0
votes
1answer
60 views

1-Cocycle of an Algebra

Is there a good definition of a 1-Cocycle of an algebra A that is relatively easy to understand? I am rather new to cohomology and representation theory, but it seems like this is a fundamental ...
5
votes
2answers
90 views

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 ...
3
votes
1answer
95 views

First group cohomology and composition factors

Let $G$ be a finite group. Let $k$ be a field ($\text{char}(k)=p>0$). Let $P(k)$ be the projective cover of $k$. Assume that for any nontrivial simple $kG$-module $M$ we have $H^1(G,M)=0$. Does it ...
3
votes
1answer
135 views

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 ...
25
votes
2answers
551 views

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 ...
1
vote
2answers
142 views

modern treatment of the triviality of the multiplicator of the finite special linear group

Given your favorite version of the Heisenberg group one can prove the Stone-Von Neumann theorem. It is then not to hard to construct a family of representations on a central extension of $Sp\left(2n, ...