Representations of the double Burnside algebra and cohomology of the extraspecial p-group II
aa r X i v : . [ m a t h . A T ] M a y REPRESENTATIONS OF THE DOUBLE BURNSIDE ALGEBRA ANDCOHOMOLOGY OF THE EXTRASPECIAL p -GROUP II AKIHIKO HIDA AND NOBUAKI YAGITA
Abstract.
Let E be the extraspecial p -group of order p and exponent p where p isan odd prime. We determine the mod p cohomology H ∗ ( X, F p ) of a summand X in thestable splitting of p -completed classifying space BE . In the previous paper [Represen-tations of the double Burnside algebra and cohomology of the extraspecial p -group, J.Algebra 409 (2014) 265-319], we determined these cohomology modulo nilpotence. Inthis paper, we consider the whole part of the cohomology. Moreover, we consider thestable splittings of BG for some finite groups with Sylow p -subgroup E related with thethree dimensional linear group L ( p ). Introduction
Let p be an odd prime and E = p the extraspecial p -group of order p and exponent p . In the previous paper [7], we determined the composition factor of H ∗ ( E ) = ( F p ⊗ H ∗ ( E, Z )) / p (0) as a right A p ( E, E )-module, where A p ( E, E ) is a double Burnside algebraof E over F p = Z /p Z . 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 struc-ture is very complicated. We shall study H ∗ ( E, F p ) through the integral cohomologyring 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 = p (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 induces anisomorphism H odd ( E, Z ) ∼ = ( y v, y v ) H ∗ ( E ) where ( y v, y v ) H ∗ ( E ) is the ideal of H ∗ ( E )generated by y v, y v ∈ H p +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 arefinite dimensional for every n . We write as M ↔ L if M n and L n have same compo-sition factors (with same multiplicity), that is, [ M n ] = [ L n ] in the Grothendieck group K ( 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 v, y v ) H ∗ ( E )[ − p ] . (2) As A p ( E, E ) -modules, H odd ( E, F p ) ↔ ( y v, y v ) H ∗ ( E )[ − p + 1] ⊕ ( N ⊕ H + ( E ))[ − . Here, for a graded F p -subspace M of H ∗ ( E ), we denote by M [ i ] the graded vectorspace with M [ i ] n = M n − i . Since the composition factors of N and ( y v, y v ) H ∗ ( E ) aredetermined 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 -completedclassifying space BE ∧ p correspond to primitive idempotents in A p ( E, E ). Moreover theycorresponds to simple A p ( E, E )-modules. We simply write BE for BE ∧ p . Let X be asummand in BE which corresponds to a simple A p ( E, E )-module S . Then the multiplicityof X in BE is equal to the dimension of S as an F p -vector space since F p is a splittingfield for A p ( E, E ). By results above, we can get the cohomology H ∗ ( X, F p ) (See Remark3.6).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 whoseSylow p -subgroup is E and p -local finite groups on E . In this paper, we consider thestable splitting for groups related with the linear group L ( p ) which were not treated in[15] in general. We use some simple A p ( E, E )-submodules of H ∗ ( E ) and determine themultiplicity of summands in BG for G = L ( p ), L ( p ) : 2, L ( p ) . L ( p ) .S (Theorem4.17, 4.18, 4.19, 4.20).Combining these results and results in [15], we have the complete information on thestable splitting of finite groups or p -local finite groups which have at least two F -radicalmaximal elementary abelian p -subgroups in E , by the classification in [12], where F is afusion system of G .In particular, for p = 7, we obtain a diagram which describes inclusions of some fusionsystems 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 section3, we prove Theorem 1.1 and determine the structures of ideals N and ( y v, y v ) H ∗ ( E ).In section 4, we consider H ∗ ( G ) and the stable splitting for finite group G which has aSylow p -subgroup E . Finally, in section 5, we consider the case p = 3 and state someremarks. 2. Preliminary results on H ∗ ( E )In this section, we quote some results from [7]. Let p be an odd prime. Let E = h a, b, c | [ a, b ] = c, a p = b p = c p = [ a, c ] = [ b, c ] = 1 i be the extraspecial p -group of order p and exponent p . Let A i = h c, ab i i for 0 ≤ i ≤ p − A ∞ = h c, b i . Then A ( E ) = { A , A , . . . , A p − , 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 )) / p (0) is generated by y , y , C, v with deg y i = 2 , deg C = 2 p − , deg v = 2 p subject to the following relations: y p y − y y p = 0 , Cy i = y pi , C = y p − + y p − − y p − y p − . We set V = v p − and Y i = y p − i . OHOMOLOGY OF EXTRASPECIAL p -GROUP II 3 Let R be a subalgebra of H ∗ ( E ) and x , . . . , x r elements of H ∗ ( E ). We set R { x , . . . , x r } = r X i =1 Rx i if x , . . . , x r are linearly independent over R . Moreover, if W = P ri =1 F p x i is a F p -vectorspace spanned by x , . . . , x r , then we set R { W } = R { x , . . . , x r } . We consider the action of Out( E ) = GL ( F p ) on H ∗ ( E ). Let S i be the homogeneouspart of degree 2 i in F p [ y , y ]. Then p ( p −
1) simple F p Out( E )-modules S i v q ∼ = S i ⊗ (det) q (0 ≤ i ≤ p − , ≤ q ≤ p − F p Out( E )-modules. Letus write CA = F p [ C, V ]and DA = F p [ D , D ]where D = C p + V , D = CV . Then CA = H ∗ ( E ) Out( E ) , the Out( E )-invariants, and therestriction map induces an isomorphism DA ∼ −→ H ∗ ( A ) Out( A ) for all A ∈ A ( E ).Let T i = F p { y p − y i , y p − y i +12 , . . . , y i y p − } for 1 ≤ i ≤ p −
2. Then S p − i = CS i + T i . The F p -subspace CS i is a GL ( F p )-submoduleof CS i + T i and ( CS i + T i ) /CS i ∼ = ( S p − − i ⊗ det i ) . Moreover we have the following expression [7, Theorem 4.4]: H ∗ ( E ) = F p [ C, v ] { ( p − M i =0 S i ) ⊕ ( p − M i =1 T i ) } = CA { p − M i =0 p − M q =0 ( S i v q ⊕ T i v q )) } where S = F p and T = S p − .Let C p be a cyclic group of order p and let U i = H i ( C p , F p ) (0 ≤ i ≤ p − U i aresimple F p Out( C p )-modules. Let A ∈ A ( E ) be a maximal elementary abelian p -subgroupof E . Let S ( A ) i = H i ( A ). Then S ( A ) i ⊗ det q (0 ≤ i ≤ p −
1, 0 ≤ q ≤ p −
2) are simplemodules for Out( A ) = GL ( 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 deter-mined the multiplicity of each summand. In particular, their result implies the classifica-tion of simple A p ( E, E )-modules.
