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Given a finite field $\mathbb{F}$ with $|\mathbb{F}|=q=p^m\geq4$ where $p=\text{char}(\mathbb{F})$, I'm wondering is there a characterization of the kernel of the map $f:H_3(\text{GL}_3(\mathbb{F}))\...

(I asked some people this question in person and got the answer "no", but wanted to see if the Internet had more to say)$ \newcommand{\FinGrp}{\mathbf{FinGrp}} $ Way back in my first group theory ...

Let $G$ be a finite group with finite-dimensional irreducible representations $\rho_i:G\to\mathrm{GL}_{n_i}(k)$ over a field $k$ indexed by $i=1,...,m$. These compose with the canonical map $\mathrm{...