Powers in finite unitary groups

Abstract

Let $\text{U}(n,{\mathbb{F}}_{q^2})$ denote the subgroup of unitary matrices of the general linear group $\text{GL}(n,{\mathbb{F}}_{q^2})$ which fixes a Hermitian form and $M\ge 2$ an integer. This is a companion paper to the previous works where the elements of the groups $\text{GL}(n,{\mathbb{F}}_q)$, $\text{Sp}(2n,{\mathbb{F}}_q)$, ${\text{O}}^{\pm }(2n,{\mathbb{F}}_q)$ and $\text{O}(2n+1,{\mathbb{F}}_q)$ which has an $M$-th root in the concerned group, have been described. Here we will describe the $M$-th powers in unitary groups for the regular semisimple, semisimple and cyclic elements. Our methods are parallel to those of the Memoir ``A generating function approach to the enumeration of matrices in classical groups over finite fields" by Fulman, Neumann and Praeger.

11 pages; preliminary version; comments are welcome