International Journal of Group Theory (Dec 2020)
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 $\myro_3$.
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