University of IsfahanInternational Journal of Group Theory2251-76509420201201Groups with many roots2612762449910.22108/ijgt.2020.119870.1582ENSarahHartBirbeck, University of London0000-0003-3612-0736DanielMcVeaghDepartment of Economics, Mathematics and Statistics,
Birkbeck, University of LondonJournal Article20191030Given 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 $myro_3$.https://ijgt.ui.ac.ir/article_24499_8550c2e0f9c5a0af741e2b46b04e2ceb.pdf