ON FIXED POINT SETS AND LEFSCHETZ MODULES FOR SPORADIC SIMPLE GROUPS

John Maginnis^{1} and Silvia Onofrei^{2}

Department of Mathematics, Kansas State University, 138 Cardwell Hall, Manhattan, KS 66506 January 30, 2008

Abstract. We consider 2-local geometries and other subgroup complexes for sporadic simple groups. For six groups, the fixed point set of a noncentral involution is shown to be equivariantly homotopy equivalent to a standard geometry for the component of the centralizer. For odd primes, fixed point sets are computed for sporadic groups having an extraspecial Sylow p-subgroup of order p^{3}, acting on the complex of those p-radical subgroups containing a p-central element in their centers. Vertices for summands of the associated reduced Lefschetz modules are described.

1. Introduction

In this paper we study various subgroup complexes of the sporadic simple groups, including 2-local geometries, the distinguished Bouc complex and the complex of p-centric and p-radical subgroups. We find information on the structure of the associated reduced Lefschetz modules, such as vertices and their distribution into the blocks of the group ring. Robinson [27] reformulated a theorem of Burry-Carlson [9] and Puig [25] to relate the indecomposable summands with given vertex of the reduced Lefschetz module to the indecomposable summands with the same vertex of a corresponding Lefschetz module for the subcomplex fixed by the action of the vertex group. Therefore the nature of the fixed point sets is essential for the understanding of the properties of the reduced Lefschetz modules. Our approach will combine this theorem with standard results from modular representation theory.

The best known example of a Lefschetz module is the Steinberg module of a Lie group G in defining characteristic p. This module is irreducible and projective and it is the homology module of the associated building. For finite groups in general, the Brown complex of inclusion chains of nontrivial p-subgroups (as well as any complex which is G-homotopy equivalent to it) has projective reduced Lefschetz module; see [26, Corollary 4.3]. For a Lie group in defining characteristic, the Brown complex is G-homotopy equivalent to the building [26, Theorem 3.1], and thus its Lefschetz module is equal to the corresponding Steinberg module.

Ryba, Smith and Yoshiara [31] proved projectivity for Lefschetz modules of 18 sporadic geometries. The characters of these projective modules (where available) and the decomposition into projective covers of irreducibles are also given. Smith and Yoshiara [35] approached the study of projective Lefschetz modules in a more systematic way. The projectivity results are obtained via homotopy equivalence with the Quillen complex of nontrivial elementary abelian p-subgroups. Their results show that the 2-local geometries for the sporadic groups of local characteristic 2 have projective Lefschetz modules.

1Email address: maginnis@math.ksu.edu.

2Email address: onofrei@math.ksu.edu.

The collection of nontrivial p-centric and p-radical subgroups (see Section 3 for precise definitions) is relevant to both modular representation theory as well as mod-p cohomology of the underlying group. Sawabe [32, Section 4] showed that its reduced Lefschetz module is projective relative to the collection of p-subgroups which are p-radical but not p-centric.

In Section 2, we assume p = 2 and discuss six sporadic simple groups: M_{12}, J_{2}, HS, Ru, Suz and Co_{3}. Each has two classes of involutions; the 2-central involutions are closed under commuting products. The fixed point sets of the 2-central involutions are contractible; we determine the structure of the fixed point sets of the noncentral involutions and we relate these fixed point sets to sporadic geometries and buildings associated to a component of the centralizer of the noncentral involution. We also determine the vertices and the distribution of the nonprojective indecomposable summands of the Lefschetz module into the 2-blocks of the group ring. These results are contained in Theorems 2.1 and 2.11.

We will refer to a p-group as being distinguished if it contains a p-central element in its center; see the paper [21] for further information on distinguished collections. For odd primes, we analyze in detail the distinguished p-radical complexes for those sporadic simple groups which have a Sylow p-subgroup of order p^{3}. The main results are given in Theorem 3.1. The homotopy type of the fixed point sets of the subgroups of order p are determined, and information on the structure of the corresponding Lefschetz modules is given. Our computations provide several examples of reduced Lefschetz modules which are acyclic and contain nonprojective indecomposable summands in the principal block of the group ring; these examples are obtained for p = 3 (He and M_{24}), p = 7 and F i′_{24} and for p = 13 and M, the Monster group.

In the second part of Section 3 we determine the properties of the reduced Lefschetz modules associated to the complex of p-centric and p-radical subgroups. The groups discussed are those sporadic groups with extraspecial Sylow p-subgroup and which do not have parabolic characteristic p; the groups J_{2}, M_{24} and He, with p = 3 fall into this category.

We use the standard notation for finite groups as in the Atlas [11]. 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. For p odd, p_{+}^{1+2} stands for the extraspecial group of order p^{3} and exponent p. Also H.K denotes an extension of the group H by a group K and H : K denotes a split extension. Regarding the conjugacy classes of elements of order p we follow the Atlas [11] 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 7AB stands for a group of order 7 which contains elements from both classes 7A and 7B. A great deal of information on the p-local structure of the sporadic simple groups used in Section 3 can be found in [15, Section 5.3]. In order to determine if certain classes of p-elements are closed under taking commuting products we compute the class multiplication coefficient denoted ξ(x, y, z) using the built-in function in GAP [14].

For both characters and Brauer characters, we use the notation from the Modular Atlas Homepage [24], where ϕ_{i} denotes the Brauer character of a simple module in characteristic p and χ_{i} is the character of a simple module in characteristic zero, also given in [11]. Then P_{G}(ϕ_{i}) denotes the projective cover of ϕ_{i}.

2. Sporadic groups which satisfy the closure property for p = 2

2.1. Fixed point sets. Let Co_{3} be the third Conway group, one of the 26 sporadic simple groups. There is a 2-local geometry Δ of Co_{3}, described by Ronan and Stroth [29], with three types of objects having isotropy groups equal to three of the maximal 2-local subgroups, 2^{.}Sp_{6}(2), 2^{2+6}.3.(S_{3} × S_{3}), and 2^{4}^{.}A_{8}. The authors have shown in [20, Proposition 6.1] that the fixed point set Δ^{z} of a 2central involution z is contractible. However, the fixed point set Δ^{t} of a noncentral involution t is not contractible (this is related to the fact that the mod-2 reduced Lefschetz module of Δ is not projective). The centralizer of a noncentral involution is C_{Co_{3}}(t) = 2 × M_{12}, so that the Mathieu group M_{12} acts on the fixed point set Δ^{t}. Also M_{12} has its own 2-local geometry, with two types of objects, stabilized by 2_{+}^{1+4}.S_{3} and 4^{2} : (2 × S_{3}); see for example [7, Section 8.27].

In the thesis of P. Grizzard [16], the character of the Lefschetz module is computed for Co_{3} and seven other sporadic groups. This yields the Euler characteristics of the fixed point sets. Grizzard noticed that the fixed point set Δ^{t} and the 2-local geometry for M_{12} have the same Euler characteristic, and conjectured that they are homotopy equivalent. With equal Euler characteristics as evidence, Grizzard conjectures that for six of the eight sporadic groups he investigated, the fixed point set for a noncentral involution is homotopy equivalent to a standard 2-local geometry for the component of the centralizer of the noncentral involution. The purpose of this section is to prove this conjecture.

Theorem 2.1. Let G be one of the following six sporadic simple groups: the Mathieu group M_{12}, the Hall-Janko-Wales group J_{2}, the Higman-Sims group HS, the third Conway group Co_{3}, the Rudvalis group Ru, or the Suzuki group Suz. Let H ≤ G be the component of the centralizer C_{G}(t) of a noncentral involution t ∈ G. Then H and G have standard 2-local geometries such that the geometry for H is equivariantly homotopy equivalent to the fixed point set for the involution t acting on the geometry for G.

A 2-local geometry for G is a simplicial complex whose vertex stabilizers are maximal 2-local subgroups of G. If G is one of the six sporadic groups mentioned in Theorem 2.1, the term standard indicates the fact that we are working with the 2-local geometries considered in Benson and Smith [7]. If G is a Lie group in defining characteristic, then the standard geometry is the corresponding Tits building.

The six groups in Theorem 2.1 have several important similarities. There are two conjugacy classes of involutions; one class consists of the 2-central involutions, those lying in the center of a Sylow 2-subgroup. The collection of 2-central involutions is closed, in the sense that the product of two distinct commuting 2-central involutions is always an involution of central type. The centralizer of a noncentral involution has a single component H, which in these cases is a simple group, normal in the centralizer. Table 2.1 describes the centralizer C_{G}(t) of a noncentral involution t ∈ G, and its component H.

