# Questions tagged [group-cohomology]

In mathematics, group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group G in an associated G-module M to elucidate the properties of the group.

756 questions
Filter by
Sorted by
Tagged with
160 views

1 vote
58 views

### Injectivity of a module over tensor product of group algebras

Let $R$ be a commutative ring and $G$ be a finite group that is the direct product $G=H_1*H_2$ of two subgroups $H_1$ and $H_2$ with $H_1$ Abelian. Let $R[G]$ be the group ring and $M$ be a left $RG$-...
152 views

### On the linearizability of the action of a finite group on a formal polydisc

$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Gl{Gl}$Let $\mathcal{D}=\mathbb{C}[[t_{1},\dotsc , t_{n}]]$ be a formal polydisc over $\mathbb{C}$, and $G$ be a finite group. On Lemma 7.8 of ...
1 vote
90 views

### Tensor product of group rings [closed]

Let $R$ be a commutative ring with unity and $\mathbb{Z}_n$ be the cyclic group of integers modulo $n$. If $R\mathbb{Z}_n$ is the group ring of $\mathbb{Z}_n$ formed over $R$, then I want to compute ...
94 views

### Tate cohomology and cup product: functoriality in $G$

Let $G$ be a finite group, and let $A, B$ be $G$-modules. See Atiyah and Wall [AW] for the definition of the Tate cohomology groups $H^q(G,A)$ for all $q\in\mathbb Z$ and of the cup product pairings \...
1 vote
84 views

173 views

### Is action of MCG on the curve complex computable for closed surfaces? [Yes: Birman Exact Sequence]

$\DeclareMathOperator\MCG{MCG}$Let $\Gamma=\MCG(S_{g, 0,0 })$ be mapping class group of closed hyperbolic surfaces. Let $V=C^0(S)$ be the set of vertices of the simplicial curve complex. We are ...
1 vote
188 views

### Number of generators of a group

Given a (finite) group $G$. Is there any bounds on the minimum number of generators $d(G)$? For example, it is clear that $d(G) \geqslant d(G^{ab})$. Where the right hand side can be easily computed. ...
76 views

### Virtual cohomological dimension of mapping class group and braid group of punctured surfaces

$\DeclareMathOperator\MCG{MCG}\DeclareMathOperator\vcd{vcd}\DeclareMathOperator\cd{cd}$Let $B_{k}(S_{g}),$ $\MCG(S_{g};k)$ and $\MCG(S_{g}))$ denote the braid group, the mapping class group (relative ...
236 views

$\DeclareMathOperator\cd{cd}$Recall that the rational cohomological dimension of a group $G$ is the supremum of the set of integers $k$ such that there exists a $\mathbb{Q}[G]$-module $M$ with $H^k(G;... 4 votes 1 answer 159 views ### Central extensions of torsion groups by$\mathbb{R}^n$Let$\Gamma$be a torsion group (i.e. every element has finite order). I am interested in understanding central extensions of the form:$\require{AMScd}$\begin{CD} 0 @>>> \mathbb{R}^n @>\... 4 votes 0 answers 129 views ### Structure of$\bigwedge^{2}_{\mathbb{Z}}(A)$with$A$a local integral domain I am trying to see the structure of$\bigwedge^{2}_{\mathbb{Z}}(A)$where$A$is a local integral domain with small residue field. Let$A$be a local integral domain with maximal ideal$M$, residue ... 8 votes 1 answer 348 views ### Finite group with squarefree order has periodic cohomology? Is it true that a finite group with squarefree order has periodic group cohomology (with trivial coefficients)? I cannot see why this would be the case, but I'm looking at a paper which seems to ... 8 votes 0 answers 114 views ### What are the stable cohomology classes of the "orthogonal groups" of finite abelian groups? Let$A$be a finite abelian group, and equip it with a nondegenerate symmetric bilinear form$\langle,\rangle : A \times A \to \mathrm{U}(1)$. Then you can reasonably talk about the "orthogonal ... 8 votes 0 answers 203 views ### Is there a finite group with nontrivial$H^2$but vanishing$H^4$,$H^5$, and$H^6$? Is there a finite group$G$such that the group cohomology$\mathrm{H}^2_{\mathrm{gp}}(G; \mathbb{Z}/2)$is nontrivial but$\mathrm{H}^4_{\mathrm{gp}}(G; \mathbb{Z}/2)$,$\mathrm{H}^5_{\mathrm{gp}}(G;...
I am looking for a reference for the assertion in the title. In more detail, let $\Gamma=\{1,\gamma\}$ be a group of order 2. Let $A$ be a $\Gamma$-module (an abelian group on which $\Gamma$ acts). ...