AKIHIKO HIDA AND NOBUAKI YAGITA
Proposition 2.1 ([5], [7, Proposition 10.1]) . The simple A p ( E, E ) -modules are given asfollows: (1) S ( E, E, S i ⊗ det q ) for ≤ i ≤ p − , ≤ q ≤ p − , dim S ( E, E, S i ⊗ det q ) = i + 1 . (2) S ( E, A, S ( A ) p − ⊗ det q ) for ≤ q ≤ p − , dim S ( E, A, S ( A ) p − ⊗ det q ) = p + 1 . (3) S ( E, C p , U i ) for ≤ i ≤ p − , dim S ( E, C p , U i ) = ( p + 1 ( i = 0) i + 1 (1 ≤ i ≤ p − . (4) S ( E, , F p ) , dim S ( E, , F p ) = 1 . To describe the composition factor of H ∗ ( E ) as an A p ( E, E )-module, we need thefollowing 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 -subspaceof H ∗ ( E ) : (1) If S = S ( E, C p , U i ) , then Γ S = (cid:26) F p [ C ] { F p C + S p − } ( i = 0) F p [ C ] { S i } (1 ≤ i ≤ p − . (2) If S = S ( E, A, S ( A ) p − ⊗ det q ) , then Γ S = (cid:26) DA {⊕ ≤ j ≤ p − D C j ( F p C + S p − )) } ( q = 0) DA {⊕ ≤ j ≤ p − v q C j ( CS q + T q ) } (1 ≤ q ≤ p − . (3) Γ S = DA + ( S = S ( E, E, S )) CA { v q } ( S = S ( E, E, det q ) , ≤ q ≤ p − DA { V S p − } ( S = S ( E, E, S p − ) CA { v q S p − } ( S = S ( E, E, S p − ⊗ det q ) , ≤ q ≤ p − Let S = S i v q , T = T p − i − v s for ≤ i ≤ p − , ≤ q ≤ p − , where s ≡ i + q (mod p − , ≤ s ≤ p − . Let Γ S ( E,E,S i ⊗ det q ) be the following F p -subspace: CA { V S } ⊕ DA { V T } ( q ≡ i ≡ CA { V S } ⊕ CA { T } ( q ≡ , i DA { S } ⊕ DA { V T } ( i = q, i ≡ DA { S } ⊕ CA { T } ( i = q, i CA { S } ⊕ DA { V T } ( q = 0 , i = q, q + 2 i ≡ CA { S } ⊕ CA { T } ( q = 0 , i = q, q + 2 i . (5) Γ S ( E, , F p ) = F p = H ( E ) . OHOMOLOGY OF EXTRASPECIAL p -GROUP II 5 The following theorem is the main result of [7]. If S is a simple A p ( E, E )-module, thenthere exists an idempotent e S such that Se S = S and S ′ e S = 0 for any simple module S ′ = S . 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 n ( E ) for some n ≤ ( p + 2)( p − . Let Γ nS = Γ S ∩ H n ( E ) be the degree n part of Γ S . Then by Theorem 2.3, X S dim Γ nS = 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 rep-resentatives of the isomorphism classes of simple A p ( E, E ) -modules.Proof. By Theorem 2.3, it suffices to show that H ∗ ( E ) = P Γ S . We shall show that CA { S i v q } (0 ≤ i ≤ p − , ≤ q ≤ p −
2) and CA { T i v q } (1 ≤ i ≤ p − , ≤ q ≤ p − P Γ S .First consider CA { S i v q } . If ( i, q ) = (0 , CA = F [ C, D ] = F [ D ] ⊕ CA { C } = F [ D ] ⊕ F [ C ] { C } ⊕ CA { D } = F [ D ] ⊕ F [ C ] { C } ⊕ p − X j =0 DA { D C j +1 } ⊕ DA { D } = DA ⊕ F [ C ] { C } ⊕ p − X j =0 DA { D 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 − , q ), 1 ≤ q ≤ p −
2, then CA v q and CA S p − v q arecontained in the subspace of Definition 2.2 (3). If ( i, q ) = ( p − , CA = F [ C ] ⊕ CA { V } = F [ C ] ⊕ DA { V } ⊕ DA { D , D C, . . . , D C p − } and we have CA { S p − } = F p [ C ] { S p − } ⊕ DA { V S p − } ⊕ ( ⊕ p − j =0 DA { D C j S p − } ) . This is contained in the sum of subspaces of Definition 2.2 (1)(2)(3).Consider ( i, q ) (1 ≤ i ≤ p − , ≤ q ≤ p − 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 } ⊕ ( ⊕ p − j =0 DA { C j +1 S i v i } ) AKIHIKO HIDA AND NOBUAKI YAGITA 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 − i = p − k − q ≡ m + k mod ( p − ≤ q ≤ p − s = m . Then T k v m = T p − i − v s where 1 ≤ i ≤ p −
2, 0 ≤ q ≤ p − s ≡ i + q mod ( p −
1) and q + 2 i ≡ m + k + 2( p − k − ≡ m − k mod ( p − . If k = m , then q + 2 i CA { T k v m } is contained in the subspace of Definition 2.2(4). If k = m , then q + 2 i ≡ CA { T k v m } = DA {⊕ p − j =0 T k C j v m } ⊕ DA { T k V v m } since CA = DA { , C, . . . , C p − , V } . This is contained in the sum of the subspaces ofDefinition 2.2 (2)(4). (cid: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 shallprove Theorem 1.1. The even degree part H even ( E, Z ) of integral cohomology ring isgenerated by y , y , b , . . . , b p − , C, v with deg y i = 2 , deg b i = 2 i subject to the following relations: py i = pb j = pC = 0 , p v = 0 ,y y p − y p y = 0 ,y i b k = b k b j = Cb j = 0 ,y i C = y pi , C = y p − + y p − − y p − y p − by [10] or [8, Theorem 3] (see [13] also). In particular, p ( H n ( E, Z )) = 0 for any n > H odd ( E, Z ) isannihilate by p and so it is considered as an F p [ y , y , v ]-module. As an F p [ y , y , v ]-module, H odd ( E, Z ) is generated by two elements a and a with deg a i = 3 subject to thefollowing relations: y a − y a = 0 , y p a − y p a = 0 . Let H ∗ ( E, Z ) −→ H ∗ ( E, F p ) be the natural map induced by Z −→ F p . We use thesame letters for the images of y i , b j , C, v in H ∗ ( E, F p ). Then N = F p [ v ] { b , . . . , b p − } = √ H even ( E, Z ) /pH even ( E, Z ) = F p ⊗ Z H even ( E, Z ) . Since H ∗ ( E ) = ( F p ⊗ Z H ∗ ( E, Z )) / √ 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 ) −→ . OHOMOLOGY OF EXTRASPECIAL p -GROUP II 7 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 )[ − −→ . Let ( y v, y v ) H even ( E, Z ) (resp. ( y v, y v ) H ∗ ( E )) be the ideal of H even ( E, Z ) (resp. H ∗ ( E )) generated by y v and y v . Since py i = 0 and y i N = 0, it follows that( y v, y v ) H even ( E, Z ) ∼ = ( y v, y v ) H ∗ ( E ) . Here we use Milnor’s primitive operator Q = P β − βP on H ∗ ( − , F p ). This operatorinduces a map Q on H ∗ ( − , Z ) such that the following diagram commutes: H odd ( E, Z ) Q −−−→ H even ( E, Z ) y y H odd ( E, F p ) Q −−−→ H even ( E, F p ) . Moreover, Q induces an isomorphism of A p ( E, E )-modules, Q : H odd ( E, Z ) ∼ −→ ( y v, y v ) H even ( E, Z ) ∼ = ( y v, y v ) H ∗ ( E ) Q ( a i ) = y i v, (see [14, section 1]). Then we have(3.3) H odd ( E, Z ) ∼ = ( y v, y v ) H ∗ ( E )[ − p + 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 [ − −→ . Let H = H even ( E, Z ) ∩ H + ( E, Z ). Since p 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 EA ( b ′ i ) = 0 for any i and any maximalelementary abelian p -subgroup A of E . Since the action of g ∈ GL ( 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:
AKIHIKO HIDA AND NOBUAKI YAGITA
Proposition 3.1.
