On the cohomology of certain groups

1961 ◽  
Vol 57 (4) ◽  
pp. 731-733 ◽  
Author(s):  
C. T. C. Wall
Keyword(s):  
Finite Group ◽  
Differential Operator ◽  
Spectral Sequence ◽  
Large Class ◽  
Recent Work ◽  
Finite Groups ◽  
Classifying Spaces ◽  
Representation Ring ◽  
Grothendieck Rings ◽  
A Finite Group

Recent work of Atiyah (1) on the Grothendieck rings of classifying spaces of finite groups has yielded, among many other interesting results, a spectral sequence relating the simple integer cohomology of a finite group to its representation ring. He has determined the first differential operator of the spectral sequence which affects the p-part of the cohomology, and the question naturally arises, when is it the only non-zero one. My object in this note is to show that this is so for a large class of groups.

HOMOLOGY DECOMPOSITIONS FOR CLASSIFYING SPACES OF FINITE GROUPS ASSOCIATED TO MODULAR REPRESENTATIONS

2001 ◽  
Vol 64 (2) ◽  
pp. 472-488 ◽  
Author(s):  
D. NOTBOHM
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Homotopy Theory ◽  
Modular Representation ◽  
Modular Representations ◽  
Classifying Spaces ◽  
Modular Representation Theory ◽  
Classifying Space ◽  
Homotopy Colimit ◽  
A Finite Group

For a prime p, a homology decomposition of the classifying space BG of a finite group G consist of a functor F : D → spaces from a small category into the category of spaces and a map hocolim F → BG from the homotopy colimit to BG that induces an isomorphism in mod-p homology. Associated to a modular representation G → Gl(n; [ ]p), a family of subgroups is constructed that is closed under conjugation, which gives rise to three different homology decompositions, the so-called subgroup, centralizer and normalizer decompositions. For an action of G on an [ ]p-vector space V, this collection consists of all subgroups of G with nontrivial p-Sylow subgroup which fix nontrivial (proper) subspaces of V pointwise. These decomposition formulas connect the modular representation theory of G with the homotopy theory of BG.

scholarly journals Regulator constants of integral representations of finite groups

2018 ◽  
Vol 168 (1) ◽  
pp. 75-117 ◽  
Author(s):  
ALEX TORZEWSKI
Keyword(s):  
Finite Group ◽  
Elliptic Curves ◽  
Finite Groups ◽  
Isomorphism Class ◽  
Integral Representations ◽  
Dihedral Groups ◽  
Representation Ring ◽  
Representations Of Finite Groups ◽  
Cohomological Invariant ◽  
A Finite Group

AbstractLet G be a finite group and p be a prime. We investigate isomorphism invariants of $\mathbb{Z}_p$[G]-lattices whose extension of scalars to $\mathbb{Q}_p$ is self-dual, called regulator constants. These were originally introduced by Dokchitser–Dokchitser in the context of elliptic curves. Regulator constants canonically yield a pairing between the space of Brauer relations for G and the subspace of the representation ring for which regulator constants are defined. For all G, we show that this pairing is never identically zero. For formal reasons, this pairing will, in general, have non-trivial kernel. But, if G has cyclic Sylow p-subgroups and we restrict to considering permutation lattices, then we show that the pairing is non-degenerate modulo the formal kernel. Using this we can show that, for certain groups, including dihedral groups of order 2p for p odd, the isomorphism class of any $\mathbb{Z}_p$[G]-lattice whose extension of scalars to $\mathbb{Q}_p$ is self-dual, is determined by its regulator constants, its extension of scalars to $\mathbb{Q}_p$, and a cohomological invariant of Yakovlev.

scholarly journals Brauer relations in finite groups II: Quasi-elementary groups of order paq

2014 ◽  
Vol 17 (3) ◽  
Author(s):  
Alex Bartel ◽  
Tim Dokchitser
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Burnside Ring ◽  
Rational Representation ◽  
Generators And Relations ◽  
Representation Ring ◽  
A Finite Group

Abstract.This is the second in a series of papers investigating the space of Brauer relations of a finite group, the kernel of the natural map from its Burnside ring to the rational representation ring. The first paper classified all primitive Brauer relations, that is those that do not come from proper subquotients. In the case of quasi-elementary groups the description is intricate, and it does not specify groups that have primitive relations in terms of generators and relations. In this paper we provide such a classification in terms of generators and relations for quasi-elementary groups of order

scholarly journals POWER MAP PERMUTATIONS AND SYMMETRIC DIFFERENCES IN FINITE GROUPS

2011 ◽  
Vol 10 (05) ◽  
pp. 947-959 ◽  
Author(s):  
MÁRTON HABLICSEK ◽  
GUILLERMO MANTILLA-SOLER
Keyword(s):  
Finite Group ◽  
Large Class ◽  
Finite Groups ◽  
Difference Method ◽  
Symmetric Difference ◽  
Quadratic Reciprocity ◽  
Equivariant Maps ◽  
Odd Order ◽  
A Finite Group

