# 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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### Trivial homology groups for p-torsion groups

Let $G$ be a group where each element has a $p$-power order. Let $M$ be a $G$-module without $p$-torsion. Here $G$ is a discrete infinite subgroup of a complete group. Then, it cannot be assumed pro-...
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### Question on a theorem about group extensions

I am reading the chapter Second cohomology groups of Continuation of the Notas de Matemàtica. Part 1 in theorem 1.2 tells us that Let $E : 1 \to A \xrightarrow{i} X \xrightarrow{f} G \to 1$ be an ...
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### What is the least $n\ge1$ for which there is an $n$-dimensional closed flat manifold with perfect fundamental group?

In answer to the question "Is there a flat manifold with trivial first homology?" I proposed choosing a finite perfect group $P$ and a surjection $\phi:F\to P$ where $F$ is a free group of ...
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### Extensions of a simple group by an elementary abelian $p$-group

Let $V$ be an elementary abelian $p$-group of size $p^n$. Let $G$ be a finite group with $V\unlhd G$ such that $G/V=H$ is simple (like $\operatorname{PSL}(m,q)$ with $q$ a power of $p$ or any other ...
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### Representative manifold of $\mathbb{Z}_4^T$

I want to know which manifold is the representative manifold (I do not know the correct terminology in math) $\mathcal{M}$ of $\mathbb{Z}_4^T$ in the following sense: $\mathcal{M}$ is unorientable, ...
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### Mayer-Vietoris sequence in group cohomology for arbitrary pushout squares of groups?

Suppose that we have (not necessarily injective) group homomorphisms $H \to G_1$ and $H \to G_2$, and we construct the pushout (i.e. amalgamated free product) $G_1 \sqcup_H G_2$. Suppose that we have ...
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### Which finite groups have low-degree essential cohomology?

Let $G$ be a finite group, $A$ some coefficients (e.g. $A = \mathbb{F}_2$ or $\mathbb{Z}$), and write $\mathrm{H}^\bullet_{\mathrm{gp}}(G; A)$ for the (ordinary) group cohomology of $G$ with ...
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### Groups homology with coefficients fitting into filtration or exact sequence

Let $G$ be a group. I have two questions about the homology of $G$: Consider a finite exact sequence $$0 \rightarrow M_1 \rightarrow \cdots \rightarrow M_m \rightarrow 0$$ of $G$-modules. How are ...
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### Classification of crossed $G$-algebras

Added later: As Viktor Ostrik points out in a comment, what I'm looking for is a classification of so-called crossed $G$-algebras corresponding to homotopy TQFTs with homotopy target space $K(G, 1)$ ...
### Cohomologically trivial module $M$ such that $M/pM$ is not cohomologically trivial for some $p\in\mathbb{N}$
I was reading Brown's Cohomology Theory of Finite groups and was wondering whether there's an example of the following. Let $n\in \mathbb{N}$. Does there exist a $\mathbb{Z}_n$ module $M$ which is ...
### The optimal ranges for the integral homological stability of $\operatorname{GL}_n(F)$'s for a field $F$
$\DeclareMathOperator\GL{GL}$ $\DeclareMathOperator\co{H}$ $\DeclareMathOperator\ko{K}$ $\DeclareMathOperator\trd{tr-deg}$ $\DeclareMathOperator{\ch}{char}$Given a field $F$ and a homological degree \$...