The Quillen complex of G is the complex of proper inclusion chains of nontrivial elementary abelian p-subgroups of G. The Benson complex is a subcomplex of the Quillen complex; its vertices are elementary abelian p-subgroups of G whose elements of order p lie in certain conjugacy classes, generated under commuting products by the p-central elements (those elements of order p which lie in the center of a Sylow p-subgroup of G). This complex was introduced by Benson [6] in order to study the mod-2 cohomology of Co_{3}.

\begin{matrix}G & C_{G}(t) & H \\ M_{12} & 2 × S_{5} & A_{5} \\ J_{2} & 2^{2} × A_{5} & A_{5} \\ HS & 2 × Aut(A_{6}) & A_{6} \\ Co_{3} & 2 × M_{12} & M_{12} \\ Ru & 2^{2} × Sz(8) & Sz(8) \\ Suz & (2^{2} × L_{3}(4)).2 & L_{3}(4) \end{matrix}

Table 2.1: Centralizers of noncentral involutions and their components

Lemma 2.2 (Benson and Smith [7]). For the six groups in Theorem 2.1, the standard 2-local geometry for G is equivariantly homotopy equivalent to the Benson complex.

Proof. This result is proved case by case in the book of Benson and Smith [7, Chapter 8]. □

For the remainder of this section, let Δ denote the Benson complex for G, where G is one of the six groups in Theorem 2.1. Since the collection of 2-central involutions is closed, the simplices of Δ are chains of elementary abelian 2-subgroups of purely central type (every involution is 2-central).

Lemma 2.3. Let t be a noncentral involution in G, and let H be the component of the centralizer C_{G}(t). Let \overline{H} = H except in the case G = HS, where \overline{H} = S_{6} ≤ Aut(A_{6}). Then the 2-central involutions of G which lie in C_{G}(t) actually lie in \overline{H}. These are precisely the 2-central involutions of \overline{H}.

Proof. For G = HS, this result is Lemma 1.7 of Aschbacher [5, p. 24]. For G = Suz, see the paragraph on p. 456 before Lemma 1 of Yoshiara [39]. For the other cases, this result follows from information in the Atlas [11]; for example, the square of an element of order 4 is 2-central. Observe that A_{5}, Sz(8) and L_{3}(4) have only one conjugacy class of involutions, and in S_{6} every involution is 2-central. □

Lemma 2.4. The standard 2-local geometry for H is equivariantly homotopy equivalent to the Benson complex for \overline{H}.

Proof. For M_{12}, this was stated in Lemma 2.2. The Benson complex equals the Quillen complex for A_{5} ≃ L_{2}(4), S_{6} ≃ Sp_{4}(2), Sz(8) and L_{3}(4). For a group of Lie type in defining characteristic, the Quillen complex is homotopy equivalent to the Tits building; see [26, Theorem 3.1]. □

Definition 2.5. Let Δ_{0}^{t} denote the subcomplex of the Benson complex Δ for G which consists of chains of purely 2-central elementary abelian 2-subgroups A satisfying t ∈ C_{G}(A) or equivalently A ≤ C_{G}(t).

By Lemma 2.3, Δ_{0}^{t} equals the Benson complex for \overline{H}, and so Δ_{0}^{t} is equivariantly homotopy equivalent to the standard 2-local geometry for H. Note that Δ_{0}^{t} is a subcomplex of the fixed point set Δ^{t}, which consists of those chains of elementary abelian 2-subgroups A of purely central type satisfying t ∈ N_{G}(A).

Lemma 2.6. The subcomplex Δ_{0}^{t} is equivariantly homotopy equivalent to Δ^{t}.

Proof. Define a map f : Δ^{t} → Δ^{t} by f(A) = A ∩ C_{G}(t), where A is a purely 2-central elementary abelian 2-subgroup of G with t ∈ N_{G}(A). Observe that the action of t on A must fix at least one nonidentity element, so that f(A) ≠ {e}. Thus f(A) ∈ Δ, which is closed under passing to nontrivial subgroups. Clearly f(A) ∈ Δ_{0}^{t} ⊆ Δ^{t}. The map f is a C_{G}(t)-equivariant poset map satisfying f(A) ≤ A and therefore Δ^{t} is C_{G}(t)-homotopy equivalent to Δ_{0}^{t}, the image of f; see [17, 2.2(3)]. □

We end the section with a summary of the arguments used to prove the theorem:

Proof of Theorem 2.1. Let G denote one of the six groups in Theorem 2.1. Each of these groups has a standard 2-local geometry which is equivariantly homotopy equivalent to the Benson complex Δ of G (Lemma 2.2). By Lemma 2.6, the fixed point set Δ^{t} of a noncentral involution t in G is equivariantly homotopy equivalent with Δ_{0}^{t}, which is the Benson complex of the component H of C_{G}(t), by Lemma 2.3 and Definition 2.5. Finally, according to Lemma 2.4, the Benson complex of H is equivariantly homotopy equivalent to the standard 2-local geometry for H. This concludes the proof of the theorem. □

2.2. Reduced Lefschetz modules. Information on the fixed point sets leads to more details on the reduced Lefschetz module:

L̃_{G}(Δ; 𝔽_{p}) = ∑_{i=−1}^{dimΔ}(−1)^{i}C_{i}(Δ; 𝔽_{p})

of the augmented chain complex of Δ; here 𝔽_{p} denotes a field of characteristic p, which is a splitting field for G and all its subgroups. The reduced Lefschetz module can also be written in terms of induced modules:

L̃_{G}(Δ : 𝔽_{p}) = ∑_{σ∈Δ}(−1)^{dimσ}Ind_{G_{σ}}^{G}(𝔽_{p}) − 𝔽_{p}.

A theorem due to Burry and Carlson [9, Theorem 5] and to Puig [25] was applied by Robinson [27, in the proof of Corollary 3.2] to Lefschetz modules to obtain the following result; also see [33, Lemma 1]:

Theorem 2.7 (Robinson [27]). The number of indecomposable summands of L̃_{G}(Δ; 𝔽_{p}) with vertex Q is equal to the number of indecomposable summands of L̃_{N_{G}(Q)}(Δ^{Q}; 𝔽_{p}) 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.

In what follows we will use this theorem in order to determine the vertices of the reduced Lefschetz module L̃_{G}(Δ; 𝔽_{2}) for the Benson complex of one of the six sporadic simple groups discussed in Section 2.1. We start with a more general result regarding the nature of the fixed point sets under the action of p-central elements:

Proposition 2.8. Let Δ denote the Benson complex for a group G, and let P be any group in the Benson collection. Then the fixed point set Δ^{P} is contractible. Further, the vertices of the indecomposable summands of L̃_{G}(Δ; 𝔽_{p}) do not contain any p-central elements.

Proof. A collection C is a G-poset which is closed under the conjugation action of G. The order complex Δ(C) of C is the simplicial complex with vertex set C and simplices proper inclusion chains in C. In what follows, C will denote the Benson collection; this is the set of p-subgroups of G which correspond to vertices in the Benson complex.

Let C_{0}^{P} = {Q ∈ C|P ≤ C_{G}(Q)}, a subposet of C^{P} = {Q ∈ C|P ≤ N_{G}(Q)}. Define a poset map F : C^{P} → C^{P} by F (Q) = Q ∩ C_{G}(P), a nontrivial group since it equals the set of elements of the p-group Q fixed under the action of the p-group P. Since F (Q) ≤ Q, we have homotopy equivalence between Δ(C^{P}) and the corresponding image under F, which equals Δ(C_{0}^{P}). If P ≤ C_{G}(Q), then P Q is also a group in the Benson collection. Then Δ(C_{0}^{P}) is conically contractible via Q ≤ P Q ≥ P. These two stages of the proof can be combined in the string, representing equivariant poset maps: Q ≥ C_{Q}(P) ≤ P · C_{Q}(P) ≥ P.

Let z be a p-central element in G, then ⟨z⟩ belongs to the Benson collection. The contractibility of Δ^{z} implies that Δ^{Q} is mod-p acyclic for any p-group Q containing z (by Smith theory), and thus the reduced Lefschetz module L̃_{N_{G}(Q)}(Δ^{Q}; 𝔽_{p}) = 0. Apply Theorem 2.7 to conclude that the vertices of the indecomposable summands of L̃_{G}(Δ; 𝔽_{p}) do not contain any p-central elements. □

Corollary 2.9. With G one of the sporadic simple groups M_{12}, J_{2}, HS, Co_{3}, Ru, Suz, the vertices of the indecomposable summands of L̃_{G}(Δ; 𝔽_{2}) do not contain any 2-central involutions.

Remark 2.10. For G = Co_{3}, the result was proved in [20, Theorem 2] using a different approach.

In what follows, we need only to consider those 2-groups Q which are purely noncentral. For five of the six groups discussed in this section (M_{12}, J_{2}, HS, Ru and Co_{3}) the square of any element of order 4 is a 2-central involution. Therefore the purely noncentral 2-groups are the purely noncentral elementary abelian 2-groups. These are classified in [1, Theorems 1.3 and 1.4], [13, Lemmas 3.3 and 3.4], [23, Table 2], [38, Section 2.5], [12, Lemma 5.10].

