The subgroups of Baer and Hughes
type: Item
instance of: scholarly article
title: The subgroups of Baer and Hughes
zbMATH Open document ID: 0820.20034
DOI: 10.1007/BF01201445
author: R. A. Bryce
published in: Archiv der Mathematik
publication date: 1995-09-11T00:00:00Z
review text: For a group \(G\) the Hughes subgroup for exponent \(n\) is defined by \(H_ n(G) = \langle x \in G;\;x^ n \neq 1\rangle\). In connection with Hughes' problem it is known that \(| G : H_ p(G)| = p\) for finite \(G\) and \(p\) prime implies \(H_ p(G)\) is nilpotent. The author deals with arbitrary groups and proves that for a prime \(p\) and soluble \(G\) of derived length \(d\) and exponent \(p^ m\) the condition \(H_ p(G) \neq G\) enforces the nilpotence of \(G\) with class bounded by a function of \(m\) and \(d\). As a consequence analogous results are obtained for soluble \(G\) with derived length \(d\) and exponent \(p^ 2\) in which the norm (= Baer's subgroup \(\kappa(G) = \{g \in G;\;U^ g = U\;\forall U \leq G\})\) is not central, and for soluble groups \(G\) of exponent \(p^ 2\) with a non-trivial power automorphism. Examples show that most of the restrictions imposed in these results cannot be weakened in general.
reviewed by: Gerhard Kowol
Mathematics Subject Classification ID: 20E34
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 493394
zbMATH Keywords: Baer subgroup
zbMATH Keywords: nilpotent groups
zbMATH Keywords: nilpotency class
zbMATH Keywords: Hughes subgroup
zbMATH Keywords: exponent
zbMATH Keywords: derived length
zbMATH Keywords: norm
zbMATH Keywords: soluble groups
zbMATH Keywords: power automorphisms
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 493394