Generating functions for the powers in $\text{GL}(n,q)$

Abstract

Consider the set of all powers $\text{GL}(n,q{)}^M=\{x^M\mid x\in \text{GL}(n,q)\}$ for an integer $M\ge 2$. In this article, we aim to enumerate the regular, regular semisimple and semisimple elements as well as conjugacy classes in the set $\text{GL}(n,q{)}^M$, i.e., the elements or classes of these kinds which are $M^{th}$ powers. We get the generating functions for (i) 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 $M$.