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Results tagged with group-cohomology
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user 65801
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.
14
votes
Accepted
Can $\text{Aut}(G)$ be extended to contain $G$?
Jesper Grodal and I once looked at this, cf. this answer. In particular Eilenberg and MacLane constructs a universal obstruction in $H^3(\text{Out}(G);Z(G))$ for the extension $1\rightarrow G\rightarr …
- 1,907
12
votes
Accepted
Elementary $p$-subgroups of a compact Lie group
This is a topic with a long history going (at least) back to Borel, Serre, Steinberg and others. The existence of non-toral elementary abelian $p$-subgroups (i.e. subgroups not contained in a maximal …
- 1,907
11
votes
Accepted
For which subgroups the transfer map kills a given element of a group?
The answer to Q1 is yes, the order of $a$ might be smaller than the gcd: Let $G=\langle x,y\mid x^8=y^2=1,x^y=x^3\rangle$ be the semidihedral group of order $16$. Let $a=[x]\in G_{\text{ab}}=C_2\times …
- 1,907
7
votes
Embedding $G$ in a $Z(G)$ extension of $\operatorname{Aut}G$
As Derek Holt points out, the existence of such an extension is equivalent to the universal cohomology class in $H^3(\text{Out}(G);Z(G))$ being zero. The Eilenberg-MacLane paper is the original refere …
- 1,907
7
votes
What is the least $n\ge1$ for which there is an $n$-dimensional closed flat manifold with pe...
For $G=A_5=\text{PSL}_2(\mathbb{F}_5)$, the minimal dimension of a flat manifold with holonomy group $G$ is 12 15 according to Theorem (V.1) in W. Plesken:``Minimal dimensions for flat manifolds with …
- 1,907
6
votes
Accepted
Non-vanishing of the Tate-Shafarevich kernel in group cohomology
I think the following is an example of $Ш(G,M(G,H,\Bbb{F}_2))\neq 0$: Take $G=A_4$ and $H$ of order $2$. Then $M$ has dimension $5$ and a (computer) calculation shows that $Ш(G,M(G,H,\Bbb{F}_2))$ has …
- 1,907
5
votes
Accepted
Group homology for a metacyclic group
The name metacyclic is normally used for a group which is cyclic-by-cyclic (ie. a group $G$ with a cyclic normal subgroup $N$ such that $G/N$ is also cyclic). I will therefore refer to a finite group …
- 1,907
4
votes
Cohomology of the adjoint representation of $\mathrm{SL}_2(k)$
(Too long for a comment.) A Magma computation shows that for $k=\mathbf{F}_p$ with $p$ prime the group $H^1(\operatorname{PSL}_2(k);k^3)$ equals $0$ for $p=3$ and $7\le p\le 17$ while the cohomology g …
- 1,907
3
votes
Compute corestriction map on group cohomology in Magma
I based my quick answer (see comments above) on
https://magma.maths.usyd.edu.au/magma/handbook/text/814
which states:
Cocycles
Before invoking the functions in this section, it is necessary to first i …
- 1,907