Let N q = F p { v k b ′ i | k ≥ , k + i ≡ q mod ( p − } for ≤ q ≤ p − . Then N = M ≤ q ≤ p − N q and N q ∼ = M S ( E, E, det q ) as A p ( E, E ) -modules. Next, we shall consider the structure of the ideal ( y v, y v ) H ∗ ( E ). Let I = ( y v, y 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 , y , C ] ⊕ F p [ V ] { v, . . . , v p − , Cv, . . . , Cv p − } ⊕ F p [ D ] { D } ⊕ F p [ D ] { D } where F p [ y , y , C ] is the subalgebra of H ∗ ( E ) generated by y , y and C . Then H ∗ ( E ) = I ⊕ L. Proof.
Let ( y , y , C ) be the ideal of H ∗ ( E ) generated by y , y and C . Since D = C p + V ≡ V mod ( y , y , C ) , we have H ∗ ( E ) = F p [ v ] ⊕ ( y , y , C )= F p ⊕ F p [ V ] { v, . . . , v p − } ⊕ F [ V ] { V } ⊕ ( y , y , C )= F p ⊕ F p [ V ] { v, . . . , v p − } ⊕ F [ D ] { D } ⊕ ( y , y , C ) . On the other hand, since D D = CV ( C p + V ) ≡ D V mod I, it follows that F p ⊕ ( y , y , C )= F p [ y , y , C ] ⊕ F p [ v ] { Cv } ⊕ I = F p [ y , y , C ] ⊕ F p [ V ] { Cv, . . . , Cv p − } ⊕ F p [ V ] { CV } ⊕ I = F p [ y , y , C ] ⊕ F p [ V ] { Cv, . . . , Cv p − } ⊕ F p [ D ] { D } ⊕ I and we have H ∗ ( E ) = I ⊕ L . (cid:3) Lemma 3.3.
For each simple A p ( E, E ) -module S , we have Γ S = ( I ∩ Γ S ) ⊕ ( L ∩ Γ S ) OHOMOLOGY OF EXTRASPECIAL p -GROUP II 9 where I = ( y v, y 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 − ⊗ det q ) , then I ∩ Γ S = Γ S = (cid:26) DA { L ≤ j ≤ p − D C j ( F p C + S p − ) } ( q = 0) DA { L ≤ j ≤ p − v q C j ( CS q + T q ) } (1 ≤ q ≤ p − . (3) I ∩ Γ S = DA { D } ( S = S ( E, E, S )) CA { C v q } ( S = S ( E, E, det q ) , ≤ q ≤ p − DA { V S p − } ( S = S ( E, E, S p − )) CA { v q S p − } ( S = S ( E, E, S p − ⊗ det q ) , ≤ q ≤ p − Let S = S i v q , T = T p − i − v s for ≤ i ≤ p − , ≤ q ≤ p − , where s ≡ i + q (mod p − , ≤ s ≤ p − . Then I ∩ Γ S ( E,E,S i ⊗ det q ) is the following F p -subspace: CA { V S } ⊕ DA { V T } ( q ≡ i ≡ CA { V S } ⊕ CA { T } ( q ≡ , i DA { S } ⊕ DA { V T } ( i = q, i ≡ DA { S } ⊕ CA { T } ( i = q, i , i DA { S } ⊕ CA { V T } ( i = q, i , i ≡ CA { S } ⊕ DA { V T } ( q = 0 , i = q, q + 2 i ≡ CA { S } ⊕ CA { T } ( q = 0 , i = q, q + 2 i , i + q CA { S } ⊕ CA { V T } ( q = 0 , i = q, q + 2 i , i + q ≡ . (5) I ∩ Γ S (1 , , F p ) = 0 .Proof. (1) If S = S ( E, C p , U i ) (0 ≤ i ≤ p − I ∩ Γ S = 0 and Γ S ⊂ L by Definition2.2.(2) If S = S ( E, A, S p − ⊗ det q ) (0 ≤ q ≤ p − S ⊂ I since D C = C V ∈ I .(3) If S = S ( E, E, S ), then Γ S = DA + , DA + = DA { D } ⊕ F p [ D ] { D , D } . Since DA { D } ⊂ I and F p [ D ] { D , D } ⊂ L , we have DA + = ( I ∩ Γ S ) ⊕ ( L ∩ Γ S )and I ∩ Γ S = DA { D } .If S = S ( E, E, det q ) (1 ≤ q ≤ p − S = CA { v q } , CA { v q } = CA { C v q } ⊕ F p [ V ] { v q , Cv q } . Since C 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 v q } .If S = S ( E, E, S p − ), then Γ S = DA { V S p − } . If S = S ( E, E, S p − ⊗ det q ) (1 ≤ q ≤ p − S = CA { v q S p − } . In these cases, Γ S ⊂ I .(4) Let S = S i v q (1 ≤ i ≤ p −
2, 0 ≤ q ≤ p − 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 − v s , s ≡ i + q mod ( p − ≤ 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, i , i ≡
0, or q = 0 , i = q, q + 2 i , 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 } . (cid:3) Lemma 3.4.
Let I = ( y v, y v ) H ∗ ( E ) . Then I = M S ( I ∩ Γ S ) . Proof.
First, H ∗ ( E ) = ⊕ S Γ S = I ⊕ L by Proposition 2.5 and Lemma 3.2. On the otherhand, 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 = M S ( I ∩ Γ S ) . (cid:3) 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 . Thendim I n = X S dim( I n ) e S ≥ X S dim( I n ∩ Γ S ) e S = X 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 . (cid:3) OHOMOLOGY OF EXTRASPECIAL p -GROUP II 11 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 whichcorrespond to S as above. Then we can get the F p -vector space H ∗ ( E, F p ) e S ∼ = H ∗ ( ∨ X, F p ) ∼ = d M H ∗ ( X, F p )where ∨ X is a wedge sum of d = dim S copies of X , from Theorem 1.1, Proposition 3.1and Theorem 3.5. Corollary 3.7.