Now consider the Suzuki group G = Suz. There are two classes of involutions, denoted 2A (2-central) and 2B (noncentral). The centralizer of a noncentral involution t_{0} = 2B is C_{G}(t_{0}) = (2^{2} ×L_{3}(4)) : 2. There are four classes of elements of order 4, and the class 4D has its square equal to a noncentral involution 2B. The maximal purely noncentral elementary abelian 2-subgroups have order four, and there are 3 classes of such subgroups, denoted V_{1}, V_{2} and V_{3} in [38]. Their centralizers are C_{G}(V_{1}) = 2^{2} × L_{3}(4), C_{G}(V_{2}) = 2^{2} × 2^{2+4}, and C_{G}(V_{3}) = 2^{2} × 3^{2} : Q_{8}. Denoting V_{1} = ⟨t_{0}, t_{1}⟩, we have V_{2} = ⟨t_{0}, t_{1}y⟩ with y an involution in L_{3}(4), and V_{3} = ⟨t_{0}, z⟩ for z an involution in C_{G}(t_{0}) \ C_{G}(V_{1}). Acting as an outer automorphism on L_{3}(4), z centralizes P SU_{3}(2) = 3^{2} : Q_{8}. The element t_{1}z is of type 4D, and D_{8} = ⟨t_{0}, t_{1}, z⟩ is the dihedral group of order 8. Let Q ≤ G be a purely noncentral 2-group, and denote E = Ω_{1}(Z(Q)). Thus E is a purely noncentral elementary abelian 2-group and Q ≤ C_{G}(E). If E ≃ 2^{2} has order four, then Q = E since every involution of L_{3}(4) is 2-central in G. Assume E = 2 = ⟨t_{0}⟩; then Q ∩ C_{G}(V_{1}) equals either E, V_{1}, or V_{2}. Either Q ≤ C_{G}(V_{1}), or Q is isomorphic to some extension of the form 2.2 or 2^{2}.2. In the first case, Q is either 4 = ⟨4D⟩ or Q = 2^{2} = V_{3}. In the second case Q is dihedral since there are no purely noncentral 2^{3}, and Ω_{1}(Z(2 × 4)) = 2^{2}. Up to conjugacy, the dihedral group contains V_{3} = ⟨t_{0}, z⟩, and either Q = D_{8} = ⟨t_{0}, t_{1}, z⟩ or Q = D_{8}^{∗} = ⟨t_{0}, t_{1}y, z⟩.

The chart in Table 2.2 lists the conjugacy classes of purely noncentral 2-subgroups in each of the six sporadic groups (second column) together with their normalizers (third column). The fixed point sets (given in the fourth column) are computed using Δ^{Q} = (Δ^{t})^{Q} for t an involution in Z(Q).

In most cases Δ^{Q} is homotopy equivalent to a building and the reduced Lefschetz module is the associated Steinberg module, or an “extended” Steinberg module; see [34]. The case of 2×M_{12} ≤ Co_{3} will require more discussion, and we need to know the projective summands of the corresponding reduced Lefschetz module for M_{12}.

G Q N_{G}(Q)  Δ^{Q}  vertex of
up to homotopy  L̃_{N_{G}(Q)}(Δ^{Q}; 𝔽_{p})
M_{12} 2 2 × S_{5}  5 points = L_{2}(4) building  2^{2}
2^{2} A_{4} × S_{3}  3 points = L_{2}(2) building  2^{2} = Q
J_{2} 2 2^{2} × A_{5}  5 points = L_{2}(4) building  2^{2}
2^{2} A_{4} × A_{5}  5 points = L_{2}(4) building  2^{2} = Q
2^{2} 2^{4} : 3  contractible
HS 2 2 × A_{6}.2^{2}  Sp_{4}(2) building  2^{2}
2^{2} 2^{2} × 5 : 4  Sz(2) building  2^{2} = Q
Ru 2 2^{2} × Sz(8)  65 points = Sz(8) building 2^{2}
2^{2} (2^{2} × Sz(8)) : 3 65 points = Sz(8) building 2^{2} = Q
Co_{3} 2 2 × M_{12}  geometry for M_{12}  2^{3} and 2 = Q
2^{2} A_{4} × S_{5}  5 points = L_{2}(4) building  2^{3}
2^{3} (2^{3} × S_{3}).(7 : 3) 3 points = L_{2}(2) building  2^{3} = Q
Suz 2 (2^{2} × L_{3}(4)).2 L_{3}(4) building  D_{8}
2^{2} (A_{4} × L_{3}(4)).2 L_{3}(4) building  D_{8}
2^{2} 2^{2} × 2^{2+4}  contractible
2^{2} D_{8} × 3^{2} : Q_{8}  9 points = P SU_{3}(2) building D_{8}
4 D_{8} × 3^{2} : Q_{8}  9 points = P SU_{3}(2) building D_{8}
D_{8} D_{8} × 3^{2} : Q_{8}  9 points = P SU_{3}(2) building D_{8} = Q
D_{8}^{∗} D_{8}^{∗} × 2  contractible

Table 2.2: Fixed point sets and vertices of L̃_{G}(Δ; 𝔽_{p})

Theorem 2.11. (a) Let G be either M_{12}, J_{2}, HS, or Ru. Then the reduced Lefschetz module L̃_{G}(Δ; 𝔽_{2}) has precisely one nonprojective summand, and that summand has vertex 2^{2} and lies in a block with defect group 2^{2}.

(b) Let G = Suz. Then L̃_{G}(Δ; 𝔽_{2}) has precisely one nonprojective summand, and that summand has vertex D_{8} and lies in a block with defect group D_{8}.

(c) Let G = Co_{3}. Then L̃_{G}(Δ; 𝔽_{2}) has either two or three nonprojective summands, all lying in a block with defect group 2^{3}. One summand has vertex 2^{3}. There is either one or two summands with vertex 2.

Proof. The results for part (a) and part (b) follow from Theorem 2.7 and the information gathered in Table 2.2. The only case that requires further discussion is the summands with vertex 2 for G = Co_{3} in part (c); this case follows from the next Proposition 2.12 concerning projective summands of the reduced Lefschetz module for M_{12}. □

Proposition 2.12. For the sporadic simple group M_{12}, the reduced Lefschetz module L̃_{M_{12}}(Δ; 𝔽_{2}) associated to the Benson complex has either one or two projective summands.

Proof. Recall that, by Lemma 2.2 the Benson complex and the 2-local geometry of M_{12} are equivariantly homotopy equivalent; therefore they have equal Lefschetz modules. The geometry Δ for M_{12} is a graph with two orbits of vertices, stabilized by P_{1} = 2_{+}^{1+4} : S_{3} and P_{2} = 4^{2} : D_{12}. An edge is stabilized by a Sylow 2-subgroup P of G, a group of order 64. Then

L̃_{M_{12}}(Δ; 𝔽_{2}) = Ind_{P_{1}}^{M_{12}}(1) + Ind_{P_{2}}^{M_{12}}(1) − Ind_{P}^{M_{12}}(1) − 1.

We will apply three results on projective summands of induced modules.

The following result is due to Robinson [28, Theorem 3] and Webb [36, Proposition 5.3]:

Lemma 2.13 (Robinson, Webb). The number of summands in Ind_{H}^{G}(1) isomorphic to the projective cover P_{G}(S) of a simple 𝔽_{p}G-module S equals the number of summands in Res_{H}^{G}(S) isomorphic to P_{H}(1).

If P_{G}(S) is a summand of Ind_{H}^{G}(1), then dim(S) must at least be the cardinality of a Sylow psubgroup of H. The simple 𝔽_{2}M_{12}-modules have dimensions 1, 10, 16, 16, 44 and 144. Therefore the only possible projective summand of L̃_{M_{12}}(Δ; 𝔽_{2}) is the projective cover of φ_{6} = 144. Note Ind_{P}^{M_{12}}(1) contains at most two summands isomorphic to P_{M_{12}}(φ_{6}).

The next Lemma, due to Robinson [28, Proposition 6], will give an upper bound on the number of projective summands of L̃_{M_{12}}(Δ; 𝔽_{2}).

Lemma 2.14 (Robinson). Let H ≤ G, g ∈ G and let {S_{i} : 1 ≤ i ≤ t} be a full set of isomorphism types of simple 𝔽_{p}G-modules. The number of double cosets HgH for which p does not divide |H ∩H^{g}| is at least ∑_{i,j=1}^{t} c_{ij}m_{i}m_{j}, where (c_{ij}) is the Cartan matrix of 𝔽_{p}G, and m_{i} denotes the multiplicity of P_{G}(S_{i}) as a summand of Ind_{H}^{G}(1).

