Minimal classes and maximal class in \(p\)-groups
type: Item
instance of: scholarly article
title: Minimal classes and maximal class in \(p\)-groups
zbMATH Open document ID: 0947.20015
DOI: 10.1007/BF02808177
author: P. Longobardi
author: M. Maj
author: Avinoam Mann
published in: Israel Journal of Mathematics
publication date: 2000-11-05T00:00:00Z
review text: The authors study finite non-commutative \(p\)-groups \(G\) with the property that the number of minimal classes (i.e., the smallest non-singleton conjugacy classes) is \(p-1\). Among others, they prove the following results. For such a group, \(Z_2(G)\) is an elementary Abelian \(p\)-group, \(|Z_2(G):Z(G)|=p\) and the minimal classes are exactly the nontrivial cosets of \(Z(G)\) in \(Z_2(G)\). A metabelian group \(G\) has this property if and only if \(G\) is a CF-group (i.e., the lower central factors of \(G\), except the first one, have order \(p\)), the degree of commutativity of \(G\) is one, the nilpotence class of \(G\) is \(c\geq 4\) and \(C(G')\cap Z_{c-1}(G)=G'\) holds. A \(p\)-group of maximal class \(G\) of order \(p^n\) has the investigated property if and only if \([G_1,G_{n-3}]\neq 1\) (where \(G_i\) for \(i\geq 2\) denotes the \(i\)th term of the lower central series of \(G\) and \(G_1=C(G_2/G_4)\)). In addition, some results about metabelian CF-groups are proved as well, for example, that a metabelian \(p\)-group \(G\) of nilpotence class at least \(3\) is a CF-group if and only if \(|Z_2(G)\cap G'|=p^2\).
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20F14
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20E45
zbMATH DE Number: 1310634
zbMATH Keywords: breadths
zbMATH Keywords: \(p\)-groups
zbMATH Keywords: minimal classes
zbMATH Keywords: conjugacy classes
zbMATH Keywords: lower central factors
zbMATH Keywords: degrees of commutativity
zbMATH Keywords: nilpotence classes
zbMATH Keywords: \(p\)-groups of maximal class
zbMATH Keywords: lower central series
zbMATH Keywords: metabelian CF-groups
reviewed by: Péter P. Pálfy
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1310634