Every simple A p ( E, E ) -module appears as a composition factor in H n ( E, F p ) for some n ≤ p − .Proof. Let H ( p −
2) = ⊕ p − n =0 H n ( E, F p ) . For a simple A p ( E, E )-module S , let 2 γ ( S )be the lowest degree such that (Γ S ∩ I ) γ ( S ) = 0. If 2 γ ( S ) ≤ p + 2)( p − γ ( S ) − p ) ≤ p − . Since S appears in the degree 2( γ ( S ) − p ) part of I [ − p ], it followsthat S appears in H γ ( S ) − p ) ( E, F p ) by Theorem 1.1. Hence S appears in H ( p − S ≤ I , then S appears in H ( p −
2) by Corollary 2.4. This implies thatsimple modules S ( E, A, S ( A ) p − ⊗ det q )(0 ≤ q ≤ p − , S ( E, E, S p − ⊗ det q )(0 ≤ q ≤ p − H ( p − C 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 S p − = 2( p + 2)( p − S ( E, E, det q ), (1 ≤ q ≤ p −
2) and S ( E, E, S i ⊗ det q ), (1 ≤ i ≤ p −
2, 0 ≤ q ≤ p − H ( p − S ( E, C p , U i ) (0 ≤ i ≤ p −
2) appears in H ( E ) ⊕ · · · ⊕ H p − ( E ), itappears in H ( p − S ( E, E, S ). Since it appears in H p ( p − ( E )and 2 p ( p − D ) ≤ p − S ( E, E, S ) appears in H ( p − (cid:3) Stable splitting of groups related to L ( 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 ( p ) and its extensions.Benson and Fechbach [2], Martino and Priddy [11] prove the following theorem oncomplete 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 completestable 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 )) / p (0)for a finite group G . From Corollary 2.4 and Corollary 3.7, we have the following. Corollary 4.2.
Let G , G have the same p -Sylow subgroup E . Suppose that G ≤ G . (1) If dim H n ( G ) = dim H n ( G ) for all ≤ n ≤ ( p + 2)( p − , then BG ∼ BG . (2) If dim H n ( G , F p ) = dim H n ( G , F p ) for all ≤ n ≤ p − , then BG ∼ BG . In general, the computation of these dim S ( G, Q, V ) is not so easy. Hence we study theway to compute it from the information on the cohomology H ∗ ( G ). (In fact, in [15], mostdirect 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 ecG -rad be the set of F ecG -radical maximal elementary abelian p -subgroups of E . If A is a maximal elementaryabelian p -subgroup of E , then A ∈ F ecG -rad if and only if W G ( A ) = N G ( A ) /C G ( A ) =Out F G ( A ) ≥ SL ( 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 ecG - rad(res EA ) − ( 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 ecG - rad(res EA ) − ( 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. (cid:3) Let X i,q be the indecomposable summand in the stable splitting of BE which cor-responds 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 − ⊗ det q ) (resp. S ( E, C p , U q )). We set M (2) = L (1 , ∨ L (2 , BG is written as BG ∼ ( ∨ i,q n ( G ) i,q X i,q ) ∨ ( ∨ q m ( G, q L (2 , q )) ∨ ( ∨ q m ( G, q L (1 , q )) . Recall that H q ( E ) ∼ = ( S ( E, C p , U i ) (1 ≤ q ≤ p − S ( E, C p , U ) ( q = p − Lemma 4.4 ([15, Corollary 4.6]) . The multiplicity m ( G, q for L (1 , q ) is given by m ( G, q = ( dim H q ( G ) (1 ≤ q ≤ p − H p − ( G ) ( q = 0) . 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 ( F p ) and GL ( 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. OHOMOLOGY OF EXTRASPECIAL p -GROUP II 13 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, q for the summand L (2 , q ) in BG . Wecan prove, Lemma 4.6 ([15, Proposition 4.9]) . The multiplicity of L (2 , in BG is given by m ( G, = ♯ G ( A ) − ♯ G ( F ec A ) where ♯ G ( A ) (resp. ♯ G ( F ec A ) ) is the number of G -conjugacy classes of rank two elementaryabelian p -subgroups in E (resp. subgroups in F ecG - rad ). Lemma 4.7 ([15, Corollary 4.10]) . The multiplicity of L (1 , in BG is given by m ( G, = dim H p − ( G ) = ♯ G ( A ) − ♯ G ( F ec A ) . Remark 4.8.
By Lemma 4.6 and 4.7, m ( G, = m ( G, , namely, L (1 ,
0) and L (2 , BG as M (2) = L (1 , ∨ L (2 , S ( E, A, S p − ) appear in H n ( E ) for n ≤ p −
1. Note that theminimal n such that S ( E, E, S p − ) appears in H n ( E ) is p − deg( V S p − ). Hencewe may replace the bound ( p + 2)( p −
1) by p − m ( G, 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, q = 0. Lemma 4.9 ([15, Lemma 4.11]) . Let ξ ∈ F ∗ p be a primitive ( p − -th root of . Supposethat G ⊃ E : h diag( ξ, ξ ) i . If ξ k = 1 , then BG does not contain the summand L (2 , k ) ,i.e., m ( G, k = 0 . Let ξ be the multiplicative generator of F ∗ p as above. Let w = (cid:18) (cid:19) ∈ GL ( F p ) . Let T = h diag( ξ, ξ ) , diag( ξ, i be a torus in GL ( F p ). Then w normalizes T . Let T h w i be the semidirect product of T by h w i Lemma 4.10.
Assume that ≤ l ≤ p − , ≤ k ≤ p − . Then ( S l v k ) T = F p y i y i v p − − i ( l = 2 i, k = p − − i, ≤ i ≤ p − ) F p y p − ⊕ F p y p − ( l = p − , k = 0)0 ( otherwise ) and ( S l v k ) T h w i = F p y j y j v p − − j ( l = 4 j, k = p − − j, ≤ j ≤ p − ) F p ( y p − + y p − ) ( l = p − , k = 0)0 ( otherwise ) . Proof.
We consider the action of diag( ξ,
1) and diag(1 , ξ ). For 0 ≤ i ≤ l , we havediag( ξ, y i y l − i v k = ξ i + k y i y l − i v k diag(1 , ξ ) y i y l − i v k = ξ l − i + k y i y l − i v k . Hence y i y l − i v k is T -invariant if and only if i + k ≡ l − i + k ≡ p −
1, namely, l ≡ i mod p − i + k ≡ p −
1. If k = 0, then i = 0 , p − l = p −
1. If k >
0, then i = p − − k , l = 2 i . Since 1 ≤ l ≤ p −
1, it follows that i ≤ p − .Next consider the action of w . Since w interchanges y and y , wv = − v , y i y i v p − − i is w -invariant if and only if i is even. (cid:3) Now assume that p − m . Note that m is even. We set H = h diag( ξ, ξ ) , diag( ξ , i . Then w normalizes H . Let H h w i be the semidirect product of H by h w i . Lemma 4.11.
