Asymptotic Bounds for the Size of Hom$(A,{\rm GL}_n(q))$

Abstract

Fix an arbitrary finite group $A$ of order $a$, and let $X(n,q)$ denote the set of homomorphisms from $A$ to the finite general linear group ${\rm GL}_n(q)$. The size of $X(n,q)$ is a polynomial in $q$. In this note it is shown that generically this polynomial has degree $n^2(1-a^{-1}) - \epsilon_r$ and leading coefficient $m_r$, where $\epsilon_r$ and $m_r$ are constants depending only on $r := n \mod a$. We also present an algorithm for explicitly determining these constants.