classes 2A, 2B and 2C of involutions. In F i_{22} their centralizers have orders 2^{16} · 3^{6} · 5 · 7 · 11, 2^{17} · 3^{4} · 5 and 2^{16} · 3^{3} respectively. In F i_{23}, the centralizers of involutions have orders 2^{18}· 3^{9}· 5^{2}· 7 · 11 · 13, 2^{18}· 3^{6}· 5 · 7 · 11 and 2^{18}· 3^{5}· 5. This is a contradiction to P ≤ C_{G}(z), and therefore Q = O_{C} contains central elements of type 3B. The condition (N3) concerning the triple of Sylow subgroups is trivially satisfied in every case except one, namely C(3A) in F i′_{24}. In this case the stronger condition is satisfied: every central element of type 3B which lies in C(3A) = 3 × O_{8}^{+} (3) : 3 actually lies in 3 × O_{8}^{+} (3). This follows from the discussion in [Wil87, p. 83] about a symplectic form defined using commutators and lifts to the group 3.F i′_{24} in M. The lift of 3 × O_{8}^{+} (3) : 3 is O_{8}^{+} (3) : 3^{1+2} . Theorems 3.6 and 5.4 now imply the results of this Proposition. Note that for G = HN,BM or M the reduced Lefschetz module may have other nonprojective summands with vertex a cyclic group of order three. □

1 2  3  4  5 6  7  8  9  10
G C_{G}(t)  H  D_{2}(G)^{t}  V V C_{G}(V)  N_{G}(V)  H_{V} D_{2}(G)^{V} Ref.

M_{12} C(2A) = 2 × A_{5}.2  L_{2}(4) A_{1}  2^{2} 2^{2}× S_{3}  A_{4}× S_{3}  L_{2}(2)  A_{1}  [MO09]

J_{2} C(2B) = 2^{2}× A_{5}  L_{2}(4) A_{1}  2^{2} 2^{2}× A_{5}  A_{4}× A_{5}  L_{2}(4)  A_{1}  [MO09]

HS C(2B) = 2 × A_{6}.2^{2}  Sp_{4}(2) C_{2}  2^{2} 2^{2}× 5 : 4  2^{2}× 5 : 4  Sz(2)  A_{1}  [MO09]

Ru C(2B) = 2^{2}× Sz(8)  Sz(8) A_{1}  2^{2} 2^{2}× Sz(8)  (2^{2}× Sz(8)) : 3  Sz(8)  A_{1}  [MO09]

Suz C(2B) = (2^{2}× L_{3}(4)) : 2  L_{3}(4) A_{2}  D_{8} D_{8}× 3^{2} : Q_{8}  D_{8}× 3^{2} : Q_{8}  U_{3}(2)  C_{1}  [MO09]

F i_{22} C(2A) = 2^{.}U_{6}(2)  U_{6}(2) C_{3}  2 2^{.}U_{6}(2)  2^{.}U_{6}(2)  U_{6}(2)  C_{3}  [MO10]
C(2C) = 2^{5+8} : (S_{3}× 3^{2} : 4)  contr.

He C(2A) = (2^{2} ^{.}L_{3}(4)) : 2  L_{3}(4) A_{2}  D_{8} D_{8}× L_{3}(2)  D_{8}× L_{3}(2)  L_{3}(2)  A_{2}  [Hel69]

Co_{1} C(2B) = (2^{2}× G_{2}(4)) : 2  G_{2}(4) G_{2}  D_{8} D_{8}× G_{2}(2)  D_{8}× G_{2}(2)  G_{2}(2)  G_{2}  [Saw99]
C(2C) = 2^{11} : M_{12}.2  contr.

BM C(2A) = 2^{.} ^{2}E_{6}(2) : 2  ^{2}E_{6}(2) F_{4}  2^{2} 2^{2}× F_{4}(2)  (2^{2}× F_{4}(2)).2  F_{4}(2)  F_{4}  [Yos05b]
C(2C) = (2^{2}× F_{4}(2)) : 2  F_{4}(2) F_{4}  D_{8} D_{8}× ^{2}F_{4}(2)  D_{8}× ^{2}F_{4}(2)  ^{2}F_{4}(2)  I_{2}  [Seg91]
C(2D) = 2^{9}.2^{16}O_{8}^{+} (2).2  contr.

