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

TL;DR

This paper derives generating functions to count powers and conjugacy classes in the general linear group over finite fields, focusing on regular, semisimple, and all elements for various conditions on M and q.

Contribution

It provides explicit generating functions for counting M-th powers and conjugacy classes in GL(n,q) under different algebraic conditions, extending previous enumeration methods.

Findings

Generated functions for regular and semisimple elements when (q,M)=1

Derived formulas for semisimple elements when M is a prime power and (q,M)=1

Established generating functions for all elements when M is a prime and q is a power of M

Abstract

Consider the set of all powers $\text{GL}(n ,q)^M = \{x^M \mid x\in \text{GL}(n, q)\}$ for an integer $M\geq 2$. 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 $M$.