Assume that p − m . Let m = 2 n . Assume that ≤ l ≤ p − , ≤ k ≤ p − . Then ( S l v k ) H is equal to the following vector space. F p { y p − , y m y m , y m y m , y p − } ( l = p − , k = 0) F p { y i − n y i + n v n − i , y i + n y i − n v n − i } ( l = 2 i, k = 3 n − i, n ≤ i < n ) F p { y i y i v m − i } ( l = 2 i, k = ( p − − i, ≤ i < m ) F p { y i − m y i + m v m − i , y i y i v m − i , y i + m y i − m v m − i } ( l = 2 i, k = ( p − − i, m ≤ i ≤ n )0 ( otherwise ) . Proof.
We consider the action of diag( ξ, ξ ) and diag( ξ ,
1) on y j y l − j v k for 0 ≤ j ≤ l . Sincediag( ξ, ξ ) y j y l − j v k = ξ l +2 k y j y l − j v k and diag( ξ , y j y l − j v k = ξ j +3 k y j y l − j v k ,y j y l − j v k is H -invariant if and only if ( l + 2 k ≡ p − j + k ≡ m. Note that the condition l + 2 k ≡ p − l is even, so we set l = 2 i ,1 ≤ i ≤ n . Then y j y l − j v k ∈ ( S l v k ) H if and only if ( i + k ≡ nj + k ≡ m. Since 0 ≤ k ≤ p − k ≡ − i mod 3 n if and only if k = 3 n − i or ( p − − i. First, assume k = 3 n − i . Then j ≡ − k = i − n mod m if and only if j = i + sn forsome integer s ∈ Z such that s ≡ ≤ j ≤ l = 2 i ⇐⇒ ≤ i + sn ≤ i ⇐⇒ − i ≤ sn ≤ i ⇐⇒ | s | ≤ in . OHOMOLOGY OF EXTRASPECIAL p -GROUP II 15 If 1 ≤ i < n , namely, in <
1, then there is no s ∈ Z such that | s | ≤ in with s ≡ n ≤ i ≤ n . If i = 3 n , namely, l = p − k = 0, then, since in = 3, | s | ≤ in if and only if s = − , − , ,
3. Then j = i + sn = (3 + s ) n = 0 , m, m, m. If n ≤ i < n , namely, 1 ≤ in <
3, then | s | ≤ in if and only if s = − , s ≡ j = i + sn = i ± n. Next we consider the case k = ( p − − 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 = 2 i ⇐⇒ − i ≤ sm ≤ i ⇐⇒ | s | ≤ im . If 1 ≤ i < m , then this implies s = 0 and j = i . If m ≤ i ≤ n , namely, 1 ≤ im ≤ < s = 0 , ±
1. Hence we have j = i, i ± m . This completes the proof. (cid:3) Lemma 4.12.
Assume that p − m . Let m = 2 n . Assume that ≤ l ≤ p − , ≤ k ≤ p − . Then ( S l v k ) H h w i is equal to the following vector space. F p { y p − + y p − , y m y m + y m y m } ( l = p − , k = 0) F p { ( y i − n y i + n + ( − n − i y i + n y i − n ) v n − i } ( l = 2 i, k = 3 n − i, n ≤ i < n ) F p { y i y i v m − i } ( l = 2 i, k = ( p − − i, ≤ i < m, i :even ) F p { ( y i − m y i + m + y i + m y i − m ) v m − i , y i y i v m − i } ( l = 2 i, k = ( p − − i, m ≤ i ≤ n, i :even ) F p { ( y i − m y i + m − y i + m y i − m ) v m − i } ( l = 2 i, k = ( p − − i, m ≤ i ≤ n, i :odd )0 ( otherwise ) . Proof.
Since w interchanges y and y , vw = (det w ) v = − v , this lemma follows from theprevious lemma. (cid:3) Let p − m . We consider the multiplicity m ( G, m and m ( G, m in some cases.Recall that ( CS q + T q ) v q ∼ = S ( E, A, S ( A ) p − ⊗ det q ) for 1 ≤ q ≤ p − y i y j ) v q , ( i = 0 , q ≤ i ≤ p − q, j = p − q − i ) . Note that y i y j , ( i = 0 , p ≤ i ≤ p − q, j = p − q − i )is a basis of CS q . On the other hand, y i y j , ( q ≤ i ≤ p − , j = p − q − i )is a basis of T q . Moreover, if q = m or q = 2 m , then all elements in ( CS q + T q ) v q arediag( ξ, ξ )-invariant sincediag( ξ, ξ )( y i y j v q ) = ξ i + j +2 q ( y i y j v q ) = ξ p − q ( y i y j v q ) = ( y i y j v q ) . Lemma 4.13.
Let M = ( CS m + T m ) v m . (1) M T has a basis y m y m v m . (2) M H has a basis y m v m , y m y m v m , y m y m v m , y m y m v m , y m v m . (3) M T h w i has a basis y m y m v m . (4) M H h w i has a basis ( y m + y m ) v m = C ( y m + y m ) v m , y m y m ( y m + y m ) v m , y m y m v m . Proof.
Since diag( ξ, y i y j v m ) = ξ i + m ( y i y j v m )and diag( ξ , y i y j v m ) = ξ i + m ) ( y i y j v m ) = ξ i ( y i y j v m ) ,y i y j v m is T -invariant if and only if i + m ≡ p −
1. Moreover, y i y j v m is H -invariant i ≡ m .(1) Since i = 0 or m ≤ i ≤ p − m = 4 m , i + m ≡ p − i = 2 m .(2) Since i = 0 or m ≤ i ≤ p − m = 4 m , i ≡ m if and only if i =0 , m, m, m, m .(3) (4) Since m is even, w acts on v m trivially. On the other hand w interchanges y and y . Hence the results follows from (1) and (2). (cid:3) Similarly, we have the following.
Lemma 4.14.
Let M = ( CS m + T m ) v m . (1) M T has a basis Cy m y m v m = y m y m v m . (2) M H has a basis y m v m , Cy m y m v m = y m y m v m , y m y m v m , y m y m v m , y m v m . (3) M T h w i has a basis Cy m y m v m = y m y m v m . (4) M H h w i has a basis ( y m + y m ) v m = C ( y m + y m ) v m , y m y m ( y m + y m ) v m , Cy m y m v m . 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 EA i ( y ) = y A i , res EA i ( y ) = iy A i for i ∈ F p and res EA ∞ ( y ) = 0 , res EA ∞ ( y ) = y A ∞ . Moreover, res EA ( C ) = y p − A , res EA ( v ) = u pA − y p − A u for any A ∈ A ( E ) (see [7, section 4]). OHOMOLOGY OF EXTRASPECIAL p -GROUP II 17 Lemma 4.15.
