REPRESENTATIONS OF THE DOUBLE BURNSIDE ALGEBRA AND COHOMOLOGY OF THE EXTRASPECIAL p-GROUP II

AKIHIKO HIDA AND NOBUAKI YAGITA

Abstract. Let E be the extraspecial p-group of order p^{3} and exponent p where p is an odd prime. We determine the mod p cohomology H^{∗}(X, 𝔽_{p}) of a summand X in the stable splitting of p-completed classifying space BE. In the previous paper [Representations of the double Burnside algebra and cohomology of the extraspecial p-group, J. Algebra 409 (2014) 265-319], we determined these cohomology modulo nilpotence. In this paper, we consider the whole part of the cohomology. Moreover, we consider the stable splittings of BG for some finite groups with Sylow p-subgroup E related with the three dimensional linear group L_{3}(p).

1. Introduction

Let p be an odd prime and E = p_{+}^{1+2} the extraspecial p-group of order p^{3} and exponent p. In the previous paper [7], we determined the composition factor of H^{∗}(E) = (𝔽_{p} ⊗ H^{∗}(E, ℤ))/\sqrt{(0)} as a right A_{p}(E, E)-module, where A_{p}(E, E) is a double Burnside algebra of E over 𝔽_{p} = ℤ/pℤ. In this paper, we consider the whole part of the cohomology H^{∗}(E, 𝔽_{p}) and determine the composition factor of H^{∗}(E, 𝔽_{p}) as an A_{p}(E, E)-module.

The mod p cohomology ring H^{∗}(E, 𝔽_{p}) of E is completely known by [9], but the structure is very complicated. We shall study H^{∗}(E, 𝔽_{p}) through the integral cohomology ring H^{∗}(E, ℤ) as in [13] and [14]. Let H^{even}(E, ℤ) (resp. H^{odd}(E, ℤ)) be the even (resp. odd) degree part of H^{∗}(E, ℤ). Let N = \sqrt{(0)} in 𝔽_{p} ⊗ H^{even}(E, ℤ). Then we have that 𝔽_{p} ⊗ H^{even}(E, ℤ)/N ≅ H^{∗}(E). On the other hand, the Milnor operator Q_{1} induces an isomorphism H^{odd}(E, ℤ) ≅ (y_{1}v, y_{2}v)H^{∗}(E) where (y_{1}v, y_{2}v)H^{∗}(E) is the ideal of H^{∗}(E) generated by y_{1}v, y_{2}v ∈ H^{2p+2}(E) (see the first part of section 2).

Let M = ⊕M^{n} and L = ⊕L^{n} be graded A_{p}(E, E)-modules such that M^{n} and L^{n} are finite dimensional for every n. We write as M ↔ L if M^{n} and L^{n} have same composition factors (with same multiplicity), that is, [M^{n}] = [L^{n}] in the Grothendieck group K_{0}(A_{p}(E, E)).

Using this notation, the structure of H^{∗}(E, 𝔽_{p}) can be stated as follows.

Theorem 1.1. (1) As A_{p}(E, E)-modules,

H^{even}(E, 𝔽_{p}) ↔ H^{∗}(E) ⊕ N ⊕ (y_{1}v, y_{2}v)H^{∗}(E)[−2p].

(2) As A_{p}(E, E)-modules,

H^{odd}(E, 𝔽_{p}) ↔ (y_{1}v, y_{2}v)H^{∗}(E)[−2p + 1] ⊕ (N ⊕ H^{+}(E))[−1].

Here, for a graded 𝔽_{p}-subspace M of H^{∗}(E), we denote by M[i] the graded vector space with M[i]^{n} = M^{n−i}. Since the composition factors of N and (y_{1}v, y_{2}v)H^{∗}(E) are determined in Proposition 3.1 and Theorem 3.5, we can get the composition factors of H^{∗}(E, 𝔽_{p}) completely.

The indecomposable summands in the complete stable splitting of the p-completed classifying space BE_{p}^{∧} correspond to primitive idempotents in A_{p}(E, E). Moreover they corresponds to simple A_{p}(E, E)-modules. We simply write BE for BE_{p}^{∧}. Let X be a summand in BE which corresponds to a simple A_{p}(E, E)-module S. Then the multiplicity of X in BE is equal to the dimension of S as an 𝔽_{p}-vector space since 𝔽_{p} is a splitting field for A_{p}(E, E). By results above, we can get the cohomology H^{∗}(X, 𝔽_{p}) (See Remark 3.6).

Let G be a finite group with Sylow p-subgroup E. Then the multiplicity of X in BG is equal to the dimension of S[G] where [G] is an element of A_{p}(E, E) corresponds to the (E, E)-biset G. See [1], [2], [11] for details.

In [15], the second author studied the splitting of BG for various finite groups G whose Sylow p-subgroup is E and p-local finite groups on E. In this paper, we consider the stable splitting for groups related with the linear group L_{3}(p) which were not treated in [15] in general. We use some simple A_{p}(E, E)-submodules of H^{∗}(E) and determine the multiplicity of summands in BG for G = L_{3}(p), L_{3}(p) : 2, L_{3}(p).3, L_{3}(p).S_{3} (Theorem 4.17, 4.18, 4.19, 4.20).

Combining these results and results in [15], we have the complete information on the stable splitting of finite groups or p-local finite groups which have at least two F-radical maximal elementary abelian p-subgroups in E, by the classification in [12], where F is a fusion system of G.

In particular, for p = 7, we obtain a diagram which describes inclusions of some fusion systems and stable splitting (Theorem 4.23). This result supplements the results of [15, section 9], in which the splitting of sporadic simple groups are mainly studied.

In section 2, we review the main results of [7] which will be used in section 3. In section 3, we prove Theorem 1.1 and determine the structures of ideals N and (y_{1}v, y_{2}v)H^{∗}(E). In section 4, we consider H^{∗}(G) and the stable splitting for finite group G which has a Sylow p-subgroup E. Finally, in section 5, we consider the case p = 3 and state some remarks.

2. Preliminary results on H^{∗}(E)

In this section, we quote some results from [7]. Let p be an odd prime. Let

E = ⟨a, b, c | [a, b] = c, a^{p} = b^{p} = c^{p} = [a, c] = [b, c] = 1⟩

be the extraspecial p-group of order p^{3} and exponent p. Let A_{i} = ⟨c, ab^{i}⟩ for 0 ≤ i ≤ p − 1 and A_{∞} = ⟨c, b⟩. Then

A(E) = {A_{0}, A_{1}, …, A_{p−1}, A_{∞}}

is the set of all maximal elementary abelian p-subgroups of E.

The cohomology of E is known by [8], [9], [15]. In particular, H^{∗}(E) = (𝔽_{p} ⊗ H^{∗}(E, ℤ))/\sqrt{(0)} is generated by

y_{1}, y_{2}, C, v

with

deg y_{i} = 2, deg C = 2p − 2, deg v = 2p

subject to the following relations:

y_{1}^{p}y_{2} − y_{1}y_{2}^{p} = 0, Cy_{i} = y_{i}^{p}, C^{2} = y_{1}^{2p−2} + y_{2}^{2p−2} − y_{1}^{p−1}y_{2}^{p−1}.

We set V = v^{p−1} and Y_{i} = y_{i}^{p−1}.

COHOMOLOGY OF EXTRASPECIAL p-GROUP II 3

Let R be a subalgebra of H^{∗}(E) and x_{1}, …, x_{r} elements of H^{∗}(E). We set

R{x_{1}, …, x_{r}} = ∑_{i=1}^{r} Rx_{i}

if x_{1}, …, x_{r} are linearly independent over R. Moreover, if W = ∑_{i=1}^{r} 𝔽_{p}x_{i} is a 𝔽_{p}-vector space spanned by x_{1}, …, x_{r}, then we set

R{W } = R{x_{1}, …, x_{r}}.

We consider the action of Out(E) = GL_{2}(𝔽_{p}) on H^{∗}(E). Let S^{i} be the homogeneous part of degree 2i in 𝔽_{p}[y_{1}, y_{2}]. Then p(p − 1) simple 𝔽_{p} Out(E)-modules

S^{i}v^{q} ≅ S^{i} ⊗ (det)^{q} (0 ≤ i ≤ p − 1, 0 ≤ q ≤ p − 2)

give the complete set of representatives of nonisomorphic simple 𝔽_{p} Out(E)-modules. Let us write

ℂ𝔸 = 𝔽_{p}[C, V]

and

𝔻𝔸 = 𝔽_{p}[D_{1}, D_{2}]

where D_{1} = C^{p} + V, D_{2} = CV. Then ℂ𝔸 = H^{∗}(E)^{Out(E)}, the Out(E)-invariants, and the restriction map induces an isomorphism

𝔻𝔸 ⟶^{∼} H^{∗}(A)^{Out(A)}

for all A ∈ A(E).

Let

T^{i} = 𝔽_{p}{y_{1}^{p−1}y_{2}^{i}, y_{1}^{p−2}y_{2}^{i+1}, …, y_{1}^{i}y_{2}^{p−1}}

for 1 ≤ i ≤ p−2. Then S^{p−1+i} = CS^{i} +T^{i}. The 𝔽_{p}-subspace CS^{i} is a GL_{2}(𝔽_{p})-submodule of CS^{i} + T^{i} and

(CS^{i} + T^{i})/CS^{i} ≅ (S^{p−1−i} ⊗ det^{i}).

Moreover we have the following expression [7, Theorem 4.4]:

H^{∗}(E) = 𝔽_{p}[C, v]{(⨁_{i=0}^{p−1} S^{i}) ⊕ (⨁_{i=1}^{p−2} T^{i})} = ℂ𝔸{⨁_{i=0}^{p−2} ⨁_{q=0}^{p−2}(S^{i}v^{q} ⊕ T^{i}v^{q}))}

where S^{0} = 𝔽_{p} and T^{0} = S^{p−1}.

Let C_{p} be a cyclic group of order p and let U_{i} = H^{2i}(C_{p}, 𝔽_{p}) (0 ≤ i ≤ p−2). Then U_{i} are simple 𝔽_{p} Out(C_{p})-modules. Let A ∈ A(E) be a maximal elementary abelian p-subgroup of E. Let S(A)^{i} = H^{2i}(A). Then S(A)^{i} ⊗ det^{q} (0 ≤ i ≤ p − 1, 0 ≤ q ≤ p − 2) are simple modules for Out(A) = GL_{2}(𝔽_{p}).

Let P be a general finite p-group and A_{p}(P, P) the double Burnside algebra of P over 𝔽_{p}. The simple A_{p}(P, P)-modules corresponds to some pairs (Q, V) where Q ≤ P and V is a simple 𝔽_{p} Out(Q)-module, see [2], [4], [11]. In this paper, we denote the simple A_{p}(P, P)-module corresponds to the pair (Q, V) by S(P, Q, V).

On the other hand, Dietz and Pridy [5] studied the stable splitting of BE and determined the multiplicity of each summand. In particular, their result implies the classification of simple A_{p}(E, E)-modules.

Proposition 2.1 ([5], [7, Proposition 10.1]). The simple A_{p}(E, E)-modules are given as follows:

(1) S(E, E, S^{i} ⊗ det^{q}) for 0 ≤ i ≤ p − 1, 0 ≤ q ≤ p − 2,

dim S(E, E, S^{i} ⊗ det^{q}) = i + 1.

(2) S(E, A, S(A)^{p−1} ⊗ det^{q}) for 0 ≤ q ≤ p − 2,

dim S(E, A, S(A)^{p−1} ⊗ det^{q}) = p + 1.

(3) S(E, C_{p}, U_{i}) for 0 ≤ i ≤ p − 2,

dim S(E, C_{p}, U_{i}) = \begin{cases}p + 1 & (i = 0) \\ i + 1 & (1 ≤ i ≤ p − 2). \end{cases}

(4) S(E, 1, 𝔽_{p}), dim S(E, 1, 𝔽_{p}) = 1.