F i′_{24} C(2A) = 2^{.}F i_{22} : 2  F i_{22} D_{2}(F i_{22}) D_{8} D_{8}× Sp_{6}(2)  D_{8}× Sp_{6}(2)  Sp_{6}(2)  C_{3}  [KY02]
2^{2} 2^{2}× O_{8}^{+} (2).3 (A_{4}× O_{8}^{+} (2).3).2 O_{8}^{+} (2)  D_{4}

HN C(2A) = 2^{.}HS: 2  HS D_{2}(HS) SD_{16} SD_{16}× 5 : 4  SD_{16}× 5 : 4  Sz(2)  A_{1}  [Yos02]
2^{2} 2^{2}× A_{8}  (A_{4}× A_{8}).2  L_{4}(2)  A_{3}  [NW86]

M C(2A) = 2^{.}BM  BM D_{2}(BM) SD_{16} SD_{16}× ^{2}F_{4}(2) SD_{16}× ^{2}F_{4}(2)  ^{2}F_{4}(2)  I_{2}  [Yos05b]

Table 6.1: Information for L̃_{G}(D_{2}(G);𝔽_{2})

1 2  3  4  5 6  7  8  9
G C_{G}(T)  H  D_{3}(G)^{T}  V V C_{G}(V)  N_{G}(V)  D_{3}(G)^{V} Ref.

M_{12} C(3B) = 3 × A_{4}  L_{2}(3) A_{1}  3 3 × A_{4}  S_{3}× A_{4}  A_{1}  [MO09]

Co_{3} C(3B) = 3^{5} : (2 × A_{5})  contr.  [MO10]
C(3C) = 3 × L_{2}(8) : 3  R(3) A_{1}  3 3 × R(3)  S_{3}× R(3)  A_{1}

J_{3} C(3B) = 3 × A_{6}  L_{2}(9) A_{1}  3 3 × A_{6}  (3 × A_{6}).2  A_{1}  [MO09]

Co_{2} C(3B) = 3 × Sp_{4}(3) : 2  Sp_{4}(3) C_{2}  3 3 × Sp_{4}(3) : 2 S_{3}× Sp_{4}(3) : 2  C_{2}

F i_{22} C(3A) = 3 × U_{4}(3) : 2  U_{4}(3) C_{2}  3 3 × U_{4}(3) : 2  S_{3}× U_{4}(3) : 2  C_{2}  [KY02]
C(3C) = 3^{5} : 2(A_{4}× A_{4}).2  contr.  [Wil84]
C(3D) = 3^{3}.3.3^{2} : 2A_{4}  contr.

F i_{23} C(3A) = 3 × O_{7}(3)  O_{7}(3) B_{3}  3 3 × O_{7}(3)  S_{3}× O_{7}(3)  B_{3}  [AO99]
C(3C) = 3^{6} : (2 × O_{5}(3))  contr.  [Wil87]
C(3D) = [3^{9}].(2 × 2A_{4})  contr.

F i′_{24} C(3A) = 3 × O_{8}^{+} (3) : 3  O_{8}^{+} (3) D_{4}  3^{2} 3^{2}× G_{2}(3)  (3^{2} : 2 × G_{2}(3)).2  G_{2}  [KY02]
C(3C) = 3^{7}.2.U_{4}(3)  contr.  [Wil87]
C(3D) = 3^{2}.3^{4}.3^{6}.(A_{4}× 2A_{4})  contr.
C(3E) = 3^{2}× G_{2}(3)  G_{2}(3) G_{2}  3^{2} 3^{2}× G_{2}(3)  (3^{2} : 2 × G_{2}(3)).2  G_{2}

T h C(3A) = 3 × G_{2}(3)  G_{2}(3) G_{2}  3 3 × G_{2}(3)  (3 × G_{2}(3)) : 2  A_{1}  [Wil88b]
C(3C) = 3 × 3^{4} : 2A_{6}  contr.

HN C(3A) = 3 × A_{9}  A_{9} D_{3}(A_{9})  3^{2} 3^{2}× A_{6}  (3^{2} : 4 × A_{6})2^{2}  A_{1}  [Yos05a]

BM C(3A) = 3 × F i_{22} : 2  F i_{22} D_{3}(F i_{22}) 3^{2} 3^{2}× U_{4}(3) : 2^{2} (3^{2} : D_{8}× U_{4}(3) : 2^{2}).2  C_{2}  [Yos05a]

M C(3A) = 3^{.}F i′_{24}  F i′_{24} D_{3}(F i′_{24}) 3^{1+2} 3^{1+2} × G_{2}(3) (3^{1+2} : 2^{2}× G_{2}(3)).2  G_{2}  [AW10]
C(3C) = 3 × T h  T h D_{3}(T h) 3^{2} 3^{2}× G_{2}(3)  (3^{1+2} : 2 × G_{2}(3)).2  G_{2}  [Wil88a]

Table 6.2: Information for L̃_{G}(D_{3}(G);𝔽_{3})

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Department of Mathematics Department, Kansas State University, 137 Cardwell Hall, Manhattan, Kansas 66506 E-mail address: maginnis@math.ksu.edu

Department of Mathematics, The Ohio State University, 100 Mathematics Tower, 231 West 18^{th} Avenue, Columbus, Ohio 43210 E-mail address: onofrei@math.ohio-state.edu