Powers in finite orthogonal and symplectic groups: A generating function approach

TL;DR

This paper develops a generating function approach to classify and enumerate M-th powers in finite symplectic and orthogonal groups over finite fields, providing explicit probabilities for such elements.

Contribution

It introduces M*-power SRIM polynomials and classifies M-th power elements in classical groups using generating functions, extending previous enumeration methods.

Findings

Complete classification of M-th powers in classical groups

Explicit generating functions for enumeration

Probabilities of elements being M-th powers

Abstract

For an integer $M\geq 2$ and a finite group $G$, an element $\alpha\in G$ is called an $M$-th power if it satisfies $A^M=\alpha$ for some $A\in G$. In this article, we will deal with the case when $G$ is finite symplectic or orthogonal group over a field of order $q$. We introduce the notion of $M^*$-power SRIM polynomials. This, amalgamated with the concept of $M$-power polynomial, we provide the complete classification of the conjugacy classes of regular semisimple, semisimple, cyclic and regular elements in $G$, which are $M$-th powers, when $(M,q)=1$. The approach here is of generating functions, as worked on by Jason Fulman, Peter M. Neumann, and Cheryl Praeger in the memoir "A generating function approach to the enumeration of matrices in classical groups over finite fields". As a byproduct, we obtain the corresponding probabilities, in terms of generating functions.