To describe the composition factor of H^{∗}(E) as an A_{p}(E, E)-module, we need the following 𝔽_{p}-subspace of H^{∗}(E).

Definition 2.2. Let S be a simple A_{p}(E, E)-module. Let Γ_{S} be the following 𝔽_{p}-subspace of H^{∗}(E):

(1) If S = S(E, C_{p}, U_{i}), then

Γ_{S} = \{ \begin{matrix}𝔽_{p}[C]\{𝔽_{p}C + S^{p−1}\} & (i = 0) \\ 𝔽_{p}[C]\{S^{i}\} & (1 ≤ i ≤ p − 2). \end{matrix}

(2) If S = S(E, A, S(A)^{p−1} ⊗ det^{q}), then

Γ_{S} = \{ \begin{matrix}𝔻𝔸\{⊕_{0≤j≤p−1}D_{2}C^{j}(𝔽_{p}C + S^{p−1}))\} & (q = 0) \\ 𝔻𝔸\{⊕_{0≤j≤p−1}v^{q}C^{j}(CS^{q} + T^{q})\} & (1 ≤ q ≤ p − 2). \end{matrix}

(3)

Γ_{S} = \begin{cases}𝔻𝔸^{+} & (S = S(E, E, S^{0})) \\ ℂ𝔸\{v^{q}\} & (S = S(E, E, det^{q}), 1 ≤ q ≤ p − 2) \\ 𝔻𝔸\{V S^{p−1}\} & (S = S(E, E, S^{p−1}) \\ ℂ𝔸\{v^{q}S^{p−1}\} & (S = S(E, E, S^{p−1} ⊗ det^{q}), 1 ≤ q ≤ p − 2) \end{cases}

(4) Let

S = S^{i}v^{q}, T = T^{p−i−1}v^{s}

for 1 ≤ i ≤ p − 2, 0 ≤ q ≤ p − 2, where s ≡ i + q (mod p − 1), 0 ≤ s ≤ p − 2. Let Γ_{S(E,E,S^{i}⊗det^{q})} be the following 𝔽_{p}-subspace:

\begin{matrix}ℂ𝔸\{V S\} & ⊕ & 𝔻𝔸\{V T \} & (q ≡ 2i ≡ 0) \\ ℂ𝔸\{V S\} & ⊕ & ℂ𝔸\{T \} & (q ≡ 0, 2i ≢ 0) \\ 𝔻𝔸\{S\} & ⊕ & 𝔻𝔸\{V T \} & (i = q, 3i ≡ 0) \\ 𝔻𝔸\{S\} & ⊕ & ℂ𝔸\{T \} & (i = q, 3i ≢ 0) \\ ℂ𝔸\{S\} & ⊕ & 𝔻𝔸\{V T \} & (q ≠ 0, i ≠ q, q + 2i ≡ 0) \\ ℂ𝔸\{S\} & ⊕ & ℂ𝔸\{T \} & (q ≠ 0, i ≠ q, q + 2i ≢ 0). \end{matrix}

(5)

Γ_{S(E,1,𝔽_{p})} = 𝔽_{p} = H^{0}(E).

The following theorem is the main result of [7]. If S is a simple A_{p}(E, E)-module, then there exists an idempotent e_{S} such that Se_{S} = S and S′e_{S} = 0 for any simple module S′ ≅ S̸. We call e_{S} an idempotent which corresponds to S.

Theorem 2.3 ([7, Theorem 10.2, 10.3, 10.4, 10.5]). Let S be a simple A_{p}(E, E)-module. Then there exists an idempotent e_{S} which corresponds to S such that

H^{∗}(E)e_{S} = Γ_{S}e_{S} ≅ Γ_{S}.

If we see the minimal degree of non zero part of Γ_{S}, we have the following corollary.

Corollary 2.4. Every simple A_{p}(E, E)-module appears as a composition factor in H^{2n}(E) for some n ≤ (p + 2)(p − 1).

Let Γ_{S}^{n} = Γ_{S} ∩ H^{n}(E) be the degree n part of Γ_{S}. Then by Theorem 2.3,

∑_{S} dim Γ_{S}^{n} = dim H^{n}(E)

for any n ≥ 0. In fact we have the following.

Proposition 2.5. H^{∗}(E) is a direct sum of 𝔽_{p}-subspaces Γ_{S} where S runs over the representatives of the isomorphism classes of simple A_{p}(E, E)-modules.

Proof. By Theorem 2.3, it suffices to show that H^{∗}(E) = ∑ Γ_{S}. We shall show that ℂ𝔸{S^{i}v^{q}} (0 ≤ i ≤ p − 1, 0 ≤ q ≤ p − 2) and ℂ𝔸{T^{i}v^{q}} (1 ≤ i ≤ p − 2, 0 ≤ q ≤ p − 2) are contained in ∑ Γ_{S}.

First consider ℂ𝔸{S^{i}v^{q}}. If (i, q) = (0, 0),

ℂ𝔸 = 𝔽[C, D_{1}] = 𝔽[D_{1}] ⊕ ℂ𝔸{C}
= 𝔽[D_{1}] ⊕ 𝔽[C]{C} ⊕ ℂ𝔸{D_{2}}
= 𝔽[D_{1}] ⊕ 𝔽[C]{C} ⊕ ∑_{j=0}^{p−1} 𝔻𝔸{D_{2}C^{j+1}} ⊕ 𝔻𝔸{D_{2}}

= 𝔻𝔸 ⊕ 𝔽[C]{C} ⊕ ∑_{j=0}^{p−1} 𝔻𝔸{D_{2}C^{j+1}}

and this is contained in the sum of the subspaces of Definition 2.2 (1)(2)(3)(5). If (i, q) = (0, q), 1 ≤ q ≤ p − 2, or (i, q) = (p − 1, q), 1 ≤ q ≤ p − 2, then ℂ𝔸v^{q} and ℂ𝔸S^{p−1}v^{q} are contained in the subspace of Definition 2.2 (3). If (i, q) = (p − 1, 0), then

ℂ𝔸 = 𝔽[C] ⊕ ℂ𝔸{V }
= 𝔽[C] ⊕ 𝔻𝔸{V } ⊕ 𝔻𝔸{D_{2}, D_{2}C, …, D_{2}C^{p−1}}

and we have

ℂ𝔸{S^{p−1}} = 𝔽_{p}[C]{S^{p−1}} ⊕ 𝔻𝔸{V S^{p−1}} ⊕ (⊕_{j=0}^{p−1}𝔻𝔸{D_{2}C^{j}S^{p−1}}).

This is contained in the sum of subspaces of Definition 2.2 (1)(2)(3).

Consider (i, q) (1 ≤ i ≤ p − 2, 0 ≤ q ≤ p − 2). If q = 0, then

ℂ𝔸{S^{i}} = 𝔽_{p}[C]{S^{i}} ⊕ ℂ𝔸{V S^{i}}

and this is contained in the sum of subspaces of Definition 2.2 (1)(4). If i = q, then

ℂ𝔸{S^{i}v^{i}} = 𝔻𝔸{S^{i}v^{i}} ⊕ (⊕_{j=0}^{p−1}𝔻𝔸{C^{j+1}S^{i}v^{i}})

and this is contained in the sum of the subspaces of Definition 2.2 (2)(4). If q ≠ 0 and i ≠ q, then ℂ𝔸{S^{i}v^{q}} is contained in the subspace of Definition 2.2 (4).

Next, consider T^{k}v^{m} (1 ≤ k ≤ p − 2, 0 ≤ m ≤ p − 2). Let i = p − k − 1, q ≡ m + k mod (p − 1), 0 ≤ q ≤ p − 2, s = m. Then

T^{k}v^{m} = T^{p−i−1}v^{s}

where 1 ≤ i ≤ p − 2, 0 ≤ q ≤ p − 2, s ≡ i + q mod (p − 1) and

q + 2i ≡ m + k + 2(p − k − 1) ≡ m − k mod (p − 1).

If k ≠ m, then q + 2i ≢ 0 and ℂ𝔸{T^{k}v^{m}} is contained in the subspace of Definition 2.2 (4). If k = m, then q + 2i ≡ 0 and

ℂ𝔸{T^{k}v^{m}} = 𝔻𝔸{⊕_{j=0}^{p−1}T^{k}C^{j}v^{m}} ⊕ 𝔻𝔸{T^{k}V v^{m}}

since ℂ𝔸 = 𝔻𝔸{1, C, …, C^{p−1}, V }. This is contained in the sum of the subspaces of Definition 2.2 (2)(4). □

3. Composition factors of H^{∗}(E, 𝔽_{p})

In this section, we study the A_{p}(E, E)-module structure of H^{∗}(E, 𝔽_{p}). First, we shall prove Theorem 1.1. The even degree part H^{even}(E, ℤ) of integral cohomology ring is generated by

y_{1}, y_{2}, b_{2}, …, b_{p−2}, C, v

with

deg y_{i} = 2, deg b_{i} = 2i

subject to the following relations:

py_{i} = pb_{j} = pC = 0, p^{2}v = 0,
y_{1}y_{2}^{p} − y_{1}^{p}y_{2} = 0,
y_{i}b_{k} = b_{k}b_{j} = Cb_{j} = 0,
y_{i}C = y_{i}^{p}, C^{2} = y_{1}^{2p−2} + y_{2}^{2p−2} − y_{1}^{p−1}y_{2}^{p−1}

by [10] or [8, Theorem 3] (see [13] also). In particular, p^{2}(H^{2n}(E, ℤ)) = 0 for any n > 0.

On the other hand, the odd degree part of integral cohomology ring H^{odd}(E, ℤ) is annihilate by p and so it is considered as an 𝔽_{p}[y_{1}, y_{2}, v]-module. As an 𝔽_{p}[y_{1}, y_{2}, v]- module, H^{odd}(E, ℤ) is generated by two elements a_{1} and a_{2} with deg a_{i} = 3 subject to the following relations:

y_{1}a_{2} − y_{2}a_{1} = 0, y_{1}^{p}a_{2} − y_{2}^{p}a_{1} = 0.

Let H^{∗}(E, ℤ) ⟶ H^{∗}(E, 𝔽_{p}) be the natural map induced by ℤ ⟶ 𝔽_{p}. We use the same letters for the images of y_{i}, b_{j}, C, v in H^{∗}(E, 𝔽_{p}). Then

N = 𝔽_{p}[v]\{b_{2}, …, b_{p−2}\} = \sqrt{0}

in

H^{even}(E, ℤ)/pH^{even}(E, ℤ) = 𝔽_{p} ⊗_{ℤ} H^{even}(E, ℤ).

Since

H^{∗}(E) = (𝔽_{p} ⊗_{ℤ} H^{∗}(E, ℤ))/\sqrt{0} = (𝔽_{p} ⊗_{ℤ} H^{even}(E, ℤ))/N,

there is a short exact sequence of A_{p}(E, E)-modules,

(3.1)  0 ⟶ N ⟶ H^{even}(E, ℤ)/pH^{even}(E, ℤ) ⟶ H^{∗}(E) ⟶ 0.

On the other hand, since pH^{odd}(E, ℤ) = 0, there is a short exact sequence of A_{p}(E, E)- modules,

(3.2) 0 ⟶ H^{even}(E, ℤ)/pH^{even}(E, ℤ) ⟶ H^{even}(E, 𝔽_{p}) ⟶ H^{odd}(E, ℤ)[−1] ⟶ 0.

