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

TL;DR

This paper investigates the asymptotic behavior of the number of homomorphisms from a fixed finite group to general linear groups over finite fields, providing bounds, formulas, and an algorithm for key constants.

Contribution

It establishes asymptotic bounds for the size of Hom$(A,GL_n(q))$, including degree and leading coefficient formulas, and introduces an algorithm to compute related constants.

Findings

Polynomial size of Hom$(A,GL_n(q))$ in $q$

Degree of polynomial depends on $n$ and $a$

Algorithm for computing constants $ epsilon_r$ and $m_r$

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.