On the probability that a group element is \(p\)-singular
type: Item
instance of: scholarly article
title: On the probability that a group element is \(p\)-singular
zbMATH Open document ID: 0835.20006
author: I. M. Isaacs
author: William M. Kantor
author: Nicolas Spaltenstein
published in: Journal of Algebra
publication date: 1996-04-10T00:00:00Z
review text: Let \(G\) be a permutation group of degree \(n\) whose order is divisible by prime \(p\). The authors' Main Theorem states: (a) The probability that an element of \(G\) has order divisible by \(p\) is \(\geq 1/n\). (b) Equality occurs in (a) if and only if \(G\) is either sharply 2-transitive with \(n\) a power of \(p\) or \(S_n\) with \(n = p \geq 5\). It is observed that sufficiency in (b) is easily seen, so that the paper is devoted to the proof of the remainder of the theorem and this begins with a reduction to the case where \(G\) is almost simple and primitive. The proof is completed by investigating the various simple groups, using the classification theorem. The reduction is given in Section 2, the alternating and sporadic groups are treated in Section 3, and Sections 4-10 contain the analysis of the groups of Lie type. (The authors' note that a similar approach was used by \textit{A. Gambini} [Ph. D. thesis, Univ. Freiburg (1992)] but her results were not sharp).
Mathematics Subject Classification ID: 20B35
Mathematics Subject Classification ID: 20P05
Mathematics Subject Classification ID: 20D05
zbMATH DE Number: 802374
zbMATH Keywords: \(p\)-singular elements
zbMATH Keywords: almost simple groups
zbMATH Keywords: primitive groups
zbMATH Keywords: permutation groups
zbMATH Keywords: probability
zbMATH Keywords: simple groups
zbMATH Keywords: sporadic groups
zbMATH Keywords: groups of Lie type
zbMATH Keywords: sharply 2-transitive groups
reviewed by: W. E. Deskins
MaRDI profile type: Publication
full work available at URL: jabr.1995.1238
OpenAlex ID: W2128861872
DOI: 10.1006/JABR.1995.1238
description: scientific article; zbMATH DE number 802374