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 $myro_p(G)$‎, ‎or just $myro_p$‎, ‎for the maximum value of $frac{1}{|G|}|{x in G‎: ‎x^p=g}|$‎, ‎where $g$ ranges over the non-identity elements of $G$‎. ‎This paper studies groups for which $myro_p$ is large‎. ‎If there is an element $g$ of $G$ with more $pth$ roots than the identity‎, ‎then we show $myro_p(G) leq myro_p(P)$‎, ‎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 $myro_p(G) leq frac{1}{p}$‎, ‎while if $G$ is a $p$-group of maximal class‎, ‎then $myro_p(G) leq frac{1}{p}‎ + ‎frac{1}{p^2}$ (both these bounds are sharp)‎. ‎We classify the groups with high values of $myro_2$‎, ‎and give partial results on groups with high values of $myro_3$‎.