Large cyclic subgroups of finite groups
type: Item
instance of: scholarly article
title: Large cyclic subgroups of finite groups
zbMATH Open document ID: 0581.20022
DOI: 10.1016/0021-8693(84)90115-7
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: \textit{M. Hausman} and \textit{H. N. Shapiro} [Commun. Pure Appl. Math. 33, 635-649 (1980)] have shown that there exists a function I such that whenever G is a finite group and \[ \gamma_ G=\max imum| \{x\in G| \quad x^ d=1\}| /d \] (where the maximum is over all divisor d of \(| G|)\) then \(\gamma \geq \gamma_ G\) implies that G has a cyclic subgroup of index \(\leq I(\gamma)\). Continuing the work which they begun in a previous paper [J. Algebra 91, 520-535 (1984; Zbl 0558.20018)], the authors find estimates for the (best possible) function I. In particular they show that for \(\gamma >2,\) \(I(\gamma)<\gamma^ 2\) and \(I(\gamma)<\exp ((1+6/\log\log\gamma) \log\gamma).\) It follows that \(I(\gamma)\ll \gamma^{1+\epsilon}\) (for \(\epsilon >0)\) and in the opposite direction the authors show that \(I(\gamma)/\gamma\) is unbounded. They also obtain an estimate for the corresponding function when attention is restricted to groups whose orders have at most a given number of prime divisors.
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 3929266
zbMATH Keywords: cyclic subgroup
zbMATH Keywords: index
author: Roberto Dvornicich
author: Marco Forti
MaRDI profile type: Publication
full work available at URL: 0021-8693(84)90115-7
OpenAlex ID: W1992704512
cites work: Finite groups with few dth roots of 1
cites work: Paracyclic and quasi-cyclic groups
cites work: Endliche Gruppen I
cites work: Approximate formulas for some functions of prime numbers
description: scientific article; zbMATH DE number 3929266