Asymptotics of the powers in finite reductive groups
The word maps on finite groups of Lie type and algebraic groups have been studied extensively in the last years. Let \(G\) be a connected reductive group over an algebraic closure of a finite field \(\mathbb{F}_q\). Let \(F\) be a Frobenius map on \(G\) giving rise to a finite group of Lie type with \(\mathbb{F}_q\)-points given by \(G(\mathbb{F}_q)=G^F\). Let \(M\geq 2\) be a positive integer. Consider the power map \(G\to G\) given by \(x\to x^M\). In this paper, the authors study asymptotic estimates of the probability that a regular semisimple, semisimple and regular element is an \(M\)th power in \(G(\mathbb{F}_q)\). They prove that, as \(q\to\infty\), the set of limits for each proportion of \(M\)th powers of regular semisimple, semisimple and regular elements of \(G(\mathbb{F}_q)\) is the same and provide a formula. This generalizes the well-known results for \(M=1\) where all the limits take the same value 1. The authors also compute this more explicitly for the groups \(\mathrm{GL}(n,q)\) and \(U(n,q)\) and show that the set of limits is the same for these two group, in fact, in bijection under \(q\mapsto -q\) for a fixed \(M\).
scientific article; zbMATH DE number 7612818
scientific article