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 $M\geq 2$ 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.
Categories: math.GR math.CO