Mutually permutable products and conjugacy classes.
type: Item
instance of: scholarly article
title: Mutually permutable products and conjugacy classes.
zbMATH Open document ID: 1275.20015
author: Adolfo Ballester-Bolinches
author: John Cossey
author: Yangming Li
published in: Monatshefte für Mathematik
publication date: 2013-06-17T00:00:00Z
review text: Let \(G\) be a group, two subgroups \(H\) and \(K\) of \(G\) are `permutable' if the set \(HK\) is a subgroup of \(G\), they are `mutually permutable' if \(H\) permutes with every subgroup of \(K\) and \(K\) permutes with every subgroup of \(H\). Let the (finite) group \(G=G_1G_2\cdots G_r\) be the product of pairwise permutable subgroups \(G_1,G_2,\dots,G_r\). The aim of the paper under review is study how does the structure of the factors \(G_i\) (\(1\leq i\leq s\)) affect the structure of the whole group \(G\). In particular Theorem 1.1 states that: (1) no conjugacy class length \(|x^G|\), where \(x\) is a \(p\)-regular element of prime power order in \(\bigcup_{i=1}^rG_i\), is divisible by \(p\) if and only if \(G=O_p(G)\times O_{p'}(G)\); (2) \(|x^G|\) is not divisible by \(p\) for every element \(x\in\bigcup_{i=1}^rG_i\) if and only if \(G=O_p(G)\times O_{p'}(G)\) with \(O_p(G)\) Abelian.
reviewed by: Enrico Jabara
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 20E45
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 6176418
zbMATH Keywords: finite groups
zbMATH Keywords: mutually permutable products
zbMATH Keywords: mutually permutable subgroups
zbMATH Keywords: conjugacy classes
zbMATH Keywords: conjugacy class lengths
zbMATH Keywords: \(p\)-regular elements
MaRDI profile type: Publication
full work available at URL: s00605-012-0411-z
OpenAlex ID: W2068699021
cites work: Products of finite groups.
cites work: Mutually permutable subgroups and group classes.
cites work: Group classes and mutually permutable products.
cites work: On the length of the conjugacy classes of finite groups
cites work: Finite soluble groups
cites work: Endliche Gruppen I
cites work: Notes on the length of conjugacy classes of finite groups.
cites work: Q3403846
DOI: 10.1007/S00605-012-0411-Z
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description: scientific article; zbMATH DE number 6176418