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 $ϱ_3$‎.

Keywords

Main Subjects


[1] Y. Berkovich, Groups of prime power order, Vol. 1, Walter de Gruyter, Berlin (2008).
[2] Y. Berkovich, On the number of solutions of the equation xpk= a in a finite p-group, Proc. American Math. Soc.,
116 (1992) 585–590.
[3] N. Blackburn, Note on a paper of Berkovich, J. Algebra, 24 (1973) 323–334.
[4] G. A. Fern´andez-Alcober, An introduction to finite p-groups: regular p-groups and groups of maximal class, Mat.
Contemp., 20 (2001) 155–226.
[5] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.10.2; 2019. https://www.gap-system.
org.
[6] B. Huppert, Endliche Gruppen I, Grundlehren der Mathematischen Wissenschaften, 134, Springer-Verlag, Berlin,
(1967).
[7] T. J. Laffey, The Number of Solutions of xp = 1 in a Finite Group, Mathematical Proceedings of the Cambridge
Philosophical Society, 80 (1976) 229–31.
[8] T. J. Laffey, The Number of Solutions of x3 = 1 in a 3-group, Math. Zeitschrift., 149 (1976) 43–45.
[9] T. Y. Lam, On the number of solutions of xpk= a in a p-group, Illinois J. Math., 32 (1988) 575–583.
[10] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language. Computational algebra
and number theory (London, 1993), J. Symbolic Comput., 24 (1997) 235–265.
[11] C. T. C. Wall, On groups consisting mostly of involutions, Proc. Camb. Phil. Soc., 67 (1970) 251–262.
Volume 9, Issue 4 - Serial Number 4
Proceedings of the conference "Engel conditions in groups"- Bath-UK-2019
December 2020
Pages 261-276
  • Receive Date: 30 October 2019
  • Revise Date: 27 February 2020
  • Accept Date: 27 February 2020
  • Published Online: 01 December 2020