Groups with many roots

Abstract

Given a prime $p$, a finite group $G$ and a non-identity element $g$, what is the largest number of \pth roots $g$ can have? We write $\rho_p(G)$, or just $\rho_p$, for the maximum cardinality of the set $\{x \in G: x^p=g\}$, where $g$ ranges over the non-identity elements of $G$. This paper studies groups for which $\rho_p$ is large. If there is an element $g$ of $G$ with more \pth roots than the identity, then we show $\rho_p(G) \leq \rho_p(P)$, where $P$ is any Sylow $p$-subgroup of $G$, meaning that we can often reduce to the case where $G$ is a $p$-group. We show that if $G$ is a regular $p$-group, then $\rho_p(G) \leq \frac{1}{p}$, while if $G$ is a $p$-group of maximal class, then $\rho_p(G) \leq \frac{1}{p} + \frac{1}{p^2}$ (both these bounds are sharp). We classify the groups with high values of $\rho_2$, and give partial results on groups with high values of $\rho_3$