With H = P_{1} or H = P_{2}, there are eleven double cosets HgH in M_{12}, but for all of them |H ∩H^{g}| is even (GAP [14] computation). Therefore Ind_{P_{1}}^{M_{12}}(1) and Ind_{P_{2}}^{M_{12}}(1) contain no projective summands. For H = P ∈ Syl_{2}(M_{12}), there are 44 double cosets P gP, and 12 of them satisfy |P ∩ P^{g}| = 1. This implies that 2m^{2} ≤ 12, where m is the multiplicity of the projective cover P_{M_{12}}(φ_{6}) as a summand of Ind_{P}^{M_{12}}(1). Therefore L̃_{M_{12}}(Δ; 𝔽_{2}) contains at most two projective summands.

The third result used in this proof is due to Landrock [19, Theorem 2.3]; this will give a lower bound on the number of projective summands of L̃_{M_{12}}(Δ; 𝔽_{2}).

Lemma 2.15 (Landrock). Let P ∈ Syl_{p}(G). Then Ind_{P}^{G}(1) is projective-free if and only if for every element g ∈ G with |P ∩ P^{g}| = 1 and for every p-section C (the set of elements of G whose p′-part lies in some fixed p-regular conjugacy class), the number of P -orbits of elements in C ∩ (P gP) is a multiple of p.

Applied to G = M_{12}, we find that Ind_{P}^{M_{12}}(1) is not projective-free. A computation using GAP [14] provides an example of such a double coset and conjugacy class (of elements of order 11) with five P -orbits.

Thus we have shown that the reduced Lefschetz module for M_{12} has either one or two projective summands. This implies that the reduced Lefschetz module for Co_{3} has either one or two summands with vertex 2. □

Remark 2.16. In a private communication, Klaus Lux has shown that the reduced Lefschetz module L̃_{M_{12}}(Δ; 𝔽_{2}) has precisely two projective summands. This implies that the reduced Lefschetz module for Co_{3} has two summands with vertex 2. Applying Lemma 2.13 for G = M_{12}, H ∈ Syl_{2}(M_{12}), and S the simple 𝔽_{2}M_{12}-module of dimension 144, we want to compute the number of projective summands in Res_{H}^{M_{12}}(S). Observe that the only projective indecomposable 𝔽_{2}H-module is the regular representation P_{H}(1) = 𝔽_{2}H. The element e = ∑_{h∈H} h lies in the socle of the group ring 𝔽_{2}H, which acts trivially on any nonprojective indecomposable 𝔽_{2}H-module. For the regular representation, each h ∈ H corresponds to a permutation matrix and e acts as a matrix with every entry equal to one. Therefore the rank of e acting on an 𝔽_{2}H-module equals the number of projective summands. Using GAP, Lux computes the rank of e acting on Res_{H}^{M_{12}}(S) to be two.

Using other techniques involving GAP and MeatAxe, Lux also shows that

L̃_{M_{12}}(Δ; 𝔽_{2}) = S − 2P_{M_{12}}(S),

where S has Brauer character φ_{6} = 144.

3. Sporadic simple groups with small Sylow p-subgroups. Case p odd

In this section we are concerned with the sporadic simple groups G which have a Sylow p-subgroup of order p^{3}, for p an odd prime. In all these cases the Sylow p-subgroup is p_{+}^{1+2}, extraspecial of exponent p. A concise treatment of these groups can be found in the paper of Ruiz and Viruel [30] on fusion and linking systems.

3.1. The distinguished Bouc complex. A p-subgroup Q is p-radical if it is the largest normal p-subgroup in its normalizer N_{G}(Q). In addition, Q is called distinguished if it contains p-central elements in its center. The distinguished Bouc collection B̂_{p}(G) contains the distinguished p-radical subgroups of G. We shall denote by |B̂_{p}(G)| the distinguished Bouc complex.

We determine the nonprojective indecomposable summands, as well as their distribution into the blocks of the group ring 𝔽_{p}G, of the reduced Lefschetz modules L̃_{G}(|B̂_{p}(G)|; 𝔽_{p}) = L̃_{G}(B̂_{p}) of these sporadic simple groups. For a p-element t of noncentral type we first find the fixed point set Δ^{t} and then we apply Theorem 2.7 (Robinson’s formulation of the Burry-Carlson Theorem).

The groups Ru and J_{4} (with p = 3) and T h (with p = 5) have one class of elements of order p each. The groups McL (with p = 5) and O′N (with p = 7) have local characteristic p; all the p-local subgroups H satisfy the condition C_{H}(O_{p}(H)) ≤ O_{p}(H). In all these cases, the distinguished Bouc collection equals the whole Bouc collection and therefore the reduced Lefschetz modules are projective. For a proof of the equality of the two collections in a group of local characteristic p see [21, Lemma 4.8]. Information on the p-radical subgroups for odd primes, for G one of the sporadic simple groups, is given in [40, Table 1].

The remaining sporadic groups with Sylow p-subgroups of order p^{3} are discussed below.

Theorem 3.1. Let t ∈ G be an element of order p of noncentral type.

a) Let G be either M_{12} or J_{2}, and let p = 3. Then the fixed point set |B̂_{3}(G)|^{t} consists of four contractible components, homotopy equivalent to the building for L_{2}(3). The reduced Lefschetz module L̃_{G}(B̂_{3}) contains precisely one nonprojective summand, which has vertex 3 = ⟨t⟩ and lies in a block with the same group as defect group.

L̃_{M_{12}}(B̂_{3}) = P_{M_{12}}(ϕ_{2}) − P_{M_{12}}(ϕ_{3}) − P_{M_{12}}(ϕ_{6}) + P_{M_{12}}(ϕ_{8}) − P_{M_{12}}(ϕ_{9}) − ϕ_{7} − ϕ_{11}.

These projective covers all lie in the principal block; χ_{14} = ϕ_{7}+ϕ_{11} is not projective and lies in the block with defect group ⟨3B⟩ = ⟨t⟩. This formula is valid in the Grothendieck ring of characters.

L̃_{J_{2}}(B̂_{3}) = P_{J_{2}}(ϕ_{10}) + ϕ_{9}.

This formula is valid in the Green ring of virtual modules.

b) Let G be Ru, HS, Co_{3} or Co_{2}, and let p = 5. The group HS has two classes of noncentral elements, denoted 5B and 5C. When G = HS, let t be a noncentral element of type 5B; the fixed point set |B̂_{5}(HS)|^{5C} is contractible. Then the fixed point set |B̂_{5}(G)|^{t} consists of six contractible components, homotopy equivalent to the building for L_{2}(5). The reduced Lefschetz module L̃_{G}(B̂_{5}) contains precisely one nonprojective summand, which has vertex 5 = ⟨t⟩ and lies in a block with the same group as defect group.

c) Let G be He and let p = 7. The simple group of Held has five conjugacy classes of elements of order 7, which generate three classes of groups, denoted 7AB, 7C(of central type), and 7DE. The fixed point set of the last group 7DE is contractible. Let t denote an element of type 7A or 7B. Then the fixed point set |B̂_{7}(He)|^{t} consists of eight contractible components, homotopy equivalent to the building for L_{2}(7). The virtual module L̃_{He}(B̂_{7}) contains precisely one nonprojective summand, which has vertex 7AB = ⟨t⟩ and lies in a block with the same group as defect group.

d) Let G be either He or M_{24}, and let p = 3. Then the fixed point set |B̂_{3}(G)|^{t} consists of 28 contractible components, equivalent to the set of Sylow subgroups Syl_{3}(L_{3}(2)). L̃_{G}(B̂_{3}) contains three nonprojective summands, all having vertex 3 = ⟨t⟩. Two of these summands lie in one block, with the same group as defect group, but the third summand lies in the principal block.

e) Let G be F i′_{24} and let p = 7. Then the fixed point set |B̂_{7}(F i′_{24})|^{t} consists of 120 contractible components, equivalent to the set of Sylow subgroups Syl_{7}(A_{7}). The reduced Lefschetz module L̃_{F i′_{24}}(B̂_{7}) contains five nonprojective summands, all having vertex 7 = ⟨t⟩. These five summands lie in four blocks; four lie in blocks with the same group 7 = ⟨t⟩ as defect group, but the fifth summand lies in the principal block.

f) Let G be the Monster M and let p = 13. Then the fixed point set |B̂_{13}(M)|^{t} consists of 144 contractible components, equivalent to the set of Sylow subgroups Syl_{13}(L_{3}(3)). The virtual module L̃_{M}(B̂_{13}) contains three nonprojective summands, all having vertex 13 = ⟨t⟩. These three summands lie in three different blocks; two lie in blocks with the same group 13 = ⟨t⟩ as defect group, but the third summand lies in the principal block.

