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# 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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### Finite generation of group homology

I am reading 'Subgroups of direct products of limit groups' of Bridson, Howie, Miller and Short (http://annals.math.princeton.edu/wp-content/uploads/annals-v170-n3-p11-p.pdf) and I am finding similar ...
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### Coboundary in Kummer theory

Let $K$ be a non archimedean local field whose residue field is of characteristic $p$. Denote by $G$ the absolute Galois group of $K$. Denote by $\mu_p$ the group of $p$-roots of unity and assume it ...
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### First homology group of the general linear group

The abelianization of the general linear group $GL(n,\mathbb{R})$, defined by $$GL(n,\mathbb{R})^{ab} := GL(n,\mathbb{R})/[GL(n,\mathbb{R}), GL(n,\mathbb{R})],$$ is isomorphic to $\mathbb{R}^{\times}$....
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### Topological derivation of the Laplace formula for determinants in Euclidean space

I'm interested in trying to derive the Leibniz formula for the determinant on real Euclidean space, without first constructing $\det$ by axiomatizing its properties. We know $GL(n)$ deformation ...
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I would like to understand the topology of the moduli space of Einstein orbifold metrics on the $K3$-surface. It is known that this space is given by the bi-quotient $SO(3,19;\mathbb{Z})\setminus SO(3,... 1answer 83 views ### Higher cohomology for trivial module for finite groups of Lie type Is anything known about the cohomology past$\mathrm{H}^1$and$\mathrm{H}^2$for the trivial module for a finite group of Lie type in cross characteristic? For the moment I just care about$\dim \...
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Let $G$ be a finite group, $S\subset G$ a generating set, $|g| = |g|_S =$ word length with respect to $S$. Define the "defect" of $g,h$ to be $$\psi(g,h) = |g|+|h|-|gh|$$ Then $\psi:G\times G \... 1answer 301 views ### Abstract proof that$\lvert H^2(G,A)\rvert$counts group extensions (This question is originally from Math.SE, where it didn't receive any answers.)$\DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\ext}{...
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### How many non-isomorphic extensions with kernel $S^1$ and quotient cyclic of order $p$?

I want to determine how many non-isomorphic extensions (as group they are non-isomorphic) are possible of the form $1 \to \mathbb{S}^1 \to G \to (\mathbb{Z}_p)^k \to 1$, where $G$ is a compact lie ...
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### Reference requests: Integral cohomology of $G_2$-homogeneous spaces

Do you know a place where the integral cohomology of $G_2$-homogeneous spaces is computed? Great computational efforts using representation theory in order to determine the ...
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Let $G$ and $H$ be finite simple groups. I expect that if $G$ and $H$ are not isomorphic, then their cohomology groups with integral coefficients are not all isomorphic, that is, $H^*(G,\mathbb{Z})... 1answer 147 views ### Group cohomology of$S_3$in terms of its Sylow subgroups I am trying to understand$H^*(S_3, M)$in terms of it's Sylow$p$subgroups. From III.10.2 and III.10.3 in Brown we know that $$H^n(G,M) = \bigoplus_p H^n(H,M)^G$$ where$p\$...

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