Let (y_{1}v, y_{2}v)H^{even}(E, ℤ) (resp. (y_{1}v, y_{2}v)H^{∗}(E)) be the ideal of H^{even}(E, ℤ) (resp. H^{∗}(E)) generated by y_{1}v and y_{2}v. Since py_{i} = 0 and y_{i}N = 0, it follows that

(y_{1}v, y_{2}v)H^{even}(E, ℤ) ≅ (y_{1}v, y_{2}v)H^{∗}(E).

Here we use Milnor’s primitive operator Q_{1} = P^{1}β − βP^{1} on H^{∗}(−, 𝔽_{p}). This operator induces a map Q_{1} on H^{∗}(−, ℤ) such that the following diagram commutes:

H^{odd}(E, ℤ) ⟶^{Q_{1}} H^{even}(E, ℤ)
↓  ↓
H^{odd}(E, 𝔽_{p}) ⟶^{Q_{1}} H^{even}(E, 𝔽_{p}).

Moreover, Q_{1} induces an isomorphism of A_{p}(E, E)-modules,

Q_{1} : H^{odd}(E, ℤ) ⟶^{∼} (y_{1}v, y_{2}v)H^{even}(E, ℤ) ≅ (y_{1}v, y_{2}v)H^{∗}(E)

Q_{1}(a_{i}) = y_{i}v,

(see [14, section 1]). Then we have

(3.3)  H^{odd}(E, ℤ) ≅ (y_{1}v, y_{2}v)H^{∗}(E)[−2p + 1]

and the proof of the first part of Theorem 1.1 is completed by the exact sequences (3.1) and (3.2).

Next we consider the odd degree part H^{odd}(E, 𝔽_{p}). Let K = {x ∈ H^{even}(E, ℤ) | px = 0}. Then there exists a short exact sequence of A_{p}(E, E)-modules,

(3.4)  0 ⟶ H^{odd}(E, ℤ) ⟶ H^{odd}(E, 𝔽_{p}) ⟶ K[−1] ⟶ 0.

Let H = H^{even}(E, ℤ) ∩ H^{+}(E, ℤ). Since p^{2}H = 0, pH ⊂ K. Moreover, K, pH and H/K are A_{p}(E, E)-modules.

Since the map p : H ⟶ H is a homomorphism of A_{ℤ}(E, E)-modules,

H/K ≅ pH

as A_{ℤ}(E, E)-modules. Since these are modules for A_{p}(E, E) = A_{ℤ}(E, E)/pA_{ℤ}(E, E), these are isomorphic as A_{p}(E, E)-modules. Hence we have

K ↔ pH ⊕ K/pH ↔ H/K ⊕ K/pH ↔ H/pH.

Moreover, since

H/pH = 𝔽_{p} ⊗ H ↔ N ⊕ H^{+}(E),

the proof of the second part of Theorem 1.1 is completed by (3.3) and (3.4).

Next we shall see the structure of N. Note that res_{A}^{E}(b′_{i}) = 0 for any i and any maximal elementary abelian p-subgroup A of E. Since the action of g ∈ GL_{2}(𝔽_{p}) is given by

g^{∗}(b_{i}) = det(g)^{i}b_{i}, g^{∗}(v) = det(g)v

(see [8, Theorem 3]), N is a direct sum of simple A_{p}(E, E)-modules isomorphic to S(E, E, det^{i}) for 0 ≤ i ≤ p − 2. Hence we have the following:

8 AKIHIKO HIDA AND NOBUAKI YAGITA

Proposition 3.1. Let

N_{q} = 𝔽_{p}{v^{k}b′_{i} | k ≥ 0, k + i ≡ q mod (p − 1)}

for 0 ≤ q ≤ p − 2. Then

N = ⨁_{0≤q≤p−2} N_{q}

and

N_{q} ≅ ⨁ S(E, E, det^{q})

as A_{p}(E, E)-modules.

Next, we shall consider the structure of the ideal (y_{1}v, y_{2}v)H^{∗}(E). Let I = (y_{1}v, y_{2}v)H^{∗}(E). Let Γ_{S} be the 𝔽_{p}-subspace defined in Definition 2.2 for each simple A_{p}(E, E)-module S. We shall show that

Ie_{S} ≅ I ∩ Γ_{S}

for an idempotent e_{S} which corresponds to S and determine the 𝔽_{p}-subspace I ∩ Γ_{S} explicitly. Lemma 3.2. Let

L = 𝔽_{p}[y_{1}, y_{2}, C] ⊕ 𝔽_{p}[V]{v, …, v^{p−2}, Cv, …, Cv^{p−2}} ⊕ 𝔽_{p}[D_{1}]{D_{1}} ⊕ 𝔽_{p}[D_{1}]{D_{2}} where 𝔽_{p}[y_{1}, y_{2}, C] is the subalgebra of H^{∗}(E) generated by y_{1}, y_{2} and C. Then

H^{∗}(E) = I ⊕ L.

Proof. Let (y_{1}, y_{2}, C) be the ideal of H^{∗}(E) generated by y_{1}, y_{2} and C. Since

D_{1} = C^{p} + V ≡ V mod (y_{1}, y_{2}, C),

we have

H^{∗}(E) = 𝔽_{p}[v] ⊕ (y_{1}, y_{2}, C)
= 𝔽_{p} ⊕ 𝔽_{p}[V]{v, …, v^{p−2}} ⊕ 𝔽[V]{V } ⊕ (y_{1}, y_{2}, C)
= 𝔽_{p} ⊕ 𝔽_{p}[V]{v, …, v^{p−2}} ⊕ 𝔽[D_{1}]{D_{1}} ⊕ (y_{1}, y_{2}, C).

On the other hand, since

D_{2}D_{1} = CV (C^{p} + V) ≡ D_{2}V mod I,

it follows that

𝔽_{p} ⊕ (y_{1}, y_{2}, C)
= 𝔽_{p}[y_{1}, y_{2}, C] ⊕ 𝔽_{p}[v]{Cv} ⊕ I
= 𝔽_{p}[y_{1}, y_{2}, C] ⊕ 𝔽_{p}[V]{Cv, …, Cv^{p−2}} ⊕ 𝔽_{p}[V]{CV } ⊕ I
= 𝔽_{p}[y_{1}, y_{2}, C] ⊕ 𝔽_{p}[V]{Cv, …, Cv^{p−2}} ⊕ 𝔽_{p}[D_{1}]{D_{2}} ⊕ I

and we have H^{∗}(E) = I ⊕ L. □ Lemma 3.3. For each simple A_{p}(E, E)-module S, we have

Γ_{S} = (I ∩ Γ_{S}) ⊕ (L ∩ Γ_{S}) where I = (y_{1}v, y_{2}v)H^{∗}(E) and L is an 𝔽_{p}-subspace defined in Lemma 3.2. Moreover,

(1) If S = S(E, C_{p}, U_{i}), then I ∩ Γ_{S} = 0.

(2) If S = S(E, A, S^{p−1} ⊗ det^{q}), then

I ∩ Γ_{S} = Γ_{S} = \{ \begin{matrix}𝔻𝔸\{⨁_{0≤j≤p−1} D_{2}C^{j}(𝔽_{p}C + S^{p−1})\} & (q = 0) \\ 𝔻𝔸\{⨁_{0≤j≤p−1} v^{q}C^{j}(CS^{q} + T^{q})\} & (1 ≤ q ≤ p − 2). \end{matrix}

(3)

I ∩ Γ_{S} = \begin{cases}𝔻𝔸\{D_{2}^{2}\} & (S = S(E, E, S^{0})) \\ ℂ𝔸\{C^{2}v^{q}\} & (S = S(E, E, det^{q}), 1 ≤ q ≤ p − 2) \\ 𝔻𝔸\{V S^{p−1}\} & (S = S(E, E, S^{p−1})) \\ ℂ𝔸\{v^{q}S^{p−1}\} & (S = S(E, E, S^{p−1} ⊗ det^{q}), 1 ≤ q ≤ p − 2) \end{cases}

(4) Let

S = S^{i}v^{q}, T = T^{p−i−1}v^{s}

for 1 ≤ i ≤ p − 2, 0 ≤ q ≤ p − 2, where s ≡ i + q (mod p − 1), 0 ≤ s ≤ p − 2. Then I ∩ Γ_{S(E,E,S^{i}⊗det^{q})} is the following 𝔽_{p}-subspace:

\begin{matrix}ℂ𝔸\{V S\} & ⊕ & 𝔻𝔸\{V T \} & (q ≡ 2i ≡ 0) \\ ℂ𝔸\{V S\} & ⊕ & ℂ𝔸\{T \} & (q ≡ 0, 2i ≢ 0) \\ 𝔻𝔸\{S\} & ⊕ & 𝔻𝔸\{V T \} & (i = q, 3i ≡ 0) \\ 𝔻𝔸\{S\} & ⊕ & ℂ𝔸\{T \} & (i = q, 3i ≢ 0, 2i ≢ 0) \\ 𝔻𝔸\{S\} & ⊕ & ℂ𝔸\{V T \} & (i = q, 3i ≢ 0, 2i ≡ 0) \\ ℂ𝔸\{S\} & ⊕ & 𝔻𝔸\{V T \} & (q ≠ 0, i ≠ q, q + 2i ≡ 0) \\ ℂ𝔸\{S\} & ⊕ & ℂ𝔸\{T \} & (q ≠ 0, i ≠ q, q + 2i ≢ 0, i + q ≢ 0) \\ ℂ𝔸\{S\} & ⊕ & ℂ𝔸\{V T \} & (q ≠ 0, i ≠ q, q + 2i ≢ 0, i + q ≡ 0). \end{matrix}

(5) I ∩ Γ_{S(1,1,𝔽_{p})} = 0.

Proof. (1) If S = S(E, C_{p}, U_{i}) (0 ≤ i ≤ p − 2), then I ∩ Γ_{S} = 0 and Γ_{S} ⊂ L by Definition 2.2.

(2) If S = S(E, A, S^{p−1} ⊗ det^{q}) (0 ≤ q ≤ p − 2), then Γ_{S} ⊂ I since D_{2}C = C^{2}V ∈ I.

(3) If S = S(E, E, S^{0}), then Γ_{S} = 𝔻𝔸^{+},

𝔻𝔸^{+} = 𝔻𝔸{D_{2}^{2}} ⊕ 𝔽_{p}[D_{1}]{D_{1}, D_{2}}.

Since 𝔻𝔸{D_{2}^{2}} ⊂ I and 𝔽_{p}[D_{1}]{D_{1}, D_{2}} ⊂ L, we have

𝔻𝔸^{+} = (I ∩ Γ_{S}) ⊕ (L ∩ Γ_{S})

and I ∩ Γ_{S} = 𝔻𝔸{D_{2}^{2}}.

If S = S(E, E, det^{q}) (1 ≤ q ≤ p − 2), then Γ_{S} = ℂ𝔸{v^{q}},

ℂ𝔸{v^{q}} = ℂ𝔸{C^{2}v^{q}} ⊕ 𝔽_{p}[V]{v^{q}, Cv^{q}}.

Since C^{2}v^{q} ∈ I and 𝔽_{p}[V]{v^{q}, Cv^{q}} ⊂ L, it follows that

ℂ𝔸{v^{q}} = (I ∩ Γ_{S}) ⊕ (L ∩ Γ_{S})

and I ∩ Γ_{S} = ℂ𝔸{C^{2}v^{q}}.

If S = S(E, E, S^{p−1}), then Γ_{S} = 𝔻𝔸{V S^{p−1}}. If S = S(E, E, S^{p−1} ⊗ det^{q}) (1 ≤ q ≤ p − 2), Γ_{S} = ℂ𝔸{v^{q}S^{p−1}}. In these cases, Γ_{S} ⊂ I.

(4) Let S = S^{i}v^{q} (1 ≤ i ≤ p − 2, 0 ≤ q ≤ p − 2). If q = 0 then SV ⊂ I. If q ≠ 0 then S ⊂ I. Hence the first term of Γ_{S} is contained in I.

