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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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        6
        votes
        0answers
        121 views

        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 ...
        3
        votes
        0answers
        133 views

        Group cohomology: Why does the trivial Z coefficient produce nontrivial cohomology [closed]

        Let $G$ be a group and $M$ be a $G$-module. Then group cohomology $H^q(G,M)$ is defined as the right derived functor $\operatorname{Ext}^q_{\mathbb Z G}(\mathbb Z,M)$. Here $\mathbb Z$ is the trivial $...
        15
        votes
        1answer
        475 views

        Characteristic classes of symmetric group $S_4$

        For the symmetric group $S_3$, it is classically known that \begin{equation} H^*(S_3;\mathbb{Z})\cong \mathbb{Z}[x,y]/(2x,6y,x^2-3y), \end{equation} where $|x|=2$ and $|y|=4$. Moreover, $x$ can be ...
        8
        votes
        0answers
        321 views

        Homology of a quotient space defined by an equivalence relation

        Let $(X,x)$ be a pointed connected CW-complex. Let $f:X\rightarrow X$ be a (pointed) homeomorphism. Denote $Y=X\vee X$ and $Y^{ n}=Y\times\cdots \times Y$ $n$-times. Lets define a new homeomorphism $h:...
        2
        votes
        0answers
        72 views

        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 ...
        4
        votes
        1answer
        342 views

        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 $\mathbb{R}^{\times}$....
        3
        votes
        0answers
        99 views

        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 ...
        9
        votes
        0answers
        172 views

        Cohomology of $\operatorname{SO}(p,q;\mathbb{Z})$ with $p=3,q=19$

        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,...
        4
        votes
        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 \...
        16
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        0answers
        492 views

        How is this group theoretic construct called?

        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 \...
        7
        votes
        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}{...
        3
        votes
        2answers
        232 views

        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 ...
        11
        votes
        1answer
        179 views

        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 ...
        8
        votes
        1answer
        363 views

        Cohomology of simple finite groups remembers the group?

        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})...
        3
        votes
        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 \begin{equation}H^n(G,M) = \bigoplus_p H^n(H,M)^G\end{equation} where $p$...

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