Proof. As mentioned above these sporadic groups, with the given prime, have Sylow p-subgroups equal to the extraspecial group of order p^{3} and exponent p. Except for Held at p = 7 and HS at p = 5, they have two conjugacy classes of elements of order p (usually type A are p-central and type B are noncentral, although this notation is reversed for F i′_{24} at p = 7 and M at p = 13). The set of p-central elements is closed under taking products of commuting elements. Thus the distinguished Bouc collection B̂_{p}(G) is homotopy equivalent to the Benson collection, which consists of nontrivial purely p-central elementary abelian subgroups [21, Theorems 3.1 and 4.4]. By Proposition 2.8, the fixed point set |B̂_{p}(G)|^{z} of a p-central element z is contractible and L̃_{G}(B̂_{p}) has no summands with a vertex containing a p-central element. We need only consider purely noncentral p-subgroups, and for these Sylow p-subgroups p_{+}^{1+2} this implies that the only possible nontrivial vertex is a group of order p of noncentral type.

Usually, the first step in computing the fixed point set |B̂_{p}(G)|^{t} is to find the number of Sylow p-subgroups S which satisfy t ∈ N_{G}(S). In most of these cases, N_{G}(S) is equal to N_{G}(Z), where Z = Z(S) is a group of order p of central type. But t ∈ N_{G}(Z) iff t ∈ C_{G}(Z) iff Z ⊆ C_{G}(t). So we want to identify the p-central elements of G that lie in C_{G}(t).

a). M_{12} and p = 3 A 3-local geometry for M_{12}, originally due to Glauberman, appears in papers by Ronan and Stroth [29, Section 3] and Buekenhout [8, Section 9] and B̂_{3}(M_{12}) is its barycentric subdivision. The 3radical subgroups of M_{12} form four conjugacy classes [3, Section 2], given by 3 = ⟨t⟩, two conjugacy classes 3A_{I}^{2} and 3A_{II}^{2} of purely 3-central elementary abelian 3-subgroups of rank 2, and the Sylow 3-subgroup 3_{+}^{1+2}. The last three are distinguished; thus |B̂_{3}(M_{12})| is a graph with three types of vertices. Given a Sylow 3-subgroup S, let Z = Z(S) be its center. Then N_{M_{12}}(S) = N_{M_{12}}(Z) = 3_{+}^{1+2}.2^{2} and C_{M_{12}}(Z) = 3_{+}^{1+2}.2. Then S ∈ |B̂_{3}(M_{12})|^{t} iff t ∈ N_{M_{12}}(S) iff t ∈ C_{M_{12}}(Z) iff Z ≤ C_{M_{12}}(t). But C_{M_{12}}(t) ≃ 3 × A_{4} contains four 3-central subgroups of order 3 (eight elements of type 3A) [37, Section 3], and so |B̂_{3}(M_{12})|^{t} contains four vertices of type 3_{+}^{1+2}. The element t will also normalize each 3A^{2} ≤ 3_{+}^{1+2} (the Sylow group contains two purely 3-central 3A^{2}), but two “adjacent” Sylows will have an intersection S_{1} ∩ S_{2} = 3A^{2} containing no elements of type 3B. If 3A^{2} ∈ |B̂_{3}(M_{12})|^{t} then t ∈ N_{M_{12}}(3A^{2}) and ⟨t, 3A^{2}⟩ = 3_{+}^{1+2} ∈ |B̂_{3}(M_{12})|^{t}. Thus |B̂_{3}(M_{12})|^{t} has four components, each component equal to 2 edges and 3 vertices.

The normalizer N_{M_{12}}(⟨t⟩) ≃ S_{3} × A_{4} acts on the fixed point set |B̂_{3}(M_{12})|^{t} with the S_{3} acting trivially and so that the action of A_{4} is equivalent to the action of L_{2}(3) on its building. Thus L̃_{A_{4}}(|B̂_{3}(M_{12})|^{t}) is the Steinberg module (projective and irreducible), and L̃_{S_{3}×A_{4}}(|B̂_{3}(M_{12})|^{t}) is indecomposable with vertex 3 = ⟨t⟩. This implies that the Lefschetz module L̃_{M_{12}}(B̂_{3}) contains one indecomposable summand with vertex of type ⟨3B⟩, and it lies in a block with the same group ⟨3B⟩ as defect group.

Let H_{1} and H_{2} denote the two conjugacy classes of 3^{2} : GL_{2}(3) in M_{12} (the normalizers of the two classes of subgroups 3A_{I}^{2} and 3A_{II}^{2}), with intersection H_{1} ∩ H_{2} = H_{12} = 3_{+}^{1+2} : 2^{2}. Then the reduced Lefschetz module is:

L̃_{M_{12}}(B̂_{3}) = Ind_{H_{1}}^{M_{12}}(1a) + Ind_{H_{2}}^{M_{12}}(1a) − Ind_{H_{12}}^{M_{12}}(1a) − 1a.

The characters for the first two terms are described in the Atlas; the third was computed with GAP, using

Ind_{H_{1}}^{M_{12}}Ind_{H_{12}}^{H_{1}}(1a) = Ind_{H_{1}}^{M_{12}}(1a + 3b).

The character formula given above in Theorem 3.1(a) was obtained. Observe that the part of this formula that lies in the principal block, being projective, is valid in the Green ring of virtual modules. There are only six indecomposable modules lying in the block with the cyclic defect group ⟨3B⟩: the two simple modules ϕ_{7} and ϕ_{11}; their projective covers, which are uniserial, P_{M_{12}}(ϕ_{7}) = ⟨ϕ_{7}, ϕ_{11}, ϕ_{7}⟩ and P_{M_{12}}(ϕ_{11}) = ⟨ϕ_{11}, ϕ_{7}, ϕ_{11}⟩; and two other modules M_{1} = ⟨ϕ_{7}, ϕ_{11}⟩ and M_{2} = ⟨ϕ_{11}, ϕ_{7}⟩. Unfortunately, M_{1} and M_{2} have the same factors and the same character. One of these must be the nonprojective indecomposable summand of the reduced Lefschetz module.

J_{2} and p = 3 The Hall-Janko-Wales group J_{2} has two conjugacy classes of elements of order 3, denoted 3A and 3B. Their normalizers are N_{J_{2}}(⟨3A⟩) = 3.P GL_{2}(9) = 3.A_{6}.2 and N_{J_{2}}(⟨3B⟩) = S_{3} × A_{4}. The Sylow subgroup contains two elements of type 3A (the center) and 24 elements of type 3B. The collection of elements of type 3A is closed (a GAP computation gives that the class multiplication coefficient ξ(3A, 3A, 3B) = 0) and thus the Benson complex Δ consists of a disjoint set of 280 vertices corresponding to the groups of type ⟨3A⟩, and this Benson complex is equivariantly homotopy equivalent to |B̂_{3}(J_{2})|. To show that the fixed point set Δ^{3B} consists of four vertices, note that 3B ∈ N_{J_{2}}(⟨3A⟩) iff 3B ∈ C_{J_{2}}(3A) iff 3A ∈ C_{J_{2}}(3B) = 3 × A_{4}. The group 3 × A_{4} contains four subgroups of type ⟨3A⟩, namely Syl_{3}(A_{4}). These computations imply that the reduced Lefschetz module L̃_{J_{2}}(B̂_{3}) has precisely one indecomposable summand with vertex 3 = ⟨3B⟩ which lies in a block with the same group as defect group. Note that

L̃_{J_{2}}(B̂_{3}) = Ind_{3.A_{6}.2}^{J_{2}}(1a) − 1a = 63a + 90a + 126a = χ_{7} + χ_{10} + χ_{11} = ϕ_{9} + P_{J_{2}}(ϕ_{10})

is an ordinary module (not a virtual module). The module which affords χ_{7} = ϕ_{9} is not projective, and lies in block 3 with defect group ⟨3B⟩. The projective cover P_{J_{2}}(ϕ_{10}) = χ_{10} + χ_{11} lies in block 2 with defect group ⟨3A⟩. Since ⟨3A⟩ is not a vertex, as |B̂_{3}(J_{2})|^{3A} is contractible, the part of the reduced Lefschetz module lying in block 2 must be projective. The entire formula given for L̃_{J_{2}}(B̂_{3}) is valid on the level of modules, not just characters.

b). Ru and p = 5 The sporadic simple group of Rudvalis has two conjugacy classes of elements of order 5, the 5central elements 5A and the noncentral elements 5B. There are three conjugacy classes of radical 5-subgroups, 5 = ⟨5B⟩, a purely central 5^{2} = 5A^{2}, and the Sylow 5_{+}^{1+2}; see [4, Table 4]. Therefore |B̂_{5}(Ru)| is a graph with two types of vertices. To compute the fixed point set |B̂_{5}(Ru)|^{5B}, let S = 5_{+}^{1+2} ∈ |B̂_{5}(Ru)|^{5B} and denote by Z the center Z(S) = ⟨5A⟩. Then 5B ∈ N_{Ru}(S) = N_{Ru}(Z) ≃ 5_{+}^{1+2} : ((4 × 4) : 2) iff Z = ⟨5A⟩ ≤ C_{Ru}(5B) ≃ 5 × A_{5}. The alternating group A_{5} contains six subgroups of order 5, all of type ⟨5A⟩. Thus |B̂_{5}(Ru)|^{5B} contains the six vertices corresponding to six Sylow 5-subgroups of Ru. If 5B ∈ S, then 5B ∈ N_{Ru}(5A^{2}) for the two copies of 5A^{2} ≤ S. Conversely, if 5B ∈ N_{Ru}(5A^{2}), then ⟨5B, 5A^{2}⟩ is a Sylow 5-subgroup of Ru. Therefore |B̂_{5}(Ru)|^{5B} consists of six components, each consisting of two edges with their three vertices.

