Elements with r-th roots in finite groups

TL;DR

This paper calculates the exact probability that a random element in projective special linear groups has an r-th root, extending previous work mainly focused on the case r=2 and exploring density analogies.

Contribution

It generalizes techniques to find the probability of r-th roots in finite groups for r>2, providing exact values for projective special linear groups.

Findings

Exact probability values for r-th roots in projective special linear groups

Extension of techniques from r=2 to r>2 cases

Density results showing analogy with the r=2 case

Abstract

The probability that a randomly chosen element of a finite group is an $r$--th root (for any integer $r\geq2$) has been studied largely in case $r=2$. Certain techniques may be generalized for $r>2$ and here we find the exact value of this probability for projective special linear groups. A result of density is placed at the end, in order to show an analogy with the case $r=2$.