Classification of BG for groups with dihedral or quarternion Sylow 2-subgroups

Martino, John; Priddy, Stewart

doi:10.1016/0022-4049(91)90103-9

Cited by 26 publications

(16 citation statements)

References 4 publications

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“…(1) and (4). The structure of the mod 2 cohomology rings of groups with semidihedral Sylow 2-subgroups was determined by J. Martino (see [32]). A different proof (using representation theory) can be found in a paper of H. Sasaki [39].…”

Section: Hochschild Cohomology Of the Group Algebra K Sl 3 (Q): Via Gmentioning

confidence: 99%

Hochschild cohomology of tame blocks

Holm

2004

Journal of Algebra

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Section: Hochschild Cohomology Of the Group Algebra K Sl 3 (Q): Via Gmentioning

confidence: 99%

Hochschild cohomology of tame blocks

Holm

2004

Journal of Algebra

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“…For further work in this area, see Martino and Priddy [22,23], Nishida [26], Priddy [27,28]. Some explicit calculations appear in Dietz [11], Dietz and Priddy [12], Martino and Priddy [20]. Much remains to be done.…”

Section: This Multiplication Makes A(g G) Into a Noncommutative Noetmentioning

confidence: 99%

Stably splitting BG

Benson

1996

Bull. Amer. Math. Soc.

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Abstract. In the early nineteen eighties, Gunnar Carlsson proved the Segal conjecture on the stable cohomotopy of the classifying space BG of a finite group G. This led to an algebraic description of the ring of stable self-maps of BG as a suitable completion of the "double Burnside ring". The problem of understanding the primitive idempotent decompositions of the identity in this ring is equivalent to understanding the stable splittings of BG into indecomposable spectra. This paper is a survey of the developments of the last ten to fifteen years in this subject. The Segal conjectureThe nineteen eighties saw a number of major breakthroughs in homotopy theory. Among the most spectacular are the proofs of the nilpotence conjecture by Devinatz, Hopkins and Smith [10], the Sullivan conjecture by Haynes Miller [24], and the Segal conjecture by Gunnar Carlsson [7]. Two of these three concern the role of finite groups in homotopy theory: the Sullivan conjecture is "unstable", while the Segal conjecture is "stable" with respect to suspension. This report is about the consequences of the Segal conjecture for the stable splittings of the classifying space of a finite group.We begin by setting the scene. Around 1960, Atiyah [3] calculated the Ktheory of the classifying space of a finite group. There is a natural map from the character ring R(G) to K 0 (BG) which sends a complex representation V to the corresponding vector bundle EG× G V → BG. Atiyah proved that this map induces an isomorphismwhere the completionn is with respect to the augmentation ideal I = Ker(dim : R(G) → Z). He also showed that K 1 (BG) = 0. The Segal conjecture is the corresponding statement for stable cohomotopy, in which the representation ring is replaced by the Burnside ring A(G). This is the 1991 Mathematics Subject Classification. Primary 55P.

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“…The cohomology algebra of B is isomorphic with that of the principal block B 0 of the same type of fusion of subpairs as in B, which is isomorphic with the cohomology algebra of a finite group G with the principal block B 0 ; these cohomology algebra had already been calculated by the second author and others [3,13,15,17]. In the next section we shall give a condition for an element in the cohomology algebra of a defect group to belong to the cohomology algebra of the block, using a conjugation family for subpairs owing to Olsson [16], by which we can specify the subpairs that determine the cohomology algebras.…”

Section: Introductionmentioning

confidence: 97%

Cohomology algebras of 2-blocks of finite groups with defect groups of rank two

Kawai

Sasaki

2006

Journal of Algebra

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We shall calculate the cohomology algebras of 2-blocks of tame type and with wreathed defect groups; the cohomology algebra of B under consideration, in the sense of M. Linckelmann, is determined by particular B-subpairs, which we shall specify. In particular, depending our description of the cohomology algebras, we shall define "corestriction maps" between cohomology algebras of blocks which correspond under Brauer correspondence.

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Classification of BG for groups with dihedral or quarternion Sylow 2-subgroups

Cited by 26 publications

References 4 publications

Hochschild cohomology of tame blocks

Hochschild cohomology of tame blocks

Stably splitting BG

Cohomology algebras of 2-blocks of finite groups with defect groups of rank two

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