Finite groups with few conjugacy classes of subgroups
type: Item
instance of: scholarly article
title: Finite groups with few conjugacy classes of subgroups
zbMATH Open document ID: 0789.20020
author: Rolf Brandl
author: Libero Verardi
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1993-01-16T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108253
review text: If \(G\) is a finite group, denote by \(C(G)\) its poset of conjugacy classes of subgroups \(\{[H]\}\), where \([H] \leq [K]\) if \(H \leq K^ g\) for some \(g\in G\). The Dilworth number \(w(G)\) is defined as the maximal cardinal of an antichain of the poset \(C(G)\). The authors study the problem of determining those groups \(G\) with a ``small'' number \(w(G)\). Since for every finite group \(G\) we have \(w(G) \geq | \pi(G)|\), they consider groups \(G\) for which \(w(G) = | \pi(G)| \in \{1,2,3\}\). The case \(w(G) = 1\) is trivial, the groups in question being cyclic \(p\)- groups. In the other cases the key argument is that the structure of the Sylow \(p\)-subgroups of \(G\) is very restricted if \(w(G) = | \pi(G)|\). Using this, the authors quickly dispose of the structure of the nonsolvable groups \(G\) with \(w(G) = | \pi(G)| = 3:\) only \(\text{SL}(2,5)\) and \(\text{PSL}(2,5)\) verify the condition. The bulk of the analysis is devoted to the structure of the solvable groups with \(w(G) = | \pi(G)| \in \{2,3\}\); their list is too long to be included here.
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20D05
zbMATH DE Number: 94573
zbMATH Keywords: soluble groups
zbMATH Keywords: finite group
zbMATH Keywords: poset of conjugacy classes of subgroups
zbMATH Keywords: Dilworth number
zbMATH Keywords: antichain
zbMATH Keywords: Sylow \(p\)-subgroups
zbMATH Keywords: solvable groups
reviewed by: Marian Deaconescu
MaRDI profile type: Publication
cites work: Q4015320
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description: scientific article; zbMATH DE number 94573