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 $myro_3$.
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