# 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.

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### Profinite projective and free modules

I am studying cohomology of profinite groups and the following question came to my mind: suppose we have $G$ a pro-$p$ group which is Poincaré Dual of dimension $d$. This means that $\mathbb{Z}_p$ as ...
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### Group extensions with non-abelian kernel

If $N$ is a normal subgroup of $G$ then there is a coupling: that is, a representation of $G/N$ in $\operatorname{Out}(N)$. In that case, the extensions of $N$ by $G/N$ affording the same coupling are ...
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### Neutral cohomology classes and restriction maps for $H^2$ in group cohomology

$\DeclareMathOperator\res{res}$ Let $G$ be a profinite group. Let $A$ be a finite $G$-group (a finite group on which $G$ acts continuously), not necessarily abelian, with center $Z=Z(A)$. We say that ...
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### A road map through group cohomology

I am a PhD student in algebraic topology, and I would like to learn something about group cohomology. The final goal would be to present one or two seminars on this topic, in order to give my mates a ...
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### Finite $\Gamma$-modules with trivial $H^2$, where $\Gamma$ is a group of order 2

Let $\Gamma=\{1,\gamma\}$ be a group of order 2. Let $A$ be a finite $\Gamma$-module, that is, a finite abelian group on which $\Gamma$ acts. It is a hopeless problem to classify finite $\Gamma$-...
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### Constraints on the homology of amenable groups

Edit (March 24): My first question has been answered nicely, but I am still looking for an answer to the second one. Due to the Kan–Thurston theorem, the homology of an arbitrary group can be anything ...
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### Using HoTT, why is twisted cohomology of BG group cohomology?

I've been reading Michael Shulman's blog posts defining cohomology in homotopy type theory, and I'd like to understand (using HoTT) why cohomology of BG is group cohomology. if I understand correctly, ...
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### Standard terminology for “observable” subgroups of discrete groups

I've encountered in Bass, Lubotzky, Magid, and Mozes - The proalgebraic completion of rigid groups (Remark 1. p. 7) the following terminology: A normal subgroup $N$ of $G$ is observable if every $N$-...
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### Cohomology of finite $p$-groups over integers in local fields

Let $p$ be a prime, $G$ be a finite group of order $p^a$. Let $M$ be a $\mathbb{Z}[G]$-module. Then $H^n(G, M)$ is annihilated by $p^a$ for all $n \geq 1$ (see e.g. Brown, Corollary III.10.2). In ...
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### How many cells needed to build the classifying space $BG$?

Let $G$ be a finitely presented group of cohomological dimension $n$. Apart from the unresolved ambiguity pertaining to the Eilenberg--Ganea conjecture, it is known that we can find an $n$-dimensional ...
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### What is the meaning of this coboundary homomorphism for group hypercohomology?

$\require{AMScd}$ Let $\Gamma=\{1,\gamma\}$ be a group of order 2. In my problem from Galois cohomology of real reductive groups I came to a commutative diagram of $\Gamma$-modules (abelian groups ...