Bounds for the probability of generating the symmetric and alternating groups.
type: Item
instance of: scholarly article
title: Bounds for the probability of generating the symmetric and alternating groups.
zbMATH Open document ID: 1222.20003
author: Attila Maróti
author: M. Chiara Tamburini
published in: Archiv der Mathematik
publication date: 2011-03-31T00:00:00Z
review text: The authors give explicit, asympotically sharp bounds for the probability \(p(S_n)\) that a pair of random permutations of degree \(n\) generates either \(S_n\) or \(A_n\) and also for the probability \(p(A_n)\) that a pair of random even permutations of degree \(n\) generates \(A_n\). Precisely they prove that if \(n\geq 4\) and \(X\in\{A_n,S_n\},\) then \(1-1/n-13/{n^2}<p(X)\leq 1-1/n+2/{3n^2}\). As an application they answer a question of Wiegold in the case of alternating groups, proving that \(A_n^t\) is 2-generated if \(t\leq\sqrt{|A_n|}\) and \(n\geq 5\).
reviewed by: Andrea Lucchini
Mathematics Subject Classification ID: 20B30
Mathematics Subject Classification ID: 20P05
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 5872602
zbMATH Keywords: symmetric groups
zbMATH Keywords: alternating groups
zbMATH Keywords: probability
zbMATH Keywords: random generation
zbMATH Keywords: random permutations
describes a project that uses: GAP
MaRDI profile type: Publication
full work available at URL: s00013-010-0216-z
OpenAlex ID: W2026394746
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DOI: 10.1007/S00013-010-0216-Z
description: scientific article; zbMATH DE number 5872602