Some cohomology of finite general linear groups

TL;DR

This paper proves non-vanishing of certain cohomology groups for finite groups of Lie type over finite fields, providing explicit constructions and calculations for specific cases, advancing understanding of their algebraic structure.

Contribution

It establishes non-zero cohomology in specific degrees for finite Lie groups and explicitly constructs non-zero elements, with detailed calculations for general linear groups under certain conditions.

Findings

Non-zero cohomology in degree r(2p-3) for groups with Coxeter number ≤ p

Explicit construction of non-zero cohomology classes

Cohomology calculations for GL_n over finite fields when p=2 or r=1

Abstract

We prove that the degree $r(2p-3)$ cohomology of any (untwisted) finite group of Lie type over $\mathbb{F}_{p^r}$, with coefficients in characteristic $p$, is nonzero as long as its Coxeter number is at most $p$. We do this by providing a simple explicit construction of a nonzero element. Furthermore, for the groups $GL_n\mathbb{F}_{p^r}$, when $p=2$ or $r=1$, we calculate the cohomology in degree $r(2p-3)$, for all $n$.