# Questions tagged [group-cohomology]

a tool used to compute invariants of group actions using methods from homology theory, such as invariants, coinvariants, extensions... Use with (homology-cohomology).

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### Cap product of infinite cyclic group in degree one

I have a question about the cap-product in group cohomology. Maybe someone can help me out or clarify a mistake in my mind. Suppose we have an infinite cyclic group $G$ with fixed generator $g$. Then ...
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### The second chomology group $H^{2}(Z_{3}.M_{22},V)$, where $V$ is the 6-dimensional vectors space over GF(4)

Is there is anyone that could help me in evaluating $H^{2}(Z_{3}.M_{22},V)$, where $V$ is the 6-dimensional vector space over GF(4)? Thank you very much in advance. Goranac
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### Relations between the three different descriptions of 2nd cohomology group in Group Cohomology

I am coming at the 2nd cohomology group in Group Cohomology from the perspective of the Group Extension Problem (or rather the group central extension problem, which perhaps more closely related to ...
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### Is it possible to recover the Cartan-Leray Spectral Sequence for Group Cohomology from the Leray Spectral Sequence for Sheaf Cohomology?

Let $G$ be a discrete group acting freely and cellularily on a CW-complex $X$. I am interested in the Cartan-Leray spectral sequence from Eilenberg and Cartan's Homological Algebra, Theorem XVI.8.4, ...
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### Does every extension of a finite group by $\mathbb{R}^n$ split?

Suppose $G$ is a topological group containing a closed normal subgroup $N$ isomorphic to $(\mathbb{R}^n, +)$ such that $G/N$ is finite. Is $N$ a semidirect factor? Equivalently, does $G$ contain a ...
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### Two group cohomology are isomorphic to each other if they have the same coefficient.

Suppose $M$ and $N$ are $\mathbb{F}[G]$-modules over a field $\mathbb{F}$, and $G$ is a finite group. Claim: Suppose there is a map $f: M \rightarrow N$ such that $f$ is a $\mathbb{F}$-vector space ...
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### Restriction map in semilocal Galois cohomology

Let $F/K$ be a finite extension of number fields, let $v$ be a prime of $K$ that is totally split in $F$. Call $w_1,\dots, w_n$ the primes of $F$ above $v$. Let $T$ be a $Gal(\bar{K}/K)$-module. Then ...
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### Group Cohomology, Module Extensions, and Group Extensions, and $Ext^2_{\mathbb{Z}G}(\mathbb{Z},A)$

I've read that for some $G$-module $A$, group cohomology can be defined as $$H^{n}(G,A)=Ext^{n}_{\mathbb{Z} G}(\mathbb{Z},A).$$ I've also read that for two $R$-modules $C,D,$ $Ext^{n}_{R}(C,D)$ can be ...
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### A map in group cohomology from $H^2(G,\widehat{G})$ to $H^3(G,U(1))$

Let $G$ be an abelian finite group, and denote by $\widehat{G}$ its Pontriajin dual, which is (non-canonically) isomorphic to $G$. I found the statement that there exists a map in group cohomology \...
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### Group Cohomology and Change of Coefficients

Let $G$ be a group and $V$ be a representation of $G$ over a field $K$. Let $L$ be any field containing $K$. Consider the natural morphism $$\varphi:H^i(G,V)\otimes_KL\rightarrow H^i(G,V\otimes_KL).$$ ...
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