Let G be a finite group. For all a ∈ ℤ, such that (a, |G|) = 1, the function ρa: G → G sending g to ga defines a permutation of the elements of G. Motivated by a recent generalization of Zolotarev's proof of classic quadratic reciprocity, due to Duke and Hopkins, we study the signature of the permutation ρa. By introducing the group of conjugacy equivariant maps and the symmetric difference method on groups, we exhibit an integer dG such that [Formula: see text] for all G in a large class of groups, containing all finite nilpotent and odd order groups.

scholarly journals Finite groups with some weakly pronormal subgroups

2020 ◽  
Vol 18 (1) ◽  
pp. 1742-1747
Author(s):  
Jianjun Liu ◽  
Mengling Jiang ◽  
Guiyun Chen
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
A Finite Group

Abstract A subgroup H of a finite group G is called weakly pronormal in G if there exists a subgroup K of G such that G = H K G=HK and H ∩ K H\cap K is pronormal in G. In this paper, we investigate the structure of the finite groups in which some subgroups are weakly pronormal. Our results improve and generalize many known results.

Stable decompositions of classifying spaces of finite abelianp-groups

1988 ◽  
Vol 103 (3) ◽  
pp. 427-449 ◽  
Author(s):  
John C. Harris ◽  
Nicholas J. Kuhn
Keyword(s):  
Finite Group ◽  
Cohomology Theory ◽  
Classifying Spaces ◽  
Classifying Space ◽  
Image Position ◽  
A Finite Group

LetBGbe the classifying space of a finite groupG. Consider the problem of finding astabledecompositionintoindecomposablewedge summands. Such a decomposition naturally splitsE*(BG), whereE* is any cohomology theory.

scholarly journals A on simplicity criterion for finite groups

1969 ◽  
Vol 10 (3-4) ◽  
pp. 359-362
Author(s):  
Nita Bryce
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Odd Order ◽  
A Finite Group

M. Suzuki [3] has proved the following theorem. Let G be a finite group which has an involution t such that C = CG(t) ≅ SL(2, q) and q odd. Then G has an abelian odd order normal subgroup A such that G = CA and C ∩ A = 〈1〉.

On S-Quasinormal and C-Normal Subgroups of Prime Power Order in Finite Groups

2011 ◽  
Vol 18 (04) ◽  
pp. 685-692
Author(s):  
Xuanli He ◽  
Shirong Li ◽  
Xiaochun Liu
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Power ◽  
Maximal Subgroups ◽  
Prime Power Order ◽  
Normal Subgroups ◽  
A Finite Group

Let G be a finite group, p the smallest prime dividing the order of G, and P a Sylow p-subgroup of G with the smallest generator number d. Consider a set [Formula: see text] of maximal subgroups of P such that [Formula: see text]. It is shown that if every member [Formula: see text] of is either S-quasinormally embedded or C-normal in G, then G is p-nilpotent. As its applications, some further results are obtained.

scholarly journals Towards a classification of rigid product quotient varieties of Kodaira dimension 0

2021 ◽  
Author(s):  
Ingrid Bauer ◽  
Christian Gleissner
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Structure Theorem ◽  
Complete Classification ◽  
Kodaira Dimension ◽  
The Other ◽  
Diagonal Action ◽  
Quotient Varieties ◽  
A Finite Group

AbstractIn this paper the authors study quotients of the product of elliptic curves by a rigid diagonal action of a finite group G. It is shown that only for $$G = {{\,\mathrm{He}\,}}(3), {\mathbb {Z}}_3^2$$ G = He ( 3 ) , Z 3 2 , and only for dimension $$\ge 4$$ ≥ 4 such an action can be free. A complete classification of the singular quotients in dimension 3 and the smooth quotients in dimension 4 is given. For the other finite groups a strong structure theorem for rigid quotients is proven.

Finite Groups with Some Subgroups of Sylow Subgroups s∗-Semipermutable

2021 ◽  
Vol 58 (2) ◽  
pp. 147-156
Author(s):  
Qingjun Kong ◽  
Xiuyun Guo
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Subnormal Subgroup ◽  
Sylow Subgroups ◽  
A Finite Group

We introduce a new subgroup embedding property in a finite group called s∗-semipermutability. Suppose that G is a finite group and H is a subgroup of G. H is said to be s∗-semipermutable in G if there exists a subnormal subgroup K of G such that G = HK and H ∩ K is s-semipermutable in G. We fix in every non-cyclic Sylow subgroup P of G some subgroup D satisfying 1 < |D| < |P | and study the structure of G under the assumption that every subgroup H of P with |H | = |D| is s∗-semipermutable in G. Some recent results are generalized and unified.

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