Subgroup commutativity degrees of finite groups.
type: Item
instance of: scholarly article
title: Subgroup commutativity degrees of finite groups.
zbMATH Open document ID: 1196.20024
author: Marius Tărnăuceanu
published in: Journal of Algebra
publication date: 2009-06-30T00:00:00Z
review text: Let \(G\) be a finite group and let \(L(G)\) be the set of subgroups of \(G\). The author defines the subgroup commutativity degree of \(G\) by \(\text{sd}(G)=|L(G)|^{-2}|\{(H,K)\in L(G)^2\mid HK=KH\}|\). Clearly, \(\text{sd}(G)\) is the probability that two subgroups of \(G\) permute. The author states some simple general properties of \(\text{sd}(G)\). Then he gives a lower bound for \(\text{sd}(G)\) in terms of \(\text{sd}(N)\), \(\text{sd}(G/N)\), \(|L(N)|\), and \(|L(G/N)|\) if \(N\) is a normal subgroup of \(G\). Finally, he computes the subgroup commutativity degrees of the finite dihedral groups and uses this result to determine \(\text{sd}(G)\) for all \(2\)-groups \(G\) with a cyclic maximal subgroup. For \(p>2\), any two subgroups of such a \(p\)-group \(G\) permute and hence \(\text{sd}(G)=1\).
reviewed by: Roland Schmidt
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20P05
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 5572163
zbMATH Keywords: subgroup commutativity degrees
zbMATH Keywords: permutable subgroups
zbMATH Keywords: dihedral groups
zbMATH Keywords: lattices of subgroups
zbMATH Keywords: finite \(p\)-groups
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2009.02.010
OpenAlex ID: W2080436555
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DOI: 10.1016/J.JALGEBRA.2009.02.010
description: scientific article; zbMATH DE number 5572163