Powers in finite unitary groups. (English) Zbl 07920052
Summary: Let \(\mathrm{U}(n, \mathbb{F}_{q^2})\) denote the subgroup of unitary matrices of the general linear group \(\mathrm{GL}(n, \mathbb{F}_{q^2})\) which fix a Hermitian form and let \(M \geq 2\) be an integer. In earlier work, elements of the groups \(\mathrm{GL}(n, \mathbb{F}_q)\), \(\mathrm{Sp}(2 n, \mathbb{F}_q)\), \(\mathrm{O}^\pm(2 n, \mathbb{F}_q)\) and \(\mathrm{O}(2 n + 1, \mathbb{F}_q)\) with an \(M\)-th root have been described. Here we will describe the \(M\)-th powers in unitary groups for the regular semisimple, semisimple and cyclic elements, under the assumption that \(M\) and \(q\) are coprime. We will further describe the generating functions in the corresponding cases.
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