Powers in finite groups of Lie type

Abstract

In this thesis, we study the image of the power map on finite reductive groups. Let $G$ be a connected reductive algebraic group over an algebraically closed field $k$, of characteristic $p$. Let $G$ be defined over $\mathbb{F}_q$, where $q$ is a power of $p$ and $F$ be a Steinberg endomorphism of $G$. Let $M\geq 2$ be an integer. The power map $\omega_M:G(\mathbb{F}_q)\to G(\mathbb{F}_q)$ is defined by $g\mapsto g^M$, where $G(\mathbb{F}_q)=G^F$ is the corresponding finite group of Lie type. Denote the image of this map by $G(\mathbb{F}_q)^M$, which is the set of all $M^{th}$ powers in $G(\mathbb{F}_q)$. We study the asymptotic $(q\to \infty)$ of the probability that a randomly chosen element of $G(\mathbb{F}_q)$ is an $M^{th}$ power; that is, we find $\lim\limits_{q\to \infty}\frac{|G(\mathbb{F}_q)^M|}{|G(\mathbb{F}_q)|}$. Along the way we consider the related probabilities, $\frac{|G(\mathbb{F}_q)_{\text{rg}}^M|}{|G(\mathbb{F}_q)|}$, $\frac{|G(\mathbb{F}_q)_{\text{ss}}^M|}{|G(\mathbb{F}_q)|}$, $\frac{|G(\mathbb{F}_q)_{\text{rs}}^M|}{|G(\mathbb{F}_q)|}$, which denote the probability that a randomly chosen element from $G(\mathbb{F}_q)$ is an $M^{th}$ power regular, semisimple, and regular semisimple element respectively and show that they are asymptotically the same.

In another direction, we study the image of the power map more explicitly in the case of $\text{GL}(n,q)$, which is the group of $n\times n$ invertible matrices over $\mathbb{F}_q$. We find necessary and sufficient conditions for an invertible matrix to be an $M^{th}$ power. In an attempt to enumerate such elements, we get the generating functions for $M^{th}$ power (i) regular and regular semisimple elements (and conjugacy classes) when $(q,M)=1$, (ii) for semisimple elements and all elements (and conjugacy classes) when $M$ is a prime power and $(q,M)=1$, and (iii) for all kinds when $M$ is a prime, and $q$ is a power of $M$.