# Tagged Questions

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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### Is there a characterization of groups with the property $\forall N\unlhd G,\:\exists H\leq G\text{ s.t. }H\cong G/N$?

A common mistake for beginning group theory students is the belief that a quotient of a group $G$ is necessarily isomorphic to a subgroup of $G$. Is there a characterization of the groups in which ...
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### What is the motivation for defining both homogeneous and inhomogeneous cochains?

In my few months of studying group cohomology, I've seen two "standard" complexes that are introduced: We let $X_r$ be the free $\mathbb{Z}[G]$-module on $G^r$ (so, it has as a $\mathbb{Z}[G]$-basis ...
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### Cohomology of a group of order two with coefficients in a finite abelian group of odd order

I am looking for an elementary proof that the cohomology groups in the title are trivial in the positive degrees. In more detain, let $G=\{1,s\}$ be a group of order two, and let $A$ be an abelian ...
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### Torsion-free virtually-Z is Z

It is well known that a torsion-free group which is virtually free must be free, by works of Serre, Stallings, Swan... Is there a simple cohomological proof of the fact that a torsion-free group ...
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### Group extensions of cyclic groups

Let $A$ be an infinite cyclic group and $B$ be a cyclic group of order $n$. Suppose $$0 \to A \to G \to B \to 0$$ is a short exact sequence of abelian groups. What could $G$ be? It is clear enough ...
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### What are the homology groups of an abelian group?

What are the homology groups of an abelian group? I know there are simple answers in certain cases (e.g. I believe $H_2(A; \mathbb{Z}) = \wedge^2 A$), but it's surprisingly difficult to find any ...
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### $\text{Hom}(\mathbb{F}_p G, M)$ and $H^1(G,M)$

I'm trying to read (part of) "The Presentation Rank of a Direct Product of Finite Groups" / Cossey, Gruenberg, Kovacs (Journal Of Alegebra 28, 597-603 (1974)). Here are some basic assertions I need ...
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### Group cohomology and Shapiro's lemma

This is a stupid question about group cohomology, but it confuses me a lot. Basically I think that the problem is the fact that I do not really understand Shapiro's lemma. Say we take a profinite ...
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I´m reading the Zig Zag lemma in Cohomology and i want to prove the exactness of cohomology sequence at $H^k(A)$ and $H^k(B)$ : A short exact sequence of cochain complexes $0 \to A \ \xrightarrow{i}... 0answers 148 views ### An equivalence between group cohomology and sheaf cohomology I'm recently reading group cohomology from Serre's book local fields, and he uses there the following terminology$H^q(G,A)$the q-th degree cohomology of G with coefficent in$A$. So i started to ... 0answers 145 views ### Does$f(gh)=gf(h)+f(g)$make$H^1(G,A)$small enough? I'm currently reading the text Beginnings of Group Cohomology, trying to understand$H^1(G,A)$for$G$a group and$A$a$G$-module. In the article, the space$H^1(G,A)$is motivated as a space of ... 2answers 89 views ### Showing that$\operatorname {Br}(\Bbb F_q)=0$I want to prove that$\operatorname {Br}(\Bbb F_q)=0$using the cohomological description of the Brauer group. We have:$\operatorname {Br}(\Bbb F_q)=H^2(\operatorname {Gal}(\overline {\Bbb F_q}/\...
I've been trying to understand homomorphisms from a finite group $G$ into $\operatorname{PGL}(n,R)$ for $n$ a positive integer, and $R$ a commutative ring with 1, usually a field. I had been under ...