The set of orders of elements in a finite group
type: Item
instance of: scholarly article
title: The set of orders of elements in a finite group
zbMATH Open document ID: 0823.20024
DOI: 10.1007/BF00739417
published in: Algebra and Logic
publication date: 1995-04-20T00:00:00Z
full work available at URL: https://eudml.org/doc/187676
review text: Let \(\omega(G)\) be the set of orders of elements in a finite group \(G\). It was conjectured by Praeger and Shi that for a given set \(\pi\) of positive integers, there exists 0, 1 or infinitely many mutually nonisomorphic groups \(G\) with \(\omega(G) = \pi\). In this paper this conjecture is falsified by proving: Theorem. Let \(G\) be a group with \(\omega(G) = \omega (L_ 3(5))\). Then \(G \simeq L_ 3(5)\) or \(L_ 3(5) : 2\), where the second group is the extension of \(L_ 3(5)\) by the contragredient automorphism. In the proof the author reduces the problem to the case where \(G\) is almost simple and the classification of finite simple groups can be applied.
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D06
zbMATH DE Number: 738976
zbMATH Keywords: orders of elements
zbMATH Keywords: finite groups
zbMATH Keywords: classification of finite simple groups
author: V. D. Mazurov
reviewed by: Ulrich Dempwolff
MaRDI profile type: Publication
cites work: A Characterization of Suzuki's Simple Groups
cites work: Q3729146
cites work: Characterization of some alternating and symmetric groups
cites work: Q3684278
cites work: Q5551256
description: scientific article; zbMATH DE number 738976