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

For an integer $M\ge 2$ 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 $M^{\ast }$-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.