Let T = T^{p−i−1}v^{s}, s ≡ i + q mod (p − 1), 0 ≤ s ≤ p − 2. If s ≡ 0, then V T ⊂ I. If i + q ≢ 0, then T ⊂ I. Hence the second term of Γ_{S} is contained in I unless i = q, 3i ≢ 0, 2i ≡ 0, or q ≠ 0, i ≠ q, q + 2i ≢ 0, i + q ≡ 0. In these cases,

ℂ𝔸{T } = 𝔽_{p}[C]{T } ⊕ ℂ𝔸{V T }

where ℂ𝔸{V T } ⊂ I, 𝔽_{p}[C]{T } ⊂ L. Hence

ℂ𝔸{T } = ℂ𝔸{T } ∩ I ⊕ 𝔽_{p}[C]{T } ∩ I

and ℂ𝔸{T } ∩ I = ℂ𝔸{V T }. □

Lemma 3.4. Let I = (y_{1}v, y_{2}v)H^{∗}(E). Then

I = ⨁_{S}(I ∩ Γ_{S}).

Proof. First, H^{∗}(E) = ⊕_{S}Γ_{S} = I ⊕ L by Proposition 2.5 and Lemma 3.2. On the other hand,

Γ_{S} = (I ∩ Γ_{S}) ⊕ (L ∩ Γ_{S})

by Lemma 3.3. Hence

H^{∗}(E) = ⊕Γ_{S} = ⊕((I ∩ Γ_{S}) ⊕ (L ∩ Γ_{S}))
= (⊕(I ∩ Γ_{S})) ⊕ (⊕(L ∩ Γ_{S})) ⊂ I ⊕ L = H^{∗}(E).

Hence we have

I = ⨁_{S}(I ∩ Γ_{S}).

□

Now, we determine the 𝔽_{p}-vector space Ie_{S} for any simple A_{p}(E, E)-module S.

Theorem 3.5. Let S be a simple A_{p}(E, E)-module. Then there exists an idempotent e_{S} corresponding to S such that

Ie_{S} = (I ∩ Γ_{S})e_{S} ≅ I ∩ Γ_{S}.

Proof. By Theorem 2.3, Γ_{S} ≅ Γ_{S}e_{S} for some idempotent corresponding to S. Hence e_{S} induces an isomorphism

I ∩ Γ_{S} ≅ (I ∩ Γ_{S})e_{S}.

For a graded 𝔽_{p}-subspace M ⊂ H^{∗}(E), let M^{n} = H^{n}(E) ∩ M. Then

dim I^{n} = ∑_{S} dim(I^{n})e_{S} ≥ ∑_{S} dim(I^{n} ∩ Γ_{S})e_{S}
= ∑_{S} dim I^{n} ∩ Γ_{S} = dim I^{n}.

The last equality follows from Lemma 3.4. Hence we have

Ie_{S} = (I ∩ Γ_{S})e_{S}.
□

Remark 3.6. Let X be the indecomposable summand in the complete stable splitting of BE which corresponds to a simple A_{p}(E, E)-modules S. Let e_{S} be an idempotent which correspond to S as above. Then we can get the 𝔽_{p}-vector space

H^{∗}(E, 𝔽_{p})e_{S} ≅ H^{∗}(∨X, 𝔽_{p}) ≅ ⨁^{d} H^{∗}(X, 𝔽_{p})

where ∨X is a wedge sum of d = dim S copies of X, from Theorem 1.1, Proposition 3.1 and Theorem 3.5.

Corollary 3.7. Every simple A_{p}(E, E)-module appears as a composition factor in H^{2n}(E, 𝔽_{p}) for some n ≤ p^{2} − 2.

Proof. Let H(p^{2} − 2) = ⊕_{n=0}^{p^{2}−2}H^{2n}(E, 𝔽_{p}). For a simple A_{p}(E, E)-module S, let 2γ(S) be the lowest degree such that (Γ_{S} ∩ I)^{2γ(S)} ≠ 0. If 2γ(S) ≤ 2(p + 2)(p − 1), then 2(γ(S)−p) ≤ 2(p^{2}−2). Since S appears in the degree 2(γ(S)−p) part of I[−2p], it follows that S appears in H^{2(γ(S)−p)}(E, 𝔽_{p}) by Theorem 1.1. Hence S appears in H(p^{2} − 2). In particular, if Γ_{S} ≤ I, then S appears in H(p^{2} − 2) by Corollary 2.4. This implies that simple modules

S(E, A, S(A)^{p−1} ⊗ det^{q})(0 ≤ q ≤ p − 2), S(E, E, S^{p−1} ⊗ det^{q})(0 ≤ q ≤ p − 2)

appear in H(p^{2} − 2).

On the other hand, since the degrees of C^{2}v^{q}, (1 ≤ q ≤ p − 2) and V S^{i}, S^{i}v^{q}, (1 ≤ i ≤ p − 2, 0 ≤ q ≤ p − 2) are all smaller than deg D_{2}S^{p−1} = 2(p + 2)(p − 1), we have that S(E, E, det^{q}), (1 ≤ q ≤ p − 2) and S(E, E, S^{i} ⊗ det^{q}), (1 ≤ i ≤ p − 2, 0 ≤ q ≤ p − 2) appear in H(p^{2} − 2).

Moreover, since S(E, C_{p}, U_{i}) (0 ≤ i ≤ p − 2) appears in H^{2}(E) ⊕ ⋯ ⊕ H^{2(p−1)}(E), it appears in H(p^{2} − 2). Finally, we consider S(E, E, S^{0}). Since it appears in H^{2p(p−1)}(E) and 2p(p − 1)(= deg D_{1}) ≤ 2(p^{2} − 2), S(E, E, S^{0}) appears in H(p^{2} − 2). This completes the proof. □

4. Stable splitting of groups related to L_{3}(p)

In this section, we consider the stable splitting of BG for groups G having E as a Sylow p-subgroup, in particular the linear group L_{3}(p) and its extensions.

Benson and Fechbach [2], Martino and Priddy [11] prove the following theorem on complete stable splitting. Let P be a finite p-group. If G is a finite group which contains E, then G is considered as an (E, E)-biset. We denote by [G] the element of A_{p}(E, E) corresponding to G. Let S(P, Q, V) be the simple A_{p}(P, P)-module which corresponds to (Q, V) where Q is a subgroup of P and V is a simple 𝔽_{p} Out(P)-module.

Theorem 4.1 ([2], [11]). Let G be a finite group with Sylow p-subgroup P. The complete stable splitting of BG is given by

BG ∼ ⋁_{(Q,V)} dim(S(G, Q, V))X_{S(P,Q,V)}

where S(G, Q, V) = S(P, Q, V)[G].

Let

H^{∗}(G) = (𝔽_{p} ⊗ H^{∗}(G, ℤ))/\sqrt{(0)}

for a finite group G. From Corollary 2.4 and Corollary 3.7, we have the following.

Corollary 4.2. Let G_{1},G_{2} have the same p-Sylow subgroup E. Suppose that G_{1} ≤ G_{2}.

(1) If

dim H^{2n}(G_{1}) = dim H^{2n}(G_{2})

for all 0 ≤ n ≤ (p + 2)(p − 1), then BG_{1} ∼ BG_{2}.

(2) If

dim H^{2n}(G_{1}, 𝔽_{p}) = dim H^{2n}(G_{2}, 𝔽_{p})

for all 0 ≤ n ≤ p^{2} − 2, then BG_{1} ∼ BG_{2}.

In general, the computation of these dim S(G, Q, V) is not so easy. Hence we study the way to compute it from the information on the cohomology H^{∗}(G). (In fact, in [15], most direct summands in the stable splitting of BG are computed from H^{∗}(G).)

Let F_{G} be the fusion system on E determined by G. Let F_{G}^{ec}-rad be the set of F_{G}^{ec}- radical maximal elementary abelian p-subgroups of E. If A is a maximal elementary abelian p-subgroup of E, then A ∈ F_{G}^{ec}-rad if and only if W_{G}(A) = N_{G}(A)/C_{G}(A) = Out_{F_{G}}(A) ≥ SL_{2}(𝔽_{p}) by [12, Lemma 4.1]. Let W_{G}(E) = N_{G}(E)/EC_{G}(E) = Out_{F_{G}}(E).

Theorem 4.3 ([13, Theorem 4.3],[15, Theorem 3.1]). Let G have the Sylow p-subgroup E, then

H^{∗}(G) ≅ H^{∗}(E)[G] = H^{∗}(E)^{W_{G}(E)} ∩ (∩_{A∈F_{G}^{ec}-rad}(res_{A}^{E})^{−1}(H^{∗}(A)^{W_{G}(A)})).

Moreover, if M is an A_{p}(E, E)-submodule of H^{∗}(E), then

M[G] = M^{W_{G}(E)} ∩ (∩_{A∈F_{G}^{ec}-rad}(res_{A}^{E})^{−1}(H^{∗}(A)^{W_{G}(A)})).

Proof. The first part follows from Alperin’s fusion theorem ([3, Theorem A.10]). Let M be an A_{p}(E, E)-submodule of H^{∗}(E). Since [G][G] ∈ 𝔽_{p}[G], it follows that M[G] = M ∩ H^{∗}(E)[G]. Hence the result follows from the first part. □

Let X_{i,q} be the indecomposable summand in the stable splitting of BE which corresponds to the simple A_{p}(E, E)-module S(E, E, S^{i} ⊗ det^{q}). For 0 ≤ q ≤ p − 2, let L(2, q) (resp. L(1, q)) be the summand which corresponds to the simple A_{p}(E, E)-module S(E, A, S(A)^{p−1} ⊗ det^{q}) (resp. S(E, C_{p}, U_{q})). We set M(2) = L(1, 0) ∨ L(2, 0).

Suppose that the stable splitting of BG is written as

BG ∼ (∨_{i,q}n(G)_{i,q}X_{i,q}) ∨ (∨_{q}m(G, 2)_{q}L(2, q)) ∨ (∨_{q}m(G, 1)_{q}L(1, q)).

Recall that

H^{2q}(E) ≅ \begin{cases}S(E, C_{p}, U_{i}) & (1 ≤ q ≤ p − 2) \\ S(E, C_{p}, U_{0}) & (q = p − 1) \end{cases}

by Theorem 2.3. Hence,

Lemma 4.4 ([15, Corollary 4.6]). The multiplicity m(G, 1)_{q} for L(1, q) is given by

m(G, 1)_{q} = \begin{cases}dim H^{2q}(G) & (1 ≤ q ≤ p − 2) \\ dim H^{2(p−1)}(G) & (q = 0). \end{cases}

The multiplicity n(G)_{i,q} of X_{i,q} depends only on W_{G}(E) = N_{G}(E)/EC_{G}(E). For H ≤ GL_{2}(𝔽_{p}) and GL_{2}(𝔽_{p})-submodule M of H^{∗}(E), let

M^{H} = {m ∈ M | mh = m for any h ∈ H}

denotes the subspace consists of H-invariant elements. Then we have the following lemma.

Lemma 4.5 ([15, Lemma 4.7]). The multiplicity n(G)_{i,q} of X_{i,q} in BG is given by

n(G)_{i,q} = dim(S^{i}v^{q})^{W_{G}(E)}.

Next problem is to seek the multiplicity m(G, 2)_{q} for the summand L(2, q) in BG. We can prove,

Lemma 4.6 ([15, Proposition 4.9]). The multiplicity of L(2, 0) in BG is given by

m(G, 2)_{0} = ♯_{G}(A) − ♯_{G}(F^{ec}A)

where ♯_{G}(A) (resp.♯_{G}(F^{ec}A)) is the number of G-conjugacy classes of rank two elementary abelian p-subgroups in E (resp. subgroups in F_{G}^{ec}-rad).

