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

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.