Cohomology Of Finite Groups Research Articles

Invariants of three-manifolds from finite group cohomology

We compare on lens spaces the values of two topological invariants of three-manifolds, both built from a finite group G and a three-cocycle ω, which we conjectured to be equal up to a normalization. The first invariant is defined by triangulation—it is the partition function of the Dijkgraaf-Witten topological field theory—and the second one by surgery, using a quasi-Hopf algebra. When G is a cyclic group, we show that the first invariant reduces to a Gauss sum. Some identities satisfied by three-cocycles are derived in an appendix.

On the cohomology of finite groups of Lie type

On the cohomology of finite groups of Lie type

On the cohomology of finite groups and the applications to modular representations

On the cohomology of finite groups and the applications to modular representations

On the cohomology of finite groups with simple coefficients

For a finite group G and a field k of characteristic p we denote by C i ( G) the intersection of the centralizers of the simple kG-modules with non-vanishing ith cohomology. It is well known that C 1( G)= O p' p ( G). For certain classes of groups we obtain detailed information on C i ( G), for i≥2. It is easy to translate our results about C i ( G) into results about the minimal projective resolution of k.

On Exponents of Homology and Cohomology of Finite Groups

Let $G$ be a finite group and let $r$ be the maximum of the $p$-ranks of $G$ for all primes $p$ dividing the order to $G$. There exist positive integers $m$ and $n$ such that the exponents of ${H^n}(G,{\mathbf {Z}})$ and ${H_m}(G,{\mathbf {Z}})$ are greater than $|G{|^{1/r}}$.

On exponents of homology and cohomology of finite groups

Let G G be a finite group and let r r be the maximum of the p p -ranks of G G for all primes p p dividing the order to G G . There exist positive integers m m and n n such that the exponents of H n ( G , Z ) {H^n}(G,{\mathbf {Z}}) and H m ( G , Z ) {H_m}(G,{\mathbf {Z}}) are greater than | G | 1 / r |G{|^{1/r}} .

Representing Tate cohomology of G-spaces

Tate cohomology of finite groups [5] is very good at emphasising periodic cohomological behaviour and hence at the study of free actions on spheres [8]. Tate cohomology of spaces was introduced by Swan [10] for finite dimensional spaces to systematically ignore free actions, and hence to simplify various arguments in fixed point theory.

Modular representations and the cohomology of finite groups

Since the classifying spaces BGL( K) and BGL δ( K) + ( K= R or C ) have the same mod p cohomology, Atiyah's theorem on the completion of the representation ring of a finite group can be interpreted as a theorem in algebraic K-theory. We discuss both this result and the analogue for finite fields, using facts about H ∗( G, Z Z r ) to recover information on R F q ( G).

A Generalization of the Cohomology of Groups

Generalizations of the cohomology of finite groups, in which one considers varying families of subgroups, are presented. These groups are shown to relate to Bredon equivariant of homology of universal $G$-spaces, and to lead to necessary algebraic conditions for $G$-actions on contractible spaces.

A generalization of the cohomology of groups

Generalizations of the cohomology of finite groups, in which one considers varying families of subgroups, are presented. These groups are shown to relate to Bredon equivariant of homology of universal G-spaces, and to lead to necessary algebraic conditions for G-actions on contractible spaces.

On the nontriviality of cohomology of finite groups

On the nontriviality of cohomology of finite groups

Cohomology of finite groups of Lie type, II

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Cohomology of finite groups of lie type, I

Cohomology of finite groups of lie type, I

Cohomology of finite groups and elementary Abelian subgroups

Cohomology of finite groups and elementary Abelian subgroups

On the cohomology of modules over Hopf algebras

Let R be a commutative ring. Define an FH-algebra H to be a Hopf algebra and a Frobenius algebra over R with a Frobenius homomorphism ψ such that ∑ ( h) h (1) ψ( h (2)) = ψ( h) · 1 for all h ϵ H. This is essentially the same as to consider finitely generated projective Hopf algebras with antipode. For modules over FH-algebras we develop a cohomology theory which is a generalization of the cohomology of finite groups. It generalizes also the cohomology of finite-dimensional restricted Lie algebras. In particular the following results are shown. The complete homology can be described in terms of the complete cohomology. There is a cup-product for the complete cohomology and some of the theorems for periodic cohomology of finite groups can be generalized. We also prove a duality theorem which expresses the cohomology of the “dual” of an H-module as the “dual” of the cohomology of the module. The last section provides techniques to describe under certain conditions the cohomology of H by the cohomology of sub- and quotient-algebras of H. In particular we have a generalization of the Hochschild-Serre spectral sequence for the cohomology of groups.

Characters and cohomology of finite groups

Characters and cohomology of finite groups

Quadratic forms over arbitrary fields

Introduction. Witt [5 ] proved that two binary or ternary quadratic forms, over an arbitrary field (of characteristic not 2) are equivalent if and only if they have the same determinant and Hasse invariant. His proof is brief and elegant but uses a lot of the theory of simple algebras. The purpose of this note is to make this fundamental theorem more accessible by giving a short proof using only the general results of modern theory of cohomology of finite groups. Professor Artin (Princeton lectures, 1956) gave such a proof for fields k over which local class field theory holds. For such fields, the group of values of the quadratic norm residue symbol is cyclic of order 2 while for arbitrary fields it may be any abelian group of exponent 2. So our proof is necessarily different from his; still it owes much to his methods.

A remark on fundamental exact sequences in cohomology of finite groups

A remark on fundamental exact sequences in cohomology of finite groups

Generalization of the concept of cohomology of finite groups

Generalization of the concept of cohomology of finite groups