In the action of N_{Ru}(⟨5B⟩) ≃ (5 : 4) × A_{5} on |B̂_{5}(Ru)|^{5B}, the Frobenius group 5 : 4 acts trivially and the action of A_{5} is equivalent to the action of L_{2}(5) on its building. Thus the reduced Lefschetz module L̃_{A_{5}}(|B̂_{5}(Ru)|^{5B}) is the projective irreducible Steinberg module, and L̃_{N_{Ru}(⟨5B⟩)}(|B̂_{5}(Ru)|^{5B}) is indecomposable with vertex 5 = ⟨5B⟩. Therefore L̃_{Ru}(B̂_{5}) contains one indecomposable summand with vertex ⟨5B⟩, and it lies in a block with the same group as defect group.

G = HS, Co_{3}, Co_{2} and p = 5 These three groups have only two conjugacy classes of radical 5-subgroups, ⟨5B⟩ and the Sylow 5subgroup 5_{+}^{1+2}. Thus |B̂_{5}(G)| is a disjoint set, with vertices corresponding to Syl_{5}(G). For G = HS or G = Co_{3}, N_{G}(⟨5B⟩) ≃ (5 : 4) × A_{5}, but for G = Co_{2} we have N_{G}(⟨5B⟩) ≃ (5 : 4) × S_{5}. The fixed point set |B̂_{5}(G)|^{5B} consists of six vertices, corresponding to those groups S ∈ Syl_{5}(G) with centers ⟨5A⟩ = Z(S) lying in the A_{5} (or S_{5}) of N_{G}(⟨5B⟩). Thus L̃_{N_{G}(⟨5B⟩)}(|B̂_{5}(G)|^{5B}) is indecomposable with vertex 5 = ⟨5B⟩. (Note the Steinberg module for A_{5} extends in a natural way to one for S_{5} [34], although S_{5} has two projective modules which restrict to the Steinberg module for A_{5}.) Therefore L̃_{G}(B̂_{5}) contains one indecomposable summand with vertex ⟨5B⟩, lying in a block with the same group as defect group.

Observe that HS has another class 5C of noncentral elements; the normalizer N_{HS}(⟨5C⟩) = 5^{2} : 4. The elementary abelian 5^{2} contains a unique central subgroup ⟨5A⟩. This implies that the fixed point set |B̂_{5}(HS)|^{5C} is contractible, and ⟨5C⟩ is not a vertex of the Lefschetz module.

c). He and p = 7 The sporadic group of Held contains five conjugacy classes of elements of order 7, with 7C the 7central elements; the 7B are inverses of the 7A and the 7E are inverses of the 7D. The normalizers are N_{He}(7AB) = (7 : 3) × L_{2}(7), N_{He}(7DE) = 7^{2}.6, and N_{He}(⟨7C⟩) = N_{He}(7_{+}^{1+2}) = 7_{+}^{1+2} : (S_{3} × 3). There is one class of purely central elementary abelian 7C^{2}, as well as two mixed classes of elementary abelian 7^{2}, each containing six 7-central elements; see [10, Section 2.4]. There are three classes of 7radical subgroups, 7AB, 7C^{2}, and 7_{+}^{1+2}; the last two are distinguished [2, Section 2]. Thus |B̂_{7}(He)| is a graph with two types of vertices. Each Sylow 7-subgroup contains three copies of 7C^{2}, and each elementary abelian 7C^{2} lies in eight Sylow subgroups. Note 7AB ≤ N_{He}(7_{+}^{1+2}) = N_{He}(⟨7C⟩) iff 7C ∈ C_{He}(7AB) = 7 × L_{2}(7). Thus the fixed point set |B̂_{7}(He)|^{7AB} contains eight vertices corresponding to Sylow subgroups of Held. Such a Sylow will contain three copies of 7C^{2} also normalized by the 7AB. If 7AB ≤ N_{He}(7C^{2}), then they generate a Sylow subgroup 7_{+}^{1+2} = ⟨7AB, 7C^{2}⟩. The fixed point set |B̂_{7}(He)|^{7AB} consists of eight contractible components, and L̃_{He}(B̂_{7}) has a summand with vertex ⟨7AB⟩, lying in a block with the same group as the defect group.

If 7DE ≤ N_{He}(7_{+}^{1+2}) = N_{He}(⟨7C⟩), then 7C ∈ C_{He}(7DE). But the latter contains only one copy of ⟨7C⟩. If 7DE ≤ N_{He}(7C^{2}), they generate a Sylow subgroup 7_{+}^{1+2} = ⟨7DE, 7C^{2}⟩. Thus the fixed point set Δ^{7DE} is contractible, consisting of the star corresponding to one Sylow subgroup and its three copies of 7C^{2}. L̃_{He}(B̂_{7}) does not contain any summand with vertex ⟨7DE⟩.

d). He and p = 3 The sporadic group of Held contains two conjugacy classes of elements of order 3, denoted 3A and 3B. The Sylow 3-subgroups 3_{+}^{1+2} contain 14 elements of type 3A and 12 elements of type 3B. There are two conjugacy classes of elementary abelian 3-subgroups of rank 2, one of type 3A^{2} (a pure group, with 8 elements of type 3A; note 3_{+}^{1+2} contains two such subgroups) and one mixed 3^{2} = A_{1}B_{3} (containing 2 elements of type 3A and 6 elements of type 3B; note 3_{+}^{1+2} contains two of these groups); see [10, Section 2.2]. The nontrivial 3-radical subgroups of Held [2, Section 2] are the conjugacy classes of ⟨3A⟩, ⟨3B⟩, 3A^{2}, and 3_{+}^{1+2}. Thus the distinguished Bouc complex |B̂_{3}(He)| contains three types of vertices, corresponding to ⟨3A⟩, 3A^{2}, and 3_{+}^{1+2}; note |B̂_{3}(He)| is a 2-dimensional complex with maximal simplices corresponding to chains ⟨3A⟩ ≤ 3A^{2} ≤ 3_{+}^{1+2}. We wish to compute the fixed point set |B̂_{3}(He)|^{3B}. Clearly 3B ∈ N_{He}(⟨3A⟩) iff 3B ∈ C_{He}(3A) iff 3A ∈ C_{He}(3B) = 3 × L_{3}(2). There are precisely 28 subgroups of type ⟨3A⟩ in this group, namely Syl_{3}(L_{3}(2)). Next, 3B ∈ N_{He}(3_{+}^{1+2}) iff 3B ∈ 3_{+}^{1+2} ≤ C_{He}(3A) = 3.A_{7}, where 3A = Z(3_{+}^{1+2}). An element of type 3B lying in C_{He}(3A) has image in A_{7} a permutation in the conjugacy class of (123)(456). Such a permutation is contained in a unique Sylow 3-subgroup of A_{7}. Thus there is a unique Sylow subgroup S = 3_{+}^{1+2} in He with center ⟨3A⟩ and containing 3B. So |B̂_{3}(He)|^{3B} contains 28 edges connecting vertices of type ⟨3A⟩ to vertices of type 3_{+}^{1+2}. Clearly, if 3B ∈ 3_{+}^{1+2}, then 3B normalizes the two elementary abelian subgroups of type 3A^{2} ≤ 3_{+}^{1+2}. But if 3B ∈ N_{He}(3A^{2}), then they generate a Sylow, ⟨3B, 3A^{2}⟩ = 3_{+}^{1+2}. Thus |B̂_{3}(He)|^{3B} contains 56 vertices of type 3A^{2}, and consists of 28 components, each (contractible) component equal to two triangles sharing one common edge.

Remark 3.2. Although the multiplication class coefficient ξ(3A, 3A, 3B) = 168 is nonzero, it is still true that if two elements of type 3A commute with each other, then their product either equals the identity or is another element of type 3A. If the two commuting elements generate an elementary abelian 3^{2}, then this 3^{2} contains at least four elements of type 3A (the given generators and their inverses). But this forces the 3^{2} to be a pure 3A^{2}. (The number 168 can be explained as follows: in a fixed Sylow 3_{+}^{1+2} with a fixed element 3B ∈ 3_{+}^{1+2}, there are six ways to multiply two (noncommuting) elements of type 3A to obtain the given element 3B. Then each 3B is contained in 28 Sylow subgroups, and 168 = 6 × 28.) Thus the Benson complex Δ(He) equals the subcomplex of the distinguished Bouc complex |B̂_{3}(He)| consisting of those terms corresponding to the groups of type ⟨3A⟩ and 3A^{2}. The inclusion Δ(He) ⊆ B̂_{3}(He) induces an equivariant homotopy equivalence.

