Groups with many roots
Hart, Sarah and Mcveagh, Dan (2020) Groups with many roots. International Journal of Group Theory , ISSN 22517650. (In Press)

Text
30780a.pdf  Published Version of Record Available under License Creative Commons Attribution. Download (259kB)  Preview 
Abstract
Given a prime $p$, a finite group $G$ and a nonidentity element $g$, what is the largest number of $\pth$ roots $g$ can have? We write $\rho_p(G)$, or just $\rho_p$, for the maximum cardinality of the set $\{x \in G: x^p=g\}$, where $g$ ranges over the nonidentity elements of $G$. This paper studies groups for which $\rho_p$ is large. If there is an element $g$ of $G$ with more $\pth$ roots than the identity, then we show $\rho_p(G) \leq \rho_p(P)$, 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 $\rho_p(G) \leq \frac{1}{p}$, while if $G$ is a $p$group of maximal class, then $\rho_p(G) \leq \frac{1}{p} + \frac{1}{p^2}$ (both these bounds are sharp). We classify the groups with high values of $\rho_2$, and give partial results on groups with high values of $\rho_3$.
Metadata
Item Type:  Article 

Additional Information:  This version was the original submission; it has been accepted on 31/1/20 subject to minor revisions. The uploaded version DOES NOT YET INCLUDE those revisions. 
School:  Birkbeck Schools and Departments > School of Business, Economics & Informatics > Economics, Mathematics and Statistics 
Depositing User:  Sarah Hart 
Date Deposited:  15 Jun 2020 09:28 
Last Modified:  05 Jul 2020 05:20 
URI:  http://eprints.bbk.ac.uk/id/eprint/30780 
Statistics
Additional statistics are available via IRStats2.