p-Divisibility of the number of linear representations of an Abelian p-group

TL;DR

This paper investigates the p-divisibility properties of the number of homomorphisms from an Abelian p-group to general linear groups over finite fields, providing lower bounds and extending known results in the area.

Contribution

It establishes new lower bounds for p-divisibility of homomorphism counts from Abelian p-groups to GL_n over finite fields, generalizing previous work on symmetric groups.

Findings

Derived lower bounds for p-divisibility of homomorphism counts

Extended results analogous to those for symmetric groups

Enhanced understanding of homomorphism enumeration in finite group representations

Abstract

We establish lower bounds for the $p$-divisibility of the quantity $\#\operatorname{Hom}(G,GL_n(\mathbb{F}_q))$, the number of homomorphisms from $G$ to a general linear group, where $G$ is an Abelian $p$-group. This is in analogy to the result of Krattenthaler and M\"{u}ller \cite{MR3383810} on homomorphisms to symmetric groups.