The action of L_{3}(2) on its 28 Sylow 3-subgroups yields a 28-dimensional representation χ = χ_{1} + 2χ_{4} + χ_{5} + χ_{6} (using the Atlas [11] notation, χ_{1} = 1a, χ_{4} = 6a, χ_{5} = 7a, and χ_{6} = 8a). Note that N_{L_{3}(2)}(3) = S_{3}, so the elements of order 4 or 7 have no fixed points. The elements of order three fix precisely one subgroup 3, and each element of order two fixes four groups of order 3. The Green correspondence implies that Ind_{S_{3}}^{L_{3}(2)}(1) is a direct sum of the trivial module and a projective module. Therefore the reduced Lefschetz module L̃_{L_{3}(2)}(|B̂_{3}(He)|^{3B}) is projective, with character 2χ_{4} + χ_{5} + χ_{6}. Note that χ_{4} = φ_{4} is projective, lying in a block of defect zero. But χ_{5} + χ_{6} is the projective cover P_{L_{3}(2)}(φ_{5}), where χ_{5} = φ_{5} and χ_{6} = φ_{1} + φ_{5}, which lies in the principal block. Note that the S_{3} in N_{He}(⟨3B⟩) ≃ S_{3} × L_{3}(2) acts trivially on |B̂_{3}(He)|^{3B}. It follows that L̃_{N_{He}(⟨3B⟩)}(|B̂_{3}(He)|^{3B}) has three summands with vertex ⟨3B⟩. Therefore the reduced Lefschetz module L̃_{He}(B̂_{3}) has precisely three nonprojective summands, all with vertex 3 = ⟨3B⟩. Two of these summands lie in the same block, with defect group also equal to ⟨3B⟩, but the third summand lies in the principal block.

M_{24} and p = 3 The situation for M_{24} is very similar to that of He, with the same Sylow 3-subgroup and fusion pattern. Also, N_{M_{24}}(⟨3B⟩) ≃ S_{3} × L_{3}(2) acts on |B̂_{3}(M_{24})|^{3B} as L_{3}(2) acts on its 28 Sylow 3subgroups. Thus the reduced Lefschetz module L̃_{M_{24}}(B̂_{3}) contains three indecomposable summands with vertex 3 = ⟨3B⟩. Two of these summands lie in the same block, having defect group ⟨3B⟩. The third summand lies in the principal block.

e). Fi′_{24} and p = 7 The Fischer group F i′_{24} has two classes of elements of order 7, type 7A and 7B (the latter are 7central). The normalizers are N_{G}(⟨7A⟩) = (7 : 6) × A_{7} and N_{G}(⟨7B⟩) = N_{G}(7_{+}^{1+2}) = 7_{+}^{1+2} : (S_{3} × 6). There are four conjugacy classes of 7-radical subgroups, ⟨7A⟩, 7_{+}^{1+2}, and two classes of purely central elementary abelian 7B^{2}. The last three are distinguished. Thus |B̂_{7}(F i′_{24})| is a graph with three types of vertices, corresponding to the Sylow subgroups and the two classes of elementary abelian 7B^{2}. The fixed point set |B̂_{7}(F i′_{24})|^{7A} consists of 120 contractible components, each component a star corresponding to one Sylow subgroup 7_{+}^{1+2} and its six purely central elementary abelian 7B^{2}, three from each conjugacy class. These components correspond to the 120 Sylow 7-subgroups of the alternating group A_{7}. The action of A_{7} on Syl_{7}(A_{7}) yields the induced module

Ind_{7:3}^{A_{7}}(1a) = 1a + 10ab + 14bb + 15a + 21a + 35a

(the character values are 120, 0, 0, 6, 0, 0, 0, 1, 1; the entry 6, for example, can be thought of in terms of the fact that the permutation (123)(456) normalizes six different groups of order 7 in A_{7}). The group ring of A_{7} has five blocks at the prime p = 7, with defects 1, 0, 0, 0, 0. The module which affords χ_{6} = ϕ_{5} = 14b is projective and lies in block 3, χ_{8} = ϕ_{6} = 21a is projective and lies in block 4, and χ_{9} = ϕ_{7} = 35a is projective and lies in block 5. The projective cover P_{A_{7}}(ϕ_{3}) = 10ab + 15a = χ_{3} + χ_{4} + χ_{7} lies in the principal block. This implies that the reduced Lefschetz module L̃_{F i′_{24}}(B̂_{7}) contains five indecomposable summands with vertex 7 = ⟨7A⟩. Four of these summands lie in three blocks, having the same group as defect group (two summands are in the same block). The fifth summand lies in the principal block.

f). M and p = 13 The Monster group M has two classes of elements of order 13, type 13A and 13B (the latter are 13-central). The normalizers are N_{M}(⟨13A⟩) = (13 : 6 × L_{3}(3)).2 and N_{M}(⟨13B⟩) = N_{M}(13_{+}^{1+2}) = 13_{+}^{1+2} : (3 × 4.S_{4}). There is one class of purely central elementary abelian 13B^{2}. The groups ⟨13A⟩, 13B^{2}, and 13_{+}^{1+2} are 13-radical, and the last two are distinguished. Thus |B̂_{13}(M)| is a graph with two types of vertices; each Sylow contains six copies of 13B^{2}, and each elementary abelian 13B^{2} lies in 14 Sylow subgroups. Note that 13A ∈ N_{M}(13_{+}^{1+2}) = N_{M}(⟨13B⟩) iff 13B ∈ C_{M}(13A) = 13 × L_{3}(3). Thus the fixed point set |B̂_{13}(M)|^{13A} contains 144 vertices corresponding to Sylow subgroups of M. For each such Sylow, the 13A also normalizes its six elementary abelian 13B^{2}, and if 13A ∈ N_{M}(13B^{2}), then they generate a Sylow 13_{+}^{1+2} = ⟨13A, 13B^{2}⟩.

The group ring of L_{3}(3) has six blocks at the prime 13, of defects 1, 0, 0, 0, 0, 0. The action of L_{3}(3) on its Sylow 13-subgroups corresponds to the induced character

Ind_{13:3}^{L_{3}(3)}(1a) = 1a + 13a + 16abcd + 27a + 39a.

Note that 13a is projective and lies in block two, and 39a is projective and lies in block six. The projective cover P_{M}(ϕ_{4}) = 16abcd + 27a lies in the principal block. This implies that the reduced Lefschetz module L̃_{M}(B̂_{13}) contains three indecomposable summands with vertex 13 = ⟨13A⟩. Two of these summands lie in two blocks, having the same group as defect group. The third summand lies in the principal block. □

3.2. The complex of p-centric and p-radical subgroups. A p-subgroup Q of G p-centric if the center Z(Q) is a Sylow p-subgroup of C_{G}(Q). The collection of p-centric and p-radical subgroups B_{p}^{cen}(G) is a subcollection of the distinguished Bouc collection B̂_{p}(G); see [22, Proposition 3.1].

A group has parabolic characteristic p if all p-local subgroups which contain a Sylow p-subgroup of G have characteristic p. For the groups with this property, the two collections B̂_{p}(G) and B_{p}^{cen}(G) are equal [22, Proposition 3.5]. All but three of the sporadic groups discussed in Section 3.1 satisfy this condition. We now turn to the three cases above in which the complex of p-radical p-centric subgroups is not equal to the complex of distinguished p-radical subgroups.

Theorem 3.3. Let Δ = |B_{3}^{cen}(G)| and let L̃_{G}(Δ) be the mod-3 reduced Lefschetz module. Let z ∈ G be a 3-central element and t ∈ G be an element of order 3 of noncentral type.

a) Let G be J_{2}. Then the fixed point set Δ^{z} consists of ten vertices, equal to the building for P GL_{2}(9), and the fixed point set Δ^{t} consists of four vertices, equal to the building for L_{2}(3). The module L̃_{J_{2}}(Δ) contains two nonprojective summands. One summand has vertex 3 = ⟨t⟩ and lies in a block with the same group as defect group. The second summand has vertex the 3-central group ⟨z⟩ and lies in a block with the same group as defect group. b) Let G be M_{24}. Then the fixed point set Δ^{z} consists of ten contractible components, equivalent to

Syl_{3}(S_{6}), and the fixed point set Δ^{t} consists of 28 contractible components, equivalent to the set of Sylow 3-subgroups Syl_{3}(L_{3}(2)). The module L̃_{M_{24}}(Δ) contains four nonprojective summands. Three of these summands have vertex 3 = ⟨t⟩; two summands lie in one block, with the same group as defect group, but the third summand lies in the principal block. The fourth summand has vertex the 3-central group ⟨z⟩, and lies in a block with the same group as defect group.

c) Let G be He. Then the fixed point set Δ^{z} is connected and is homotopy equivalent to a subcomplex of the Quillen complex for the symmetric group S_{7}. The fixed point set Δ^{t} consists of 28 contractible components, equivalent to the set of Sylow subgroups Syl_{3}(L_{3}(2)). The module L̃_{He}(Δ) contains four nonprojective summands. Three of these summands have vertex 3 = ⟨t⟩; two summands lie in one block, with the same group as defect group, but the third summand lies in the principal block. The fourth summand has vertex the 3-central group ⟨z⟩, and lies in a block with defect group the purely central elementary abelian group 3A^{2}.

