On the modular cohomology of $GL_2(\mathbb{Z}/p^n)$ and $SL_2(\mathbb{Z}/p^n)$

TL;DR

This paper investigates the mod-$p$ cohomology of certain groups related to $GL_2$ and $SL_2$ over modular integers, showing stability properties in spectral sequences and describing stable elements via fusion systems.

Contribution

It proves that the $E_2$-page of the Lyndon-Hochschild-Serre spectral sequence remains independent of $n$ for specific group extensions, and characterizes the ring of stable elements using fusion systems.

Findings

Spectral sequence $E_2$-page is independent of $n>1$.

Stable elements are described via fusion systems.

Provides new insights into the cohomology of groups related to $GL_2$ and $SL_2$ over modular integers.

Abstract

Let $p$ be an odd prime. Denote a Sylow $p$-subgroup of $GL_2(\mathbb{Z}/p^n)$ and $SL_2(\mathbb{Z}/p^n)$ by $S_p(n,GL)$ and $S_p(n,SL)$ respectively. The theory of stable elements tells us that the mod-$p$ cohomology of a finite group is given by the stable elements of the mod-$p$ cohomology of it's Sylow $p$-subgroup. We prove that for suitable group extensions of $S_p(n,GL)$ and $S_p(n,SL)$ the $E_2$-page of the Lyndon-Hochschild-Serre spectral sequence associated to these extensions does not depend on $n>1$. Finally, we use the theory of fusion systems to describe the ring of stable elements.