Lemma 4.7 ([15, Corollary 4.10]). The multiplicity of L(1, 0) in BG is given by

m(G, 1)_{0} = dim H^{2(p−1)}(G) = ♯_{G}(A) − ♯_{G}(F^{ec}A).

Remark 4.8. By Lemma 4.6 and 4.7, m(G, 1)_{0} = m(G, 2)_{0}, namely, L(1, 0) and L(2, 0) always appear in BG as M(2) = L(1, 0)∨L(2, 0). On the other hand, in Corollary 2.4, all simple modules except for S(E, A, S^{p−1}) appear in H^{2n}(E) for n ≤ p^{2} − 1. Note that the minimal n such that S(E, E, S^{p−1}) appears in H^{2n}(E) is p^{2} − 1 = \frac{1}{2} deg(V S^{p−1}). Hence we may replace the bound (p + 2)(p − 1) by p^{2} − 1 in Corollary 4.2 (1).

For the number m(G, 2)_{q} for q ≠ 0, it seems that there is not a good way to find it. However we give some condition such that m(G, 2)_{q} = 0.

Lemma 4.9 ([15, Lemma 4.11]). Let ξ ∈ 𝔽_{p}^{∗} be a primitive (p − 1)-th root of 1. Suppose that G ⊃ E: ⟨diag(ξ, ξ)⟩. If ξ^{3k} ≠ 1, then BG does not contain the summand L(2, k), i.e., m(G, 2)_{k} = 0.

Let ξ be the multiplicative generator of 𝔽_{p}^{∗} as above. Let

w = (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) ∈ GL_{2}(𝔽_{p}).

Let

T = ⟨diag(ξ, ξ), diag(ξ, 1)⟩

be a torus in GL_{2}(𝔽_{p}). Then w normalizes T. Let T ⟨w⟩ be the semidirect product of T by ⟨w⟩

Lemma 4.10. Assume that 1 ≤ l ≤ p − 1, 0 ≤ k ≤ p − 2. Then

(S^{l}v^{k})^{T} = \begin{cases}𝔽_{p}y_{1}^{i}y_{2}^{i}v^{p−1−i} & (l = 2i, k = p − 1 − i, 1 ≤ i ≤ \frac{p−1}{2}) \\ 𝔽_{p}y_{1}^{p−1} ⊕ 𝔽_{p}y_{2}^{p−1} & (l = p − 1, k = 0) \\ 0 & (otherwise) \end{cases}

and

(S^{l}v^{k})^{T ⟨w⟩} = \begin{cases}𝔽_{p}y_{1}^{2j}y_{2}^{2j}v^{p−1−2j} & (l = 4j, k = p − 1 − 2j, 1 ≤ j ≤ \frac{p−1}{4}) \\ 𝔽_{p}(y_{1}^{p−1} + y_{2}^{p−1}) & (l = p − 1, k = 0) \\ 0 & (otherwise). \end{cases}

Proof. We consider the action of diag(ξ, 1) and diag(1, ξ). For 0 ≤ i ≤ l, we have

diag(ξ, 1)y_{1}^{i}y_{2}^{l−i}v^{k} = ξ^{i+k}y_{1}^{i}y_{2}^{l−i}v^{k}
diag(1, ξ)y_{1}^{i}y_{2}^{l−i}v^{k} = ξ^{l−i+k}y_{1}^{i}y_{2}^{l−i}v^{k}.

Hence y_{1}^{i}y_{2}^{l−i}v^{k} is T -invariant if and only if i+k ≡ l −i+k ≡ 0 mod p−1, namely, l ≡ 2i mod p − 1 and i + k ≡ 0 mod p − 1. If k = 0, then i = 0, p − 1, l = p − 1. If k > 0, then i = p − 1 − k, l = 2i. Since 1 ≤ l ≤ p − 1, it follows that i ≤ \frac{p−1}{2}.

Next consider the action of w. Since w interchanges y_{1} and y_{2}, wv = −v, y_{1}^{i}y_{2}^{i}v^{p−1−i} is w-invariant if and only if i is even. □

Now assume that p − 1 = 3m. Note that m is even. We set

H = ⟨diag(ξ, ξ), diag(ξ^{3}, 1)⟩.

Then w normalizes H. Let H⟨w⟩ be the semidirect product of H by ⟨w⟩.

Lemma 4.11. Assume that p − 1 = 3m. Let m = 2n. Assume that 1 ≤ l ≤ p − 1, 0 ≤ k ≤ p − 2. Then (S^{l}v^{k})^{H} is equal to the following vector space.

𝔽_{p}{y_{1}^{p−1}, y_{1}^{2m}y_{2}^{m}, y_{1}^{m}y_{2}^{2m}, y_{2}^{p−1}}  (l = p − 1, k = 0)
𝔽_{p}{y_{1}^{i−n}y_{2}^{i+n}v^{3n−i}, y_{1}^{i+n}y_{2}^{i−n}v^{3n−i}}  (l = 2i, k = 3n − i, n ≤ i < 3n)
𝔽_{p}{y_{1}^{i}y_{2}^{i}v^{3m−i}}  (l = 2i, k = (p − 1) − i, 1 ≤ i < m)
𝔽_{p}{y_{1}^{i−m}y_{2}^{i+m}v^{3m−i}, y_{1}^{i}y_{2}^{i}v^{3m−i}, y_{1}^{i+m}y_{2}^{i−m}v^{3m−i}} (l = 2i, k = (p − 1) − i, m ≤ i ≤ 3n)
0  (otherwise).

Proof. We consider the action of diag(ξ, ξ) and diag(ξ^{3}, 1) on y_{1}^{j}y_{2}^{l−j}v_{k} for 0 ≤ j ≤ l. Since

diag(ξ, ξ)y_{1}^{j}y_{2}^{l−j}v^{k} = ξ^{l+2k}y_{1}^{j}y_{2}^{l−j}v^{k}

and

diag(ξ^{3}, 1)y_{1}^{j}y_{2}^{l−j}v^{k} = ξ^{3j+3k}y_{1}^{j}y_{2}^{l−j}v^{k},

y_{1}^{j}y_{2}^{l−j}v^{k} is H-invariant if and only if

\begin{cases}l + 2k ≡ 0 & mod p − 1 \\ j + k ≡ 0 & mod m. \end{cases}

Note that the condition l + 2k ≡ 0 mod p − 1 implies that l is even, so we set l = 2i, 1 ≤ i ≤ 3n. Then y_{1}^{j}y_{2}^{l−j}v^{k} ∈ (S^{l}v^{k})^{H} if and only if

\begin{cases}i + k ≡ 0 & mod 3n \\ j + k ≡ 0 & mod m. \end{cases}

Since 0 ≤ k ≤ p − 2, k ≡ −i mod 3n if and only if

k = 3n − i or (p − 1) − i.

First, assume k = 3n − i. Then j ≡ −k = i − 3n mod m if and only if j = i + sn for some integer s ∈ ℤ such that s ≡ 1 mod 2. Then

0 ≤ j ≤ l = 2i ⟺ 0 ≤ i + sn ≤ 2i
⟺ −i ≤ sn ≤ i
⟺ |s| ≤ \frac{i}{n}.

If 1 ≤ i < n, namely, \frac{i}{n} < 1, then there is no s ∈ ℤ such that |s| ≤ \frac{i}{n} with s ≡ 1 mod 2.

Assume that n ≤ i ≤ 3n. If i = 3n, namely, l = p − 1 and k = 0, then, since \frac{i}{n} = 3, |s| ≤ \frac{i}{n} if and only if s = −3, −1, 1, 3. Then

j = i + sn = (3 + s)n = 0, m, 2m, 3m.

If n ≤ i < 3n, namely, 1 ≤ \frac{i}{n} < 3, then |s| ≤ \frac{i}{n} if and only if s = −1, 1 since s ≡ 1 mod 2. So we have

j = i + sn = i ± n.

Next we consider the case k = (p − 1) − i. Then,

j ≡ −k = i − (p − 1) mod m

if and only if j = i + sm for some s ∈ ℤ. Then,

0 ≤ j = i + sm ≤ l = 2i ⟺ −i ≤ sm ≤ i
⟺ |s| ≤ \frac{i}{m}.

If 1 ≤ i < m, then this implies s = 0 and j = i. If m ≤ i ≤ 3n, namely, 1 ≤ \frac{i}{m} ≤ \frac{3}{2} < 2, then this implies s = 0, ±1. Hence we have j = i, i ± m. This completes the proof. □

Lemma 4.12. Assume that p − 1 = 3m. Let m = 2n. Assume that 1 ≤ l ≤ p − 1, 0 ≤ k ≤ p − 2. Then (S^{l}v^{k})^{H⟨w⟩} is equal to the following vector space.

𝔽_{p}{y_{1}^{p−1} + y_{2}^{p−1}, y_{1}^{2m}y_{2}^{m} + y_{1}^{m}y_{2}^{2m}}  (l = p − 1, k = 0)
𝔽_{p}{(y_{1}^{i−n}y_{2}^{i+n} + (−1)^{3n−i}y_{1}^{i+n}y_{2}^{i−n})v^{3n−i}}  (l = 2i, k = 3n − i, n ≤ i < 3n)
𝔽_{p}{y_{1}^{i}y_{2}^{i}v^{3m−i}}  (l = 2i, k = (p − 1) − i, 1 ≤ i < m, i:even)
𝔽_{p}{(y_{1}^{i−m}y_{2}^{i+m} + y_{1}^{i+m}y_{2}^{i−m})v^{3m−i}, y_{1}^{i}y_{2}^{i}v^{3m−i}} (l = 2i, k = (p − 1) − i, m ≤ i ≤ 3n, i:even)
𝔽_{p}{(y_{1}^{i−m}y_{2}^{i+m} − y_{1}^{i+m}y_{2}^{i−m})v^{3m−i}}  (l = 2i, k = (p − 1) − i, m ≤ i ≤ 3n, i:odd)
0  (otherwise).

Proof. Since w interchanges y_{1} and y_{2}, vw = (det w)v = −v, this lemma follows from the previous lemma. □

Let p − 1 = 3m. We consider the multiplicity m(G, 2)_{m} and m(G, 2)_{2m} in some cases. Recall that (CS^{q} + T^{q})v^{q} ≅ S(E, A, S(A)^{p−1} ⊗ det^{q}) for 1 ≤ q ≤ p − 2 by [7, Corollary 9.3]. This module has a basis

(y_{1}^{i}y_{2}^{j})v^{q}, (i = 0, q ≤ i ≤ p − 1 + q, j = p − 1 + q − i).

Note that

y_{1}^{i}y_{2}^{j}, (i = 0, p ≤ i ≤ p − 1 + q, j = p − 1 + q − i)

is a basis of CS^{q}. On the other hand,

y_{1}^{i}y_{2}^{j}, (q ≤ i ≤ p − 1 , j = p − 1 + q − i)

is a basis of T^{q}. Moreover, if q = m or q = 2m, then all elements in (CS^{q} + T^{q})v^{q} are diag(ξ, ξ)-invariant since

diag(ξ, ξ)(y_{1}^{i}y_{2}^{j}v^{q}) = ξ^{i+j+2q}(y_{1}^{i}y_{2}^{j}v^{q}) = ξ^{p−1+3q}(y_{1}^{i}y_{2}^{j}v^{q}) = (y_{1}^{i}y_{2}^{j}v^{q}).

Lemma 4.13. Let M = (CS^{m} + T^{m})v^{m}.

(1) M^{T} has a basis

y_{1}^{2m}y_{2}^{2m}v^{m}.

(2) M^{H} has a basis

y_{1}^{4m}v^{m}, y_{1}^{3m}y_{2}^{m}v^{m}, y_{1}^{2m}y_{2}^{2m}v^{m}, y_{1}^{m}y_{2}^{3m}v^{m}, y_{2}^{4m}v^{m}.

