# Groups with many roots

Document Type : Proceedings of the conference "Engel conditions in groups" - Bath - UK - 2019

Authors

1 Birbeck, University of London

2 Department of Economics, Mathematics and Statistics, Birkbeck, University of London

Abstract

Given a prime $p$‎, ‎a finite group $G$ and a non-identity element $g$‎, ‎what is the largest number of $p^{th}$ roots $g$ can have? We write $ϱ_p(G)$‎, ‎or just $ϱ_p$‎, ‎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 $ϱ_p$ is large‎. ‎If there is an element $g$ of $G$ with more $p^{th}$ roots than the identity‎, ‎then we show $ϱ_p(G) \leq ϱ_p(P)$‎, ‎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 $ϱ_p(G) \leq \frac{1}{p}$‎, ‎while if $G$ is a $p$-group of maximal class‎, ‎then $ϱ_p(G) \leq \frac{1}{p}‎ + ‎\frac{1}{p^2}$ (both these bounds are sharp)‎. ‎We classify the groups with high values of $ϱ_2$‎, ‎and give partial results on groups with high values of $ϱ_3$‎.

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