@article {
author = {Hart, Sarah and McVeagh, Daniel},
title = {Groups with many roots},
journal = {International Journal of Group Theory},
volume = {9},
number = {4},
pages = {261-276},
year = {2020},
publisher = {University of Isfahan},
issn = {2251-7650},
eissn = {2251-7669},
doi = {10.22108/ijgt.2020.119870.1582},
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 $\myro_3$.},
keywords = {$\pth$ roots,square roots,cube roots},
url = {http://ijgt.ui.ac.ir/article_24499.html},
eprint = {http://ijgt.ui.ac.ir/article_24499_8550c2e0f9c5a0af741e2b46b04e2ceb.pdf}
}