# Tagged Questions

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### Known computations of certain 2-cohomology groups?

I wanted to know if there are any computations of cohomology groups $H^n(\Gamma,A^{(\Gamma)})$ in the literature for certain $n\in\mathbb{N}$, Abelian groups $A$, and infinite groups $\Gamma$. Here ...
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### Non split extension isomorphic (as a group) to a split extension

$\def\Z{\mathbb{Z}}$ Let $A$ be a finite abelian group and $G$ a finite group acting on $A$. Then the extension $0 \to A \to E \to G \to 1$ is splits if and only if the corresponding $2$-cocycle is ...
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### Group extensions isomorphic as groups

Let $G$ be a group and $A$ a $G$-module. It well know that there is a group isomorphism between the second cohomologoy group $H^2(G,A)$ and the abelian group $OpExt(G,A)$ of classes of extension ...
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### Groups such that all finite-dim representations are finitely presented

Let $G$ be an infinite group. What sorts of finiteness properties can I put on $G$ to ensure the following holds for all $M$? Let $M$ be a finite-dimensional vector space over $\mathbb{Q}$ upon ...
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### 3-cocycle representatives for the dihedral group $D_{2n}$?

I am looking for a reference for a complete list of 3-cocycle representatives for $H^3(D_{2n},\mathbb{C}^\times)$, where $$D_{2n}=\langle a, b\mid a^2=b^2=(ab)^n=e\rangle$$ is the dihedral group of ...
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### Cohomological dimension of a homomorphism

Let $G$ and $\Gamma$ be discrete groups, and let $\phi\colon\thinspace G\to \Gamma$ be a homomorphism. Define its cohomological dimension $\operatorname{cd}\phi$ to be the least integer $d$ such that ...
For any group G we can form the lower central series of normal subgroups by taking $G_0 = G$, $G_1 = [G,G]$, $G_{i+1} = [G,G_i]$. We can check this gives a normal chain $G_0 > G_1 > ... > ... 6answers 2k views ### Characterization of the transfer map in group theory Let$i : H \to G$be a subgroup of finite index. The transfer map is a special homomorphism$V(i) : G^\mathrm{ab} \to H^\mathrm{ab}$. The usual ad hoc definition uses a set of representatives of$H$... 0answers 304 views ### Inseparable Galois Cohomology First let me give a general form of my question, and then I'll give some motivation and a more specific version of it. Let$K/k$be a Galois extension of fields with Galois group$G$, and let$X$be ... 1answer 359 views ### Hirsch length and cohomological dimension It seems to be known that Hirsch length and cohomological dimension agree for (torsion-free, finitely generated) polycyclic groups. If we drop the assumption "torsion-free", then cd is of course ... 1answer 351 views ### Posets of cosets and contractibility For this question let$G$be a group, perhaps infinite, and let$H_i$for$i\in I$be a (finite) family of subgroups closed under taking intersections. I am interested in the coset poset ... 1answer 925 views ### Grothendieck's question on the Brauer group for groups Let$G$be a group, and let$M(G)=H^2(G,\mathbb{C}^*)$be the Schur multiplier of$G$. There is a group$Br(G)$of complex projective representations of$G$modulo those that can be lifted to linear ... 1answer 341 views ### Examples of p-groups exhibiting isomorphic mod-p cohomology rings. Hello, given 2 different finite$p$-groups$G$and$H$,$|G|=|H|=p^n$. It has been shown by Ian Leary that the mod-p cohomology rings do not determine the groups$G$and$H$. In fact he gave an ... 4answers 2k views ### Non-vanishing of group cohomology in sufficiently high degree Atiyah in his famous paper , Characters and cohomology of finite groups, after proving completion of representation ring in augmentation ideal is the same as$ K(BG)$, gives bunch of corollaries of ... 7answers 1k views ### universal cover of SL2(R): does it admit central extensions? Is it true that the universal cover of SL2(ℝ) has no non-trivial central extensions... as an abstract group? (that's certainly true as a Lie group) Motivation: I have a projective action of ... 3answers 391 views ### Embedding groups into groups with some vanishing homology groups Which finite subsets$S \subset \mathbb{N}$have the following property : every countable group$G$embeds into a finitely generated group$\Gamma$such that$H_i(\Gamma;\mathbb{Z})=0$for all$i \in ...
Is it true that a finitely generated subgroup of a cartesian product of free groups has a finite cohomological dimension? The same question about pro-$p$ groups: Is it true that a finitely generated ...