(3) M^{T ⟨w⟩} has a basis

y_{1}^{2m}y_{2}^{2m}v^{m}.

(4) M^{H⟨w⟩} has a basis

(y_{1}^{4m} + y_{2}^{4m})v^{m} = C(y_{1}^{m} + y_{2}^{m})v^{m}, y_{1}^{m}y_{2}^{m}(y_{1}^{2m} + y_{2}^{2m})v^{m}, y_{1}^{2m}y_{2}^{2m}v^{m}.

Proof. Since

diag(ξ, 1)(y_{1}^{i}y_{2}^{j}v^{m}) = ξ^{i+m}(y_{1}^{i}y_{2}^{j}v^{m})

and

diag(ξ^{3}, 1)(y_{1}^{i}y_{2}^{j}v^{m}) = ξ^{3(i+m)}(y_{1}^{i}y_{2}^{j}v^{m}) = ξ^{3i}(y_{1}^{i}y_{2}^{j}v^{m}),

y_{1}^{i}y_{2}^{j}v^{m} is T -invariant if and only if i+m ≡ 0 mod p−1. Moreover, y_{1}^{i}y_{2}^{j}v^{m} is H-invariant i ≡ 0 mod m.

(1) Since i = 0 or m ≤ i ≤ p − 1 + m = 4m, i + m ≡ 0 mod p − 1 if and only if i = 2m.

(2) Since i = 0 or m ≤ i ≤ p − 1 + m = 4m, i ≡ 0 mod m if and only if i = 0, m, 2m, 3m, 4m.

(3) (4) Since m is even, w acts on v^{m} trivially. On the other hand w interchanges y_{1} and y_{2}. Hence the results follows from (1) and (2). □

Similarly, we have the following.

Lemma 4.14. Let M = (CS^{2m} + T^{2m})v^{2m}.

(1) M^{T} has a basis

Cy_{1}^{m}y_{2}^{m}v^{2m} = y_{1}^{4m}y_{2}^{m}v^{2m}.

(2) M^{H} has a basis

y_{1}^{5m}v^{2m}, Cy_{1}^{m}y_{2}^{m}v^{2m} = y_{1}^{4m}y_{2}^{m}v^{2m}, y_{1}^{3m}y_{2}^{2m}v^{m}, y_{1}^{2m}y_{2}^{3m}v^{m}, y_{2}^{5m}v^{m}.

(3) M^{T ⟨w⟩} has a basis

Cy_{1}^{m}y_{2}^{m}v^{2m} = y_{1}^{4m}y_{2}^{m}v^{2m}.

(4) M^{H⟨w⟩} has a basis

(y_{1}^{5m} + y_{2}^{5m})v^{2m} = C(y_{1}^{2m} + y_{2}^{2m})v^{2m}, y_{1}^{2m}y_{2}^{2m}(y_{1}^{m} + y_{2}^{m})v^{2m}, Cy_{1}^{m}y_{2}^{m}v^{2m}.

If A ∈ A(E) is a maximal elementary abelian p=subgroup of E, then

H^{∗}(A) = 𝔽_{p}[y_{A}, u_{A}], deg y_{A} = deg u_{A} = 2.

We may assume that

res_{A_{i}}^{E}(y_{1}) = y_{A_{i}}, res_{A_{i}}^{E}(y_{2}) = iy_{A_{i}} for i ∈ 𝔽_{p}

and

res_{A_{∞}}^{E}(y_{1}) = 0, res_{A_{∞}}^{E}(y_{2}) = y_{A_{∞}}.

Moreover,

res_{A}^{E}(C) = y_{A}^{p−1}, res_{A}^{E}(v) = u_{A}^{p} − y_{A}^{p−1}u

for any A ∈ A(E) (see [7, section 4]).

Lemma 4.15. Let 1 ≤ q ≤ p − 2. Then

((CS^{q} + T^{q})v^{q})[G] = ((CS^{q} + T^{q})v^{q})^{W_{G}(E)} ∩ (∩_{A∈F_{G}^{ec}-rad} ker res_{A}^{E})

Proof. Let y = y_{A} and u = u_{A} for A ∈ A(E). Then

res_{A}^{E}((CS^{q} + T^{q})v^{q}) = 𝔽_{p}y^{p−1+q} res_{A}^{E}(v^{q}) = 𝔽_{p}y^{p−1}(yu^{p} − y^{p}u)^{q}.

If g ∈ Aut(A) = GL_{2}(𝔽_{p}), then

g(yu^{p} − y^{p}u) = (det g)(yu^{p} − y^{p}u)

and y^{p−1}(yu^{p} − y^{p}u)^{q} is not SL_{2}(𝔽_{p})-invariant, hence the result follows from Theorem 4.3. □

Proposition 4.16. Suppose that F_{G}^{ec}-rad = {A_{0}, A_{∞}}. Then m(G, 2)_{m} = m(G, 2)_{2m} and we have the following values:

\begin{matrix}W_{G}(E) & H & H⟨w⟩ & T & T ⟨w⟩ \\ m(G, 2)_{m} = m(G, 2)_{2m} & 3 & 2 & 1 & 1 \end{matrix}

Proof. Since

res_{A_{0}}^{E}(y_{1}) ≠ 0, res_{A_{0}}^{E}(y_{2}) = 0

and

res_{A_{∞}}^{E}(y_{1}) = 0, res_{A_{∞}}^{E}(y_{2}) ≠ 0,

the results follows from Lemma 4.13, Lemma 4.14 and 4.15. □

Next we study the stable splitting of BG for some G related to the linear group L_{3}(p). There are 6 saturated fusion systems related to L_{3}(p) [12, p. 46, Table 1.1].

W_{G}(E) |F_{G}^{ec}-rad| Group  p
H  1 + 1 L_{3}(p)  3 | (p − 1)
H⟨w⟩  2  L_{3}(p) : 2 3 | (p − 1)
T  1 + 1 L_{3}(p).3 3 | (p − 1)
T ⟨w⟩  2  L_{3}(p).S_{3} 3 | (p − 1)
T  1 + 1 L_{3}(p)  3 ∤ (p − 1)
T ⟨w⟩  2  L_{3}(p) : 2 3 ∤ (p − 1)

We determine the stable splittings of these 6 groups. Note that the these results, with the results in [15], give a complete information on the splitting for fusion systems on E with |F^{ec}-rad| ≥ 2 by the classification in [12].

Let

X = X_{0,0} ∨ 2X_{p−1,0} ∨ (∨_{1≤i≤(p−1)/2}X_{2i,p−1−i}) ∨ M(2)

and

X′ = X_{0,0} ∨ X_{p−1,0} ∨ (∨_{1≤j≤(p−1)/4}X_{4j,p−1−2j}) ∨ M(2).

Theorem 4.17. Suppose that W_{G}(E) = T and F_{G}^{ec}-rad = {A_{0}, A_{∞}}. If 3 ∤ p−1 then BG is stably homotopic to X. If p − 1 = 3m, then BG is stably homotopic to X ∨ L(2, m) ∨ L(2, 2m).

Proof. By Lemma 4.4 and Lemma 4.10, L(1, q) (1 ≤ q ≤ p − 2) is not contained in BG. Since A_{i} (1 ≤ i ≤ p − 1) are T -conjugate, there are three conjugacy classes of maximal elementary abelian p-subgroups and two of them consist of F_{G}^{ec}-radical subgroups. From Lemma 4.6 and Lemma 4.7, just one L(2, 0) (and one L(1, 0)) is contained in BG. Moreover if 3 does not divide p − 1, then L(2, q) is not contained in BG for each 1 ≤ q ≤ p − 2 from Lemma 4.9.

Moreover, by Lemma 4.5 and Lemma 4.10,

n(G)_{l,k} = \begin{cases}1 & (l = 2i, k = p − 1 − i, 1 ≤ i ≤ \frac{p−1}{2}) \\ 2 & (l = p − 1, k = 0) \\ 0 & (otherwise). \end{cases}

On the other hand, if p−1 = 3m, then m(G, 2)_{m} = m(G, 2)_{2m} = 1 by Proposition 4.16. This completes the proof. □

Theorem 4.18. Suppose that W_{G}(E) = T ⟨w⟩ and F_{G}^{ec}-rad = {A_{0}, A_{∞}}. If 3 ∤ p − 1 then BG is stably homotopic to X′. If p − 1 = 3m, then BG is stably homotopic to X′ ∨ L(2, m) ∨ L(2, 2m).

Proof. The proof is similar to that of previous Theorem. By Lemma 4.4 and Lemma 4.10, L(1, q) (1 ≤ q ≤ p − 2) is not contained in BG. Since A_{i} (1 ≤ i ≤ p − 1) are T -conjugate, there are two conjugacy classes of maximal elementary abelian p-subgroups and one of them consists of F_{G}^{ec}-radical subgroups. From Lemma 4.6 and Lemma 4.7, just one L(2, p − 1) (and one L(1, p − 1)) is contained in BG. Moreover if 3 does not divide p − 1, then L(2, q) is not contained in BG for each 1 ≤ q ≤ p − 2 from Lemma 4.9.

Moreover, by Lemma 4.5 and Lemma 4.10,

n(G)_{l,k} = \begin{cases}1 & (l = 4j, k = p − 1 − 2j, 1 ≤ j ≤ \frac{p−1}{4}) \\ 1 & (l = p − 1, k = 0) \\ 0 & (otherwise). \end{cases}

On the other hand, if p−1 = 3m, then m(G, 2)_{m} = m(G, 2)_{2m} = 1 by Proposition 4.16. This completes the proof. □

Next assume that p − 1 = 3m. Let m = 2n.

Theorem 4.19. Suppose that W_{G}(E) = H and F_{G}^{ec}-rad = {A_{0}, A_{∞}}. Then BG is stably homotopic to

X_{0,0} ∨ 4X_{p−1,0} ∨ 2(∨_{n≤i<3n}X_{2i,3n−i}) ∨ (∨_{1≤i<m}X_{2i,3m−i})

∨3(∨_{m≤i≤3n}X_{2i,3m−i}) ∨ 3(M(2) ∨ L(2, m) ∨ L(2, 2m)).

Proof. By Lemma 4.4 and Lemma 4.11, m(G, 1)_{q} = 0 for 1 ≤ q ≤ p − 2. On the other hand, by Lemma 4.6 and Corollary 4.7, m(G, 1)_{0} = m(G, 2)_{0} = 5 − 2 = 3. By Lemma 4.9, m(G, 2)_{k} = 0 for 1 ≤ k ≤ p − 2, k ≠ m, 2m. By Proposition 4.16, m(G, 2)_{m} = m(G, 2)_{2m} = 3. The multiplicity n(G)_{i,q} is obtained by Lemma 4.11. By Lemma 4.11,

n(G)_{i,q} = \begin{cases}4 & (l = p − 1, q = 0) \\ 2 & (l = 2i, q = 3n − i, n ≤ i < 3n) \\ 1 & (l = 2i, q = (p − 1) − i, 1 ≤ i < m) \\ 3 & (l = 2i, q = (p − 1) − i, m ≤ i ≤ 3n) \\ 0 & (otherwise). \end{cases}

□

Theorem 4.20. Suppose that W_{G}(E) = H⟨w⟩ and F_{G}^{ec}-rad = {A_{0}, A_{∞}}. Then BG is stably homotopic to

X_{0,0} ∨ 2X_{p−1,0} ∨ (∨_{n≤i<3n}X_{2i,3n−i}) ∨ (∨_{1≤j≤3n/2}X_{4j,3m−2j}) ∨ (∨_{m≤i≤3n}X_{2i,(p−1)−i})
∨2(M(2) ∨ L(2, m) ∨ L(2, 2m)).

