# Tagged Questions

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### Cohomology of finite groups with finite coefficients

I'm wondering if the group cohomology of a finite group $G$ can be made nontrivial with a nice choice of a finite $G$-module M. In other words, given a finite group $G$ and a number $n$, does there ...
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### Extensions of elementary abelian p-groups by themselves.

How many distinct extensions of $C^k_p$ by $C^l_p$ are there where $C^n_p$ is elementary abelian of order $p^n$, p a prime?
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### Non vanishing of group cohomology

Let $G$ be a finite group, then $H^n(G,\mathbb{Z})\neq 0$ for infinitely many $n$. This is not hard to see for cyclic groups. Can we prove this fact algebraically, could anyone provide a reference? ...
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### Are homomorphisms into PGL related to the Schur multiplier?

I've been trying to understand homomorphisms from a finite group $G$ into $\operatorname{PGL}(n,R)$ for $n$ a positive integer, and $R$ a commutative ring with 1, usually a field. I had been under ...
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### Finite groups such that $H^1(G,M)=0$ for any simple $G$-module $M$

I'm trying to understand for which finite groups $G$ the augmentation ideal of $\mathbb{F}_2G$ is generated by a single element over $\mathbb{F}_2G$. I'm reading a paper with a result that implies ...
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### What does the group ring $\mathbb{Z}[G]$ of a finite group know about $G$?

The group algebra $k[G]$ of a finite group $G$ over a field $k$ knows little about $G$ most of the time; if $k$ has characteristic prime to $|G|$ and contains every $|G|^{th}$ root of unity, then ...
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### Ring structure of non-modular group cohomology

I know if $k$ is a field such that $char(k)$ divides $|G|$ (a finite group), then finding the ring structure on $H^\ast(G,k)$ can be very, very hard. But what about when $k=\mathbb{Z}$? Is the ...
### All possible extensions of $S_3$ by $\mathbb{Z}$?
Write $S_3$ for the symmetric group on 3 letters. The question: What are the possible extensions of $S_3$ by $\mathbb{Z}$ (up to equivalence)? (To avoid ambiguity, by an extension of $G''$ by ...