Generating functions for the powers in GL(n, q)

Abstract

Consider the set of all powers GL(n, q)M = {xM ∣ x ∈ GL(n, q)} for an integer M ≥ 2. In this article, we aim to enumerate the regular, regular semisimple and semisimple elements as well as conjugacy classes in the set GL(n, q)M, i.e., the elements or classes of these kinds which are Mth powers. We get the generating functions (i) for regular and regular semisimple elements (and classes) when (q, M) = 1, (ii) for semisimple elements and all elements (and classes) when M is a prime power and (q, M) = 1, and (iii) for all kinds when M is a prime and q is a power of M.