Proof. By Lemma 4.4 and Lemma 4.12, m(G, 1)_{q} = 0 for 1 ≤ q ≤ p − 2. By Lemma 4.6 and Corollary 4.7, m(G, 1)_{0} = m(G, 2)_{0} = 3 − 1 = 2. By Lemma 4.9, m(G, 2)_{k} = 0 for 1 ≤ k ≤ p − 2, k ≠ m, 2m. By Proposition 4.16, m(G, 2)_{m} = m(G, 2)_{2m} = 2. The multiplicity n(G)_{i,q} is obtained by Lemma 4.12. By Lemma 4.12,

n(G)_{i,q} = \begin{cases}2 & (l = p − 1, q = 0) \\ 1 & (l = 2i, q = 3n − i, n ≤ i < 3n) \\ 1 & (l = 2i, q = (p − 1) − i, 1 ≤ i < m, i: even) \\ 2 & (l = 2i, q = (p − 1) − i, m ≤ i ≤ 3n, i: even) \\ 1 & (l = 2i, q = (p − 1) − i, m ≤ i ≤ 3n, i: odd) \\ 0 & (otherwise). \end{cases}

Moreover, consider the 3rd, 4th and 5th cases. We have

(∨_{1≤i<m,i:even}X_{2i,(p−1)−i}) ∨ 2(∨_{m≤i≤3n,i:even}X_{2i,(p−1)−i}) ∨ (∨_{m≤i≤3n,i;odd}X_{2i,(p−1)−i})

= (∨_{1≤i≤3n,i:even}X_{2i,(p−1)−i}) ∨ (∨_{m≤i≤3n}(X_{2i,(p−1)−i})
= (∨_{1≤j≤3n/2}X_{4j,(p−1)−2j}) ∨ (∨_{m≤i≤3n}X_{2i,(p−1)−i}).

This completes the proof. □

Next we consider the specific case, that is, p = 7. We give a result which supplements the result on splitting for p = 7 in [15].

Example 4.21. Let p = 7, p − 1 = 6, m = 2, n = 1. Suppose that F_{G}^{ec}-rad = {A_{0}, A_{∞}}.

(1) If W_{G}(E) = T, then

BG ∼ X_{0,0} ∨ X_{2,5} ∨ X_{4,4} ∨ 2X_{6,0} ∨ X_{6,3} ∨ M(2) ∨ L(2, 2) ∨ L(2, 4).

(2) If W_{G}(E) = T ⟨w⟩, then

BG ∼ X_{0,0} ∨ X_{4,4} ∨ X_{6,0} ∨ M(2) ∨ L(2, 2) ∨ L(2, 4).

(3) If W_{G}(E) = H, then

BG ∼ X_{0,0} ∨ 2X_{2,2} ∨ X_{2,5} ∨ 2X_{4,1} ∨ 3X_{4,4} ∨ 4X_{6,0} ∨ 3X_{6,3}
∨3(M(2) ∨ L(2, 2) ∨ L(2, 4)).

(4) If W_{G}(E) = H⟨w⟩, then

BG ∼ X_{0,0} ∨ X_{2,2} ∨ X_{4,1} ∨ 2X_{4,4} ∨ 2X_{6,0} ∨ X_{6,3}

∨2(M(2) ∨ L(2, 2) ∨ L(2, 4)).

Let G_{1} and G_{2} be finite groups with Sylow p-subgroup E. If F_{G_{1}} is (isomorphic to) a subfusion system of F_{G_{2}}, then BG_{1} ∼ BG_{2} ∨ X for some summand X of BG_{1}. In this case, we write

G_{2} ⟵^{X} G_{1}

We use same notation for fusion systems.

In [15], the second author considered the graphs related to the splitting of sporadic simple groups and some exotic fusion system for p = 7 and obtained the following.

Theorem 4.22 ([15, Theorem 9.4]). Let p = 7. We have the following two sequences:

X_{0,0} ∨ X_{4,4} ∼ RV_{3} ⟵^{X_{2,2}∨X_{6,0}} RV_{2} ⟵^{M(2)∨L(2,2)∨L(2,4)} O′N : 2 ⟵^{X_{4,1}∨X_{4,4}∨X_{6,0}∨X_{6,3}} O′N

X_{0,0} ∨ X_{4,4} ∨ X_{6,0} ∼ RV_{1} ⟵^{M(2)} F i_{24} ⟵^{X_{2,2}∨X_{6,0}∨X_{6,3}} F i′_{24} ⟵^{M(2)∨L(2,2)∨L(2,4)} He : 2

⟵^{X_{3,0}∨X_{3,3}∨X_{5,2}∨X_{5,5}∨L(1,3)∨L(2,3)} He.

where RV_{1}, RV_{2}, RV_{3} are the exotic fusion systems of Ruiz and Viruel [12].

Now we add more information on the splittings for p = 7.

Theorem 4.23. Let p = 7. We have the following diagram:

RV_{1} ⟵^{M(2)∨L̃} L_{3}(7).S_{3}  ⟵^{Y′}  L_{3}(7).3
Y ∨Z∨M(2)∨L̃↑ ^{Y ∨Z∨M(2)∨L̃}↑  ↑^{2(Y ∨Z∨M(2)∨L̃)}
O′N ⟵^{M(2)∨L̃} L_{3}(7) : 2 ⟵^{Y ∨Y′∨Z∨M(2)∨L̃} L_{3}(7)
Y ∨Z∨M(2)∨L̃↓ ^{Y ∨Z∨M(2)∨2L̃}↓  ↓^{Y ∨Y′∨2Z∨2M(2)∨3L̃}
RV_{1} ⟵^{M(2)} F i_{24}  ⟵^{Y}  F i′_{24}

where

Y = X_{2,2} ∨ X_{6,0} ∨ X_{6,3}, Y′ = X_{2,5} ∨ X_{6,0} ∨ X_{6,3}, Z = X_{4,1} ∨ X_{4,4}

L̃ = L(2, 2) ∨ L(2, 4).

Proof. We have the following table by [12, Lemma 4.9, Lemma 4.16].

group  Out_{F}(E)  F^{ec}-rad  Out_{F}(A)

(fusion system)

L_{3}(7)  6 × 2 = ⟨3I, u⟩  {A_{0}}{A_{∞}}  SL_{2}(7) : 2
L_{3}(7)  6 × 2 = ⟨3I, w⟩  {A_{1}}{A_{6}}  SL_{2}(7) : 2
L_{3}(7) : 2  (6 × 2) : 2 = ⟨3I, u, w⟩ {A_{0}, A_{∞}}  SL_{2}(7) : 2
L_{3}(7) : 2  (6 × 2) : 2 = ⟨3I, u, w⟩ {A_{1}, A_{6}}  SL_{2}(7) : 2
O′N  (6 × 2) : 2 = ⟨3I, u, w⟩ {A_{0}, A_{∞}}{A_{1}, A_{6}}  SL_{2}(7) : 2
L_{3}(7).3  6^{2} = T  {A_{0}}{A_{∞}}  GL_{2}(7)
L_{3}(7).S_{3}  6^{2} : 2 = T ⟨w⟩  {A_{0}, A_{∞}}  GL_{2}(7)
F i′_{24}  6 × S_{3} = ⟨3I, s, w⟩  {A_{1}, A_{2}, A_{4}}{A_{3}, A_{5}, A_{6}} SL_{2}(7) : 2
F i_{24}  6^{2} : 2 = T ⟨w⟩  {A_{1}, …, A_{6}}  SL_{2}(7) : 2
RV_{1}  6^{2} : 2 = T ⟨w⟩  {A_{0}, A_{∞}}{A_{1}, …, A_{6}} GL_{2}(7), SL_{2}(7) : 2

where

I = (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) , u = (\begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix}) , w = (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) , s = (\begin{matrix}2 & 0 \\ 0 & 4 \end{matrix})

and T = {diag(α, β) | α, β ∈ F_{7}^{×}} is the subgroup of all invertible diagonal matrices. The set F^{ec}-rad is separated by conjugacy classes and Out_{F}(A) is described for each representative A of conjugacy classes in F^{ec}-rad if they are different. Note that if we take the generators a and b of E suitably, we can obtained the two rows in the case of L_{3}(7) and L_{3}(7) : 2. For example, consider G = L_{3}(7). Let E be the group of all upper triangular matrices with diagonal entry 1. The subgroups in F_{G}^{ec}-rad are

\{[\begin{matrix}1 & α & β \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]| α, β ∈ 𝔽_{p}\} , \{[\begin{matrix}1 & 0 & α \\ 0 & 1 & β \\ 0 & 0 & 1 \end{matrix}]| α, β ∈ 𝔽_{p}\} .

If we take

a = [\begin{matrix}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] , b = [\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix}]

then F_{G}^{ec}-rad = {A_{0}, A_{∞}} and Out_{F_{G}}(E) = ⟨3I, u⟩ = H. On the other hand, if we take

a = [\begin{matrix}1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix}] , b = [\begin{matrix}1 & 1 & 0 \\ 0 & 1 & −1 \\ 0 & 0 & 1 \end{matrix}]

then F_{G}^{ec}-rad = {A_{1}, A_{6}} and Out_{F_{G}}(E) = ⟨3I, w⟩.

The inclusions of fusion systems are obtained by the table above. For F i′_{24} ⟵ L_{3}(7) and F i_{24} ⟵ L_{3}(7) : 2, we use the second rows of L_{3}(7) and L_{3}(7) : 2. The information on the summands are obtained by Example 4.21 and Theorem 4.22. □

Remark 4.24. As we can see from Theorem 4.22 or 4.23 above, B(F i_{24}) is a stable summand of B(O′N), B(O′N) ∼ B(F i_{24}) ∨ Y ∨ Z ∨ L̃, but the fusion system of O′N is not isomorphic to a subfusion system of fusion system of F i_{24}, namely,

F i_{24} ⟵^{Y ∨Z∨L̃} O′N

does not hold.

Let F_{0} = F_{O′N}, F_{1} = F_{F i_{24}}. By [12, Lemma 4.3], for each A_{i} ∈ F_{0}^{ec}-rad, there exists an element of order 6 in Out_{F_{0}}(E) ≤ GL_{2}(𝔽_{p}) which has an eigenvalue 3 with eigenvector

(\begin{matrix}1 \\ i \end{matrix}) ((\begin{matrix}0 \\ 1 \end{matrix}) if i = ∞) and determinant 5. Hence there exists an involution in Out_{F_{0}}(E)

which has an eigenvalue −1 with eigenvector (\begin{matrix}1 \\ i \end{matrix}) ((\begin{matrix}0 \\ 1 \end{matrix}) if i = ∞) and determinant −1.

We may assume that Out_{F_{1}}(E) = 6^{2} : 2 = T ⟨w⟩ and F_{1}^{ec}-rad = {A_{1}, …, A_{6}} as above. Suppose that K = Out_{F_{0}}(E)(≅ (6 × 2) : 2) ≤ Out_{F_{1}}(E). Then K contains exactly 4 involutions with determinant −1. Moreover K ⊃ ⟨diag(−1, 1), diag(1, −1)⟩ since ⟨diag(−1, 1), diag(1, −1)⟩◁(6^{2} : 2). Note that |F_{0}^{ec}-rad| = 4. Since diag(−1, 1) (resp.

diag(1, −1)) has an eigenvalue −1 with eigenvector (\begin{matrix}1 \\ 0 \end{matrix}) (resp. (\begin{matrix}0 \\ 1 \end{matrix})) and determinant

−1, it follows that A_{0}, A_{∞} ∈ F_{0}^{ec}-rad for any choice of K ≤ 6^{2} : 2 = T ⟨w⟩. Hence F_{0} is not isomorphic to a subfusion system of F_{1}.