Let ≤ q ≤ p − . Then (( CS q + T q ) v q )[ G ] = (( CS q + T q ) v q ) W G ( E ) ∩ ( ∩ A ∈F ecG - rad ker res EA ) Proof.
Let y = y A and u = u A for A ∈ A ( E ). Thenres EA (( CS q + T q ) v q ) = F p y p − q res EA ( v q ) = F p y p − ( yu p − y p u ) q . If g ∈ Aut( A ) = GL ( F p ), then g ( yu p − y p u ) = (det g )( yu p − y p u )and y p − ( yu p − y p u ) q is not SL ( F p )-invariant, hence the result follows from Theorem4.3. (cid:3) Proposition 4.16.
Suppose that F ecG - rad = { A , A ∞ } . Then m ( G, m = m ( G, m andwe have the following values: W G ( E ) H H h w i T T h w i m ( G, m = m ( G, m Proof.
Since res EA ( y ) = 0 , res EA ( y ) = 0and res EA ∞ ( y ) = 0 , res EA ∞ ( y ) = 0 , the results follows from Lemma 4.13, Lemma 4.14 and 4.15. (cid:3) Next we study the stable splitting of BG for some G related to the linear group L ( p ).There are 6 saturated fusion systems related to L ( p ) [12, p. 46, Table 1.1]. W G ( E ) |F ecG -rad | Group pH L ( p ) 3 | ( p − H h w i L ( p ) : 2 3 | ( p − T L ( p ) . | ( p − T h w i L ( p ) .S | ( p − T L ( p ) 3 ∤ ( p − T h w i L ( p ) : 2 3 ∤ ( p − E with |F ec -rad | ≥ X = X , ∨ X p − , ∨ ( ∨ ≤ i ≤ ( p − / X i,p − − i ) ∨ M (2)and X ′ = X , ∨ X p − , ∨ ( ∨ ≤ j ≤ ( p − / X j,p − − j ) ∨ M (2) . Theorem 4.17.
Suppose that W G ( E ) = T and F ecG - rad = { A , A ∞ } . If ∤ p − then BG is stably homotopic to X . If p − m , then BG is stably homotopic to X ∨ L (2 , m ) ∨ L (2 , m ) . 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 maximalelementary abelian p -subgroups and two of them consist of F ecG -radical subgroups. FromLemma 4.6 and Lemma 4.7, just one L (2 ,
0) (and one L (1 , BG . More-over if 3 does not divide p −
1, then L (2 , q ) is not contained in BG for each 1 ≤ q ≤ p − n ( G ) l,k = l = 2 i, k = p − − i, ≤ i ≤ p − )2 ( l = p − , k = 0)0 (otherwise) . On the other hand, if p − m , then m ( G, m = m ( G, m = 1 by Proposition 4.16.This completes the proof. (cid:3) Theorem 4.18.
Suppose that W G ( E ) = T h w i and F ecG - rad = { A , A ∞ } . If ∤ p − then BG is stably homotopic to X ′ . If p − m , then BG is stably homotopic to X ′ ∨ L (2 , m ) ∨ L (2 , m ) .Proof. The proof is similar to that of previous Theorem. By Lemma 4.4 and Lemma4.10, 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 -subgroupsand one of them consists of F ecG -radical subgroups. From Lemma 4.6 and Lemma 4.7, justone L (2 , p −
1) (and one L (1 , p − BG . Moreover if 3 does not divide p −
1, then L (2 , q ) is not contained in BG for each 1 ≤ q ≤ p − n ( G ) l,k = l = 4 j, k = p − − j, ≤ j ≤ p − )1 ( l = p − , k = 0)0 (otherwise) . On the other hand, if p − m , then m ( G, m = m ( G, m = 1 by Proposition 4.16.This completes the proof. (cid:3) Next assume that p − m . Let m = 2 n . Theorem 4.19.
Suppose that W G ( E ) = H and F ecG - rad = { A , A ∞ } . Then BG is stablyhomotopic to X , ∨ X p − , ∨ ∨ n ≤ i< n X i, n − i ) ∨ ( ∨ ≤ i By Lemma 4.4 and Lemma 4.11, m ( G, q = 0 for 1 ≤ q ≤ p − 2. On the otherhand, by Lemma 4.6 and Corollary 4.7, m ( G, = m ( G, = 5 − m ( G, k = 0 for 1 ≤ k ≤ p − k = m, m . By Proposition 4.16, m ( G, m = OHOMOLOGY OF EXTRASPECIAL p -GROUP II 19 m ( G, m = 3. The multiplicity n ( G ) i,q is obtained by Lemma 4.11. By Lemma 4.11, n ( G ) i,q = l = p − , q = 0)2 ( l = 2 i, q = 3 n − i, n ≤ i < n )1 ( l = 2 i, q = ( p − − i, ≤ i < m )3 ( l = 2 i, q = ( p − − i, m ≤ i ≤ n )0 (otherwise) . (cid:3) Theorem 4.20. Suppose that W G ( E ) = H h w i and F ecG - rad = { A , A ∞ } . Then BG isstably homotopic to X , ∨ X p − , ∨ ( ∨ n ≤ i< n X i, n − i ) ∨ ( ∨ ≤ j ≤ n/ X j, m − j ) ∨ ( ∨ m ≤ i ≤ n X i, ( p − − i ) ∨ M (2) ∨ L (2 , m ) ∨ L (2 , m )) . Proof. By Lemma 4.4 and Lemma 4.12, m ( G, q = 0 for 1 ≤ q ≤ p − 2. By Lemma4.6 and Corollary 4.7, m ( G, = m ( G, = 3 − m ( G, k = 0for 1 ≤ k ≤ p − k = m, m . By Proposition 4.16, m ( G, m = m ( G, m = 2. Themultiplicity n ( G ) i,q is obtained by Lemma 4.12. By Lemma 4.12, n ( G ) i,q = l = p − , q = 0)1 ( l = 2 i, q = 3 n − i, n ≤ i < n )1 ( l = 2 i, q = ( p − − i, ≤ i < m, i : even)2 ( l = 2 i, q = ( p − − i, m ≤ i ≤ n, i : even)1 ( l = 2 i, q = ( p − − i, m ≤ i ≤ n, i : odd)0 (otherwise) . Moreover, consider the 3rd, 4th and 5th cases. We have( ∨ ≤ i Next we consider the specific case, that is, p = 7. We give a result which supplementsthe result on splitting for p = 7 in [15]. Example 4.21. Let p = 7 , p − , m = 2 , n = 1 . Suppose that F ecG - rad = { A , A ∞ } . (1) If W G ( E ) = T , then BG ∼ X , ∨ X , ∨ X , ∨ X , ∨ X , ∨ M (2) ∨ L (2 , ∨ L (2 , . (2) If W G ( E ) = T h w i , then BG ∼ X , ∨ X , ∨ X , ∨ M (2) ∨ L (2 , ∨ L (2 , . (3) If W G ( E ) = H , then BG ∼ X , ∨ X , ∨ X , ∨ X , ∨ X , ∨ X , ∨ X , ∨ M (2) ∨ L (2 , ∨ L (2 , . (4) If W G ( E ) = H h w i , then BG ∼ X , ∨ X , ∨ X , ∨ X , ∨ X , ∨ X , ∨ M (2) ∨ L (2 , ∨ L (2 , . Let G and G be finite groups with Sylow p -subgroup E . If F G is (isomorphic to) asubfusion system of F G , then BG ∼ BG ∨ X for some summand X of BG . In thiscase, we write G X ←−−− G We use same notation for fusion systems.In [15], the second author considered the graphs related to the splitting of sporadicsimple 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 , ∨ X , ∼ RV X , ∨ X , ←−−−−−− RV M (2) ∨ L (2 , ∨ L (2 , ←−−−−−−−−−−− O ′ N : 2 X , ∨ X , ∨ X , ∨ X , ←−−−−−−−−−−−−− O ′ NX , ∨ X , ∨ X , ∼ RV M (2) ←−−− F i X , ∨ X , ∨ X , ←−−−−−−−−− F i ′ M (2) ∨ L (2 , ∨ L (2 , ←−−−−−−−−−−− He : 2 X , ∨ X , ∨ X , ∨ X , ∨ L (1 , ∨ L (2 , ←−−−−−−−−−−−−−−−−−−−−− He. where RV , RV , RV 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 M (2) ∨ ˜ L ←−−−− L (7) .S Y ′ ←−−− L (7) . Y ∨ Z ∨ M (2) ∨ ˜ L x Y ∨ Z ∨ M (2) ∨ ˜ L x x Y ∨ Z ∨ M (2) ∨ ˜ L ) O ′ N M (2) ∨ ˜ L ←−−−− L (7) : 2 Y ∨ Y ′ ∨ Z ∨ M (2) ∨ ˜ L ←−−−−−−−−−− L (7) Y ∨ Z ∨ M (2) ∨ ˜ L y Y ∨ Z ∨ M (2) ∨ L y y Y ∨ Y ′ ∨ Z ∨ M (2) ∨ L RV M (2) ←−−− F i Y ←−−− F i ′ where Y = X , ∨ X , ∨ X , , Y ′ = X , ∨ X , ∨ X , , Z = X , ∨ X , ˜ L = L (2 , ∨ L (2 , . OHOMOLOGY OF EXTRASPECIAL p -GROUP II 21 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 (7) 6 × h I, u i { A }{ A ∞ } SL (7) : 2 L (7) 6 × h I, w i { A }{ A } SL (7) : 2 L (7) : 2 (6 × 2) : 2 = h I, u, w i { A , A ∞ } SL (7) : 2 L (7) : 2 (6 × 2) : 2 = h I, u, w i { A , A } SL (7) : 2 O ′ N (6 × 2) : 2 = h I, u, w i { A , A ∞ }{ A , A } SL (7) : 2 L (7) . = T { A }{ A ∞ } GL (7) L (7) .S : 2 = T h w i { A , A ∞ } GL (7) F i ′ × S = h I, s, w i { A , A , A }{ A , A , A } SL (7) : 2 F i : 2 = T h w i { A , . . . , A } SL (7) : 2 RV : 2 = T h w i { A , A ∞ }{ A , . . . , A } GL (7) , SL (7) : 2where I = (cid:18) (cid:19) , u = (cid:18) − (cid:19) , w = (cid:18) (cid:19) , s = (cid:18) (cid:19) and T = { diag( α, β ) | α, β ∈ F × } 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 eachrepresentative A of conjugacy classes in F ec -rad if they are different. Note that if wetake the generators a and b of E suitably, we can obtained the two rows in the case of L (7) and L (7) : 2. For example, consider G = L (7). Let E be the group of all uppertriangular matrices with diagonal entry 1. The subgroups in F ecG -rad are α β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α, β ∈ F p , α β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α, β ∈ F p . If we take a = , b = then F ecG -rad = { A , A ∞ } and Out F G ( E ) = h I, u i = H . On the other hand, if we take a = , b = − 10 0 1 then F ecG -rad = { A , A } and Out F G ( E ) = h I, w i .The inclusions of fusion systems are obtained by the table above. For F i ′ ←− L (7)and F i ←− L (7) : 2, we use the second rows of L (7) and L (7) : 2. The informationon the summands are obtained by Example 4.21 and Theorem 4.22. (cid:3) Remark 4.24. As we can see from Theorem 4.22 or 4.23 above, B ( F i ) is a stablesummand of B ( O ′ N ), B ( O ′ N ) ∼ B ( F i ) ∨ Y ∨ Z ∨ ˜ L , but the fusion system of O ′ N isnot isomorphic to a subfusion system of fusion system of F i , namely, F i Y ∨ Z ∨ ˜ L ←−−−− O ′ N does not hold. Let F = F O ′ N , F = F F i . By [12, Lemma 4.3], for each A i ∈ F ec -rad, there existsan element of order 6 in Out F ( E ) ≤ GL ( F p ) which has an eigenvalue 3 with eigenvector (cid:18) i (cid:19) ( (cid:18) (cid:19) if i = ∞ ) and determinant 5. Hence there exists an involution in Out F ( E )which has an eigenvalue − (cid:18) i (cid:19) ( (cid:18) (cid:19) if i = ∞ ) and determinant − F ( E ) = 6 : 2 = T h w i and F ec -rad = { A , . . . , A } asabove. Suppose that K = Out F ( E )( ∼ = (6 × 2) : 2) ≤ Out F ( E ). Then K containsexactly 4 involutions with determinant − 1. Moreover K ⊃ h diag( − , , diag(1 , − i since h diag( − , , diag(1 , − i ⊳ (6 : 2). Note that |F ec -rad | = 4. Since diag( − , 1) (resp.diag(1 , − − (cid:18) (cid:19) (resp. (cid:18) (cid:19) ) and determinant − 1, it follows that A , A ∞ ∈ F ec -rad for any choice of K ≤ : 2 = T h w i . Hence F isnot isomorphic to a subfusion system of F .5. Some remarks on the case p = 3Recall that H ( E, Z ) = F p { a , a } . The short exact sequence0 −→ Z p −→ Z j −→ F p −→ −→ H ( E, Z ) j ∗ −→ H ( E, F p ) ˆ β −→ H ( E, Z ) −→ . Hence there exist elements a ′ , a ′ ∈ H ( E, F p ) such that ˆ β ( a ′ i ) = a i and H ( E, F p ) = F p { y , y , a ′ , a ′ } . We consider the action of Out( E ) = GL ( F p ). The sequence (5.1) is a sequence of F p GL ( F p )-modules and the map Q ˆ β induces an isomorphism of F p GL ( F p )-modules, H ( E, F p ) /H ( E, Z ) −→ H ( E, Z ) −→ F p { y v, y v } ∼ = S ⊗ det . Now, consider the sequence H p ( E, Z ) j ∗ −−−→ H p ( E, F p ) ˆ β −−−→ H p +1 ( E, Z ) p −−−→ H p +1 ( E, Z ) j ∗ −−−→ H p +1 ( E, F p ) . By taking the p -th power, we have an F p GL ( F p )-morphism, H ( E, F p ) −→ H p ( E, F p ) . Since β = j ∗ ˆ β is the Bockstein homomorphism, we have β (( a ′ i ) p ) = 0. Moreover, since j ∗ : H p +1 ( E, Z ) −→ H p +1 ( E, F p ) is injective, ˆ β (( a ′ i ) p ) = 0. Hence, it follows that( a ′ i ) p ∈ j ∗ ( H p ( E, Z )) ∼ = H p ( E ) . On the other hand , since H p ( E ) = CS + T + F p { v } ↔ S ⊕ ( S p − ⊗ det) ⊕ det , we see that if p > a ′ i ) p = 0. Since H even ( E, F p ) is generated by 1 , a ′ , a ′ as amodule over F p ⊗ H even ( E, Z ), it follows that H even ( E, F p ) / p (0) = ( F p ⊗ H even ( E, Z )) / p (0) OHOMOLOGY OF EXTRASPECIAL p -GROUP II 23 and in particular, H ∗ ( E, F p ) / p (0) = ( F p ⊗ H ∗ ( E, Z )) / p (0) = H ∗ ( E )for p > p = 3, then the sequence (5.1) does not split and F { y , y } is theunique nontrivial F GL ( F )-submodule of H ( E, F ) (cf. [9, p.74]). This implies that( a ′ i ) p n = 0 for any n > a ′ i is not nilpotent.In fact, the structure of H ∗ ( E, F ) / p (0) is known by the result of Leary [9, Theorem7] and we have the following: Proposition 5.1. Assume that p = 3 . Then H ∗ ( E, F ) / √ is generated by y , y , a ′ , a ′ , v with deg y i = deg a ′ i = 2 , deg v = 6 subject to the following relations: y y − y y = 0 a ′ a ′ = a ′ y = a ′ y = y y , ( a ′ ) = ( a ′ ) = a ′ y = a ′ y . Since H ( E, F ) / ( H ( E, F ) ∩√ 0) is spanned by y , y y , y , ( a ′ ) and dim F H ( E ) = 4, H ( E, F ) / ( H ( E, F ) ∩ √ 0) = H ( E ) . In particular, ( a ′ i ) ∈ H ( E ) and hence we have H ∗ ( E, F ) / √ H ∗ ( E ) ⊕ F [ v ] { F a ′ + F a ′ } . Since H ( E, F ) /H ( E ) ∼ = S ⊗ det as F GL ( F )-modules,( H ∗ ( E, F ) / √ /H ∗ ( E ) ∼ = F [ v ] ⊗ ( S ⊗ det)as F GL ( F )-modules. If Q is a proper subgroup of E , then H ∗ ( Q, F ) / √ H ∗ ( Q ).Hence ( H ∗ ( E, F ) / √ A ( Q, E ) A ( E, Q ) ⊂ ( H ∗ ( Q, F ) / √ A ( E, Q )= H ∗ ( Q ) A ( E, Q ) ⊂ H ∗ ( E ) . In particular, ( H ∗ ( E, F ) / √ /H ∗ ( E ) is annihilated by A ( Q, E ) A ( E, Q ) for any Q < E .Hence, every composition factors of ( H ∗ ( E, F ) / √ /H ∗ ( E ) as an A ( E, E )-module isisomorphic to S ( E, E, S i ⊗ det q ) for some i, q and we have the following: Proposition 5.2. ( H n ( E, F ) / √ /H n ( E ) ∼ = S ( E, E, S ⊗ det) ( n ≡ S ( E, E, S ) ( n ≡ otherwise ) . Corollary 5.3. Let X , 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 ) / √ e ∼ = H ∗ ( E ) e ∼ = DA + . At last of this paper, we see more closely the cohomology H ∗ ( X ) of a summand X inthe stable splitting of BG with E ∈ Syl ( G ) in the case p = 3. The lowest degree andsome of the second lowest degree ∗ > H ∗ ( X ) = 0 are given as follows: L (1 , 1) : | S | = 1 L (1 , 0) : | y | = 2 ,L (2 , 1) : | CS v | = 6 L (2 , 0) : | S D | = 10 ,X , : | V | = 6 X , : | v | = 3 ( | Cv | = 5) X , : | S V | = 7 , X , : | T | = 3 ( | S v | = 4) X , : | S V | = 8 X , : | S v | = 5where | x | = deg x for an element or a subspace of H ∗ ( E ). First note that BG alwayscontains X , . The lowest degree of nonzero elements in H ∗ ( L ( i, j )) or H ∗ ( X i,q ), ( i, q ) =(0 , 0) are all different except for X , and X , . On the other hand we see H ( X , ) = 0but H ( X , ) ∼ = F . Moreover L (1 , 0) and L (2 , 0) have same multiplicity by Lemma 4.6and 4.7. Hence we can count the numbers of L (1 , , M (2) = L (1 , ∨ L (2 , , X , , X , , X , , L (2 , , X , , X , from H ∗ ( G ) for ∗ = 1 , , , , , , , 8. Thus we have the following result which is similarto Corollary 4.2 (1) (See Remark 4.8). Theorem 5.4. Let G and G be finite groups with same Sylow -subgroup E . If dim H n ( G ) = dim H n ( G ) for n ≤ , then BG ∼ BG . For example, let G = F (2) ′ and G = J . Then by [15, Theorem 6.2], B ( F (2) ′ ) ∼ BJ ∨ X , . Hence H n ( F (2) ′ ) ∼ = H n ( J ) for n < H ( F (2) ′ ) > dim H ( J ). See [15,section 6] for details. Remark 5.5. If G has a Sylow 3-subgroup E , then BG is homotopic to the classifyingspace of one of the groups listed in [15, Theorem 6.2]. Moreover the cohomology of eachdominant summand X i,j of BE , expect for X , and X , , is deduced from the cohomologyof those finite groups.On the other hand, as we can see from the graph in [15, Theorem 6.2], X , and X , always appear as X , ∨ X , . We shall give an brief explanation of this fact. Let H = W G ( E ) = N G ( E ) /EC G ( E ) ≤ Out( E ) = GL ( F ) . Note that H is a 3 ′ -group, in fact, 2-group. The multiplicity of X ,j in the stable splittingof BG is equal to dim( S ⊗ det j ) H . 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