Proof. Since Δ is a subcomplex of |B̂_{3}(G)| we shall use some of the details given in the proof of Theorem 3.1(a, d). For these three groups, the 3-central group ⟨3A⟩ is 3-radical but not 3-centric. For J_{2} the complex Δ is a discrete set of vertices, corresponding to Syl_{3}(J_{2}). For M_{24} and He, the complex Δ is a graph with two types of vertices, corresponding to the Sylow 3-subgroups and to the purely central elementary abelian groups 3A^{2}. In all three cases, the fixed point set Δ^{t} is equivalent to that described in the previous theorem, with the same explanation for the summands of the reduced Lefschetz module which have vertex the group ⟨t⟩. In the previous theorem, the fixed point sets for the 3-central element z were contractible; this is no longer true for the complex Δ.

For G = J_{2}, the fixed point set Δ^{z} consists of ten vertices corresponding to the ten Sylow 3-subgroups containing z. The action of N_{J_{2}}(⟨z⟩) = 3.P GL_{2}(9) on these ten vertices is the action of P GL_{2}(9) on its building, with reduced Lefschetz module the Steinberg module. Then Theorem 2.7 implies that L̃_{J_{2}}(Δ) has one summand with vertex ⟨z⟩ which lies in a block with the same group as defect group. This concludes the proof of the statement in part a) of the theorem.

For G equal to either M_{24} or He, note that the collection consisting of the Sylow 3-subgroups and the purely central elementary abelian subgroups 3A^{2} is closed under passage to 3-supergroups. We have an equivariant poset map F_{1} : Δ^{z} → Δ^{z} defined by F_{1}(P) = ⟨z⟩ · P. Since this satisfies F_{1}(P) ≥ P, there is an N_{G}(⟨z⟩)-homotopy equivalence between Δ^{z} and the image of F_{1}, the subcomplex Δ_{1}^{z} determined by those groups P which contain z; see [17, 2.2(3)]. Next, F_{2} : Δ_{1}^{z} → Δ_{1}^{z} defined by F_{2}(P) = P ∩ C_{G}(z) is an equivariant poset map satisfying F_{2}(P) ≤ P. Observe that if z ∈ P = 3A^{2} then P ≤ C_{G}(z), and if z ∈ P = 3_{+}^{1+2} then either z ∈ Z(P) and P ≤ C_{G}(z), or z is contained in only one of the two purely central elementary abelian 3A^{2} in P. In this latter case, F_{2}(P) = 3A^{2} does lie in Δ_{1}^{z}. Therefore Δ_{1}^{z} is N_{G}(⟨z⟩)-homotopy equivalent to the image of F_{2}, the subcomplex Δ_{2}^{z} determined by those groups P satisfying z ∈ P ≤ C_{G}(z); of course P is either 3A^{2} or 3_{+}^{1+2}.

The centralizers of the 3-central elements are C_{M_{24}}(z) = 3.A_{6} and C_{He}(z) = 3.A_{7}. Let H denote A_{6} when G = M_{24} and A_{7} when G = He. The quotient map π : 3.H → H induces an isomorphism of the complex Δ_{2}^{z} with the subcomplex E_{3}(H) of the Quillen complex for H determined by the collection consisting of the Sylow 3-subgroups of H (elementary abelian of rank 2) and the groups of order three generated by elements in the conjugacy class of the cycle (123). Since an element of H in the conjugacy class of the product (123)(456) of two cycles lies in a unique Sylow 3-subgroup of H, the subcomplex E_{3}(H) is equivariantly homotopy equivalent to the entire Quillen complex A_{3}(H). Recall that the reduced Lefschetz module for the Quillen complex of any finite group is projective.

For G equal to either M_{24} or He, the normalizers of the 3-central elements are N_{M_{24}}(⟨z⟩) = 3.S_{6} and N_{He}(⟨z⟩) = 3.S_{7}. For S_{6}, the Sylow 3-subgroups satisfy the trivial intersection property; in M_{24}, each 3A^{2} lies in a unique S ∈ Syl_{3}(M_{24}). The induced module

Ind_{3^{2}:D_{8}}^{S_{6}}(1a) − 1a = 9a = φ_{5}

yields a projective irreducible module lying in a defect zero block of 𝔽_{3}S_{6}. Consequently, the reduced Lefschetz module L̃_{M_{24}}(Δ) contains a summand with vertex 3 = ⟨z⟩. This proves part b) of the theorem.

For the symmetric group S_{7} we have

Ind_{S_{3}×S_{4}}^{S_{7}}(1a) + Ind_{(S_{3}×S_{3}):2}^{S_{7}}(1a) − Ind_{S_{3}×S_{3}}^{S_{7}}(1a) − 1a = −(21a + 15a) = −(χ_{8} + χ_{12}) = −P_{S_{7}}(φ_{2})

is projective and indecomposable and lies in a block of defect one. Thus L̃_{He}(Δ) has a summand with vertex 3^{2}, and this concludes the proof of part c) of the theorem. □

3.3. One last example. We conclude with one further example; although the Sylow subgroup is not extraspecial, the complex of distinguished 3-radical subgroups is very simple (a disjoint collection of vertices).

Theorem 3.4. Let G = J_{3} and p = 3. Then the reduced Lefschetz module for the collection of distinguished 3-radical subgroups has precisely one nonprojective summand with vertex a group of order 3 (of noncentral type) and lying in a block with the same group as defect group.

Proof. The third Janko group J_{3} has a Sylow 3-subgroup S of order 3^{5}, and contains two conjugacy classes of elements of order 3, denoted 3A and 3B; the class 3B lies in the center of the Sylow subgroup S. The center Z(S) = 3B^{2} is an elementary abelian 3-group of rank 2 containing 8 elements of type 3B. The normalizers are N_{J_{3}}(⟨3A⟩) = 3 : P GL_{2}(9), with C_{J_{3}}(3A) = 3 × A_{6}, and N_{J_{3}}(S) = N_{J_{3}}(3B^{2}) = 3^{2}.(3 × 3^{2}) : 8. The nontrivial 3-radical subgroups of J_{3} are the groups of type ⟨3A⟩ and the Sylow 3-subgroups of J_{3}; see [18]. Thus the distinguished Bouc complex |B̂_{3}(J_{3})| consists only of the 2^{4} · 5 · 17 · 19 vertices corresponding to Syl_{3}(J_{3}). The fixed point set |B̂_{3}(J_{3})|^{3A} can be computed by noting that 3A ∈ N_{J_{3}}(S) iff 3A ∈ S = C_{J_{3}}(3B^{2}) iff 3B^{2} ≤ C_{J_{3}}(3A) = 3 × A_{6}. There are precisely ten groups of type 3B^{2} in 3 × A_{6}, namely Syl_{3}(A_{6}). The action of P GL_{2}(9) on the ten vertices of |B̂_{3}(J_{3})|^{3A} is the action on the projective line 𝔽_{9} ∪ {∞}.

Thus L̃_{P GL_{2}(9)}(|B̂_{3}(J_{3})|^{3A}) is the projective irreducible Steinberg module and L̃_{N_{J_{3}}(⟨3A⟩)}(|B̂_{3}(J_{3})|^{3A}) is indecomposable with vertex ⟨3A⟩. The reduced Lefschetz module L̃_{J_{3}}(B̂_{3}) contains one indecomposable summand with vertex ⟨3A⟩, lying in a block with the same group as defect group.

The 3-central elements in J_{3} are closed since the multiplication class coefficient ξ(3B, 3B, 3A) = 0, but the Benson complex is slightly more complicated, containing 3B and 3B^{2}. Since |B̂_{3}(J_{3})| is homotopy equivalent to the Benson complex [21, Theorem 3.1 and 4.4], |B̂_{3}(J_{3})|^{3B} is contractible. The reduced Lefschetz module L̃_{J_{3}}(B̂_{3}) contains no summands with a vertex containing a 3-central element. Since C_{J_{3}}(3B) = 3 × A_{6} and all 3-elements in A_{6} are 3-central, the group 3 = ⟨3A⟩ is the only purely noncentral subgroup of J_{3}. Thus L̃_{J_{3}}(B̂_{3}) has only one nonprojective summand. □

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