Asymptotics of the powers in finite reductive groups

Abstract

Let $G$ be a connected reductive group defined over ${\mathbb{F}}_q$. Fix an integer $M\ge 2$, and consider the power map $x\mapsto x^M$ on $G$. We denote the image of $G({\mathbb{F}}_q)$ under this map by $G({\mathbb{F}}_q{)}^M$ and estimate what proportion of regular semisimple, semisimple and regular elements of $G({\mathbb{F}}_q)$ it contains. We prove that as $q\to \infty$, all of these proportions are equal and provide a formula for the same. We also calculate this more explicitly for the groups $\text{GL}(n,q)$ and $\text{U}(n,q)$.

References in corpus (1)

• On the maximal tori in finite linear and unitary groups

Cited by in corpus (1)

• Powers in the wreath product of $G$ with $S_n$