Groups in which all subgroups are pronormal.
type: Item
instance of: scholarly article
title: Groups in which all subgroups are pronormal.
zbMATH Open document ID: 0642.20028
DOI: 10.1007/BF01057228
author: N. F. Kuzennyj
author: Igor Subbotin
published in: Ukrainian Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: A subgroup \(H\) in a group \(G\) is pronormal if for any element \(g\) in \(G\) there exists an element \(c\in N(H)\) such that \(gc\) belongs to the subgroup generated by \(H\) and \(gHg^{-1}\). Let \(G\) be a group in which any subgroup is pronormal. If \(G\) is locally solvable and non-periodic, then \(G\) is Abelian. Suppose any finitely generated subgroup \(H\neq 1\) has a subgroup \(H_ 1\) of finite index \(r>1\) and \(G\) is periodic. Then \(G\) is a semidirect product of an Abelian \(2'\)-group \(A\) and a Dedekind group \(B\). Each subgroup in \(A\) is normal in \(G\); \(\pi(B)\cap\pi(A)\) is empty; \(G'=A\times B'\); \(G\) is a semidirect product of \(A\) and any \(\pi(B)\)-subgroup in \(G\). This theorem extends results of \textit{T. A. Peng} [Proc. Am. Math. Soc. 20, 232-234 (1969; Zbl 0167.02302)] and \textit{A. Fattahi} [J. Algebra 28, 15-19 (1974; Zbl 0274.20022)].
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20F19
zbMATH DE Number: 4046011
zbMATH Keywords: pronormal subgroup
zbMATH Keