Asymptotic Bounds for the Size of Hom(A,GL_n(q))

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 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.