5. Some remarks on the case p = 3

Recall that H^{3}(E, ℤ) = 𝔽_{p}{a_{1}, a_{2}}. The short exact sequence

0 ⟶ ℤ ⟶^{p} ℤ ⟶^{j} 𝔽_{p} ⟶ 0

induces the following short exact sequence

(5.1)  0 ⟶ H^{2}(E, ℤ) ⟶^{j_{∗}} H^{2}(E, 𝔽_{p}) ⟶^{β̂} H^{3}(E, ℤ) ⟶ 0.

Hence there exist elements a′_{1}, a′_{2} ∈ H^{2}(E, 𝔽_{p}) such that β̂(a′_{i}) = a_{i} and

H^{2}(E, 𝔽_{p}) = 𝔽_{p}{y_{1}, y_{2}, a′_{1}, a′_{2}}.

We consider the action of Out(E) = GL_{2}(𝔽_{p}). The sequence (5.1) is a sequence of 𝔽_{p} GL_{2}(𝔽_{p})-modules and the map Q_{1}β̂ induces an isomorphism of 𝔽_{p} GL_{2}(𝔽_{p})-modules,

H^{2}(E, 𝔽_{p})/H^{2}(E, ℤ) ⟶ H^{3}(E, ℤ) ⟶ 𝔽_{p}{y_{1}v, y_{2}v} ≅ S^{1} ⊗ det .

Now, consider the sequence

\begin{aligned}& H^{2p}(E, ℤ) & ⟶^{j_{∗}} & H^{2p}(E, 𝔽_{p}) & ⟶^{β̂} & H^{2p+1}(E, ℤ) \\ ⟶^{p} & H^{2p+1}(E, ℤ) & ⟶^{j_{∗}} & H^{2p+1}(E, 𝔽_{p}). \end{aligned}

By taking the p-th power, we have an 𝔽_{p} GL_{2}(𝔽_{p})-morphism,

H^{2}(E, 𝔽_{p}) ⟶ H^{2p}(E, 𝔽_{p}).

Since β = j_{∗}β̂ is the Bockstein homomorphism, we have β((a′_{i})^{p}) = 0. Moreover, since j_{∗} : H^{2p+1}(E, ℤ) ⟶ H^{2p+1}(E, 𝔽_{p}) is injective, β̂((a′_{i})^{p}) = 0. Hence, it follows that

(a′_{i})^{p} ∈ j_{∗}(H^{2p}(E, ℤ)) ≅ H^{2p}(E).

On the other hand , since

H^{2p}(E) = CS^{1} + T^{1} + 𝔽_{p}{v} ↔ S^{1} ⊕ (S^{p−2} ⊗ det) ⊕ det,

we see that if p > 3 then (a′_{i})^{p} = 0. Since H^{even}(E, 𝔽_{p}) is generated by 1, a′_{1}, a′_{2} as a module over 𝔽_{p} ⊗ H^{even}(E, ℤ), it follows that

H^{even}(E, 𝔽_{p})/\sqrt{(0)} = (𝔽_{p} ⊗ H^{even}(E, ℤ))/\sqrt{(0)}

and in particular,

H^{∗}(E, 𝔽_{p})/\sqrt{(0)} = (𝔽_{p} ⊗ H^{∗}(E, ℤ))/\sqrt{(0)} = H^{∗}(E)

for p > 3.

On the other hand, if p = 3, then the sequence (5.1) does not split and 𝔽_{3}{y_{1}, y_{2}} is the unique nontrivial 𝔽_{3} GL_{2}(𝔽_{3})-submodule of H^{2}(E, 𝔽_{3}) (cf. [9, p.74]). This implies that (a′_{i})^{p^{n}} ≠ 0 for any n > 0 and in particular, we see that a′_{i} is not nilpotent.

In fact, the structure of H^{∗}(E, 𝔽_{3})/\sqrt{(0)} is known by the result of Leary [9, Theorem 7] and we have the following:

Proposition 5.1. Assume that p = 3. Then H^{∗}(E, 𝔽_{3})/\sqrt{0} is generated by

y_{1}, y_{2}, a′_{1}, a′_{2}, v

with

deg y_{i} = deg a′_{i} = 2, deg v = 6

subject to the following relations:

y_{1}^{3}y_{2} − y_{1}y_{2}^{3} = 0
a′_{1}a′_{2} = a′_{1}y_{1} = a′_{2}y_{2} = y_{1}y_{2}, (a′_{1})^{2} = (a′_{2})^{2} = a′_{1}y_{2} = a′_{2}y_{1}.

Since H^{4}(E, 𝔽_{3})/(H^{4}(E, 𝔽_{3})∩\sqrt{0}) is spanned by y_{1}^{2}, y_{1}y_{2}, y_{2}^{2}, (a′_{1})^{2} and dim_{𝔽_{3}} H^{4}(E) = 4,

H^{4}(E, 𝔽_{3})/(H^{4}(E, 𝔽_{3}) ∩ \sqrt{0}) = H^{4}(E).

In particular, (a′_{i})^{2} ∈ H^{4}(E) and hence we have

H^{∗}(E, 𝔽_{3})/\sqrt{0} = H^{∗}(E) ⊕ 𝔽_{3}[v]\{𝔽_{3}a′_{1} + 𝔽_{3}a′_{2}\}.

Since H^{2}(E, 𝔽_{3})/H^{2}(E) ≅ S^{1} ⊗ det as 𝔽 GL_{2}(𝔽_{3})-modules,

(H^{∗}(E, 𝔽_{3})/\sqrt{0})/H^{∗}(E) ≅ 𝔽_{3}[v] ⊗ (S^{1} ⊗ det)

as 𝔽_{3} GL_{2}(𝔽_{3})-modules. If Q is a proper subgroup of E, then H^{∗}(Q, 𝔽_{3})/\sqrt{0} = H^{∗}(Q). Hence

(H^{∗}(E, 𝔽_{3})/\sqrt{0})A_{3}(Q, E)A_{3}(E, Q) ⊂ (H^{∗}(Q, 𝔽_{3})/\sqrt{0})A_{3}(E, Q)
= H^{∗}(Q)A_{3}(E, Q) ⊂ H^{∗}(E).

In particular, (H^{∗}(E, 𝔽_{3})/\sqrt{0})/H^{∗}(E) is annihilated by A_{3}(Q, E)A_{3}(E, Q) for any Q < E. Hence, every composition factors of (H^{∗}(E, 𝔽_{3})/\sqrt{0})/H^{∗}(E) as an A_{3}(E, E)-module is isomorphic to S(E, E, S^{i} ⊗ det^{q}) for some i, q and we have the following:

Proposition 5.2.

(H^{n}(E, 𝔽_{3})/\sqrt{0})/H^{n}(E) ≅ \begin{cases}S(E, E, S^{1} ⊗ det) & (n ≡ 2 mod 12) \\ S(E, E, S^{1}) & (n ≡ 8 mod 12) \\ 0 & (otherwise). \end{cases}

Corollary 5.3. Let X_{0,0} be the summand which corresponding to the simple module S(E, E, 𝔽_{p}) and e be the corresponding idempotent in A_{p}(E, E). Then

(H^{∗}(E, 𝔽_{3})/\sqrt{0})e ≅ H^{∗}(E)e ≅ 𝔻𝔸^{+}.

At last of this paper, we see more closely the cohomology H^{∗}(X) of a summand X in the stable splitting of BG with E ∈ Syl_{3}(G) in the case p = 3. The lowest degree and some of the second lowest degree ∗ > 0 with H^{2∗}(X) ≠ 0 are given as follows:

\begin{aligned}L(1, 1) & : & |S^{1}| = 1 & L(1, 0) & : & |y^{2}| = 2, \\ L(2, 1) & : & |CS^{1}v| = 6 & L(2, 0) & : & |S^{2}D_{2}| = 10, \\ X_{0,0} & : & |V | = 6 & X_{0,1} & : & |v| = 3 (|Cv| = 5) \\ X_{1,0} & : & |S^{1}V | = 7, & X_{1,1} & : & |T^{1}| = 3 (|S^{1}v| = 4) \\ X_{2,0} & : & |S^{2}V | = 8 & X_{2,1} & : & |S^{2}v| = 5 \end{aligned}

where |x| = \frac{1}{2} deg x for an element or a subspace of H^{∗}(E). First note that BG always contains X_{0,0}. The lowest degree of nonzero elements in H^{2∗}(L(i, j)) or H^{2∗}(X_{i,q}), (i, q) ≠ (0, 0) are all different except for X_{0,1} and X_{1,1}. On the other hand we see H^{4}(X_{0,1}) = 0 but H^{4}(X_{1,1}) ≅ 𝔽_{3}. Moreover L(1, 0) and L(2, 0) have same multiplicity by Lemma 4.6 and 4.7. Hence we can count the numbers of

L(1, 1), M(2) = L(1, 0) ∨ L(2, 0), X_{1,1}, X_{0,1}, X_{2,1}, L(2, 1), X_{1,0}, X_{2,0}

from H^{2∗}(G) for ∗ = 1, 2, 4, 3, 5, 6, 7, 8. Thus we have the following result which is similar to Corollary 4.2 (1) (See Remark 4.8).

Theorem 5.4. Let G_{1} and G_{2} be finite groups with same Sylow 3-subgroup E. If

dim H^{2n}(G_{1}) = dim H^{2n}(G_{2})

for n ≤ 8, then BG_{1} ∼ BG_{2}.

For example, let G_{1} = ^{2}F_{4}(2)′ and G_{2} = J_{4}. Then by [15, Theorem 6.2],

B(^{2}F_{4}(2)′) ∼ BJ_{4} ∨ X_{2,0}.

Hence H^{2n}(^{2}F_{4}(2)′) ≅ H^{2n}(J_{4}) for n < 8 and dim H^{16}(^{2}F_{4}(2)′) > dim H^{16}(J_{4}). See [15, section 6] for details.

Remark 5.5. If G has a Sylow 3-subgroup E, then BG is homotopic to the classifying space of one of the groups listed in [15, Theorem 6.2]. Moreover the cohomology of each dominant summand X_{i,j} of BE, expect for X_{1,0} and X_{1,1}, is deduced from the cohomology of those finite groups.

On the other hand, as we can see from the graph in [15, Theorem 6.2], X_{1,0} and X_{1,1} always appear as X_{1,0} ∨ X_{1,1}. We shall give an brief explanation of this fact. Let

H = W_{G}(E) = N_{G}(E)/EC_{G}(E) ≤ Out(E) = GL_{2}(𝔽_{3}).

Note that H is a 3′-group, in fact, 2-group. The multiplicity of X_{1,j} in the stable splitting of BG is equal to dim(S^{1} ⊗ det^{j})^{H}. We have to show that

dim(S^{1})^{H} = dim(S^{1} ⊗ det)^{H}.

We may assume that H ≠ 1. Then dim(S^{1})^{H} = 1 if and only if H is conjugate to the subgroup ⟨diag(1, −1)⟩ in GL_{2}(𝔽_{3}). Similarly dim(S^{1} ⊗ det)^{H} = 1 if and only if H is conjugate to the subgroup ⟨diag(1, −1)⟩ in GL_{2}(𝔽_{3}). Hence we have

dim(S^{1})^{H} = dim(S^{1} ⊗ det)^{H}

and this implies that X_{1,0} and X_{1,1} appear in BG with same multiplicity.

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Akihiko Hida, Faculty of Education, Saitama University, Shimo-okubo 255, Sakura-ku, Saitama-city, Saitama, Japan

E-mail address: ahida@mail.saitama-u.ac.jp

Nobuaki Yagita, Department of Mathematics, Faculty of Education, Ibaraki University, Mito, Ibaraki, Japan

E-mail address: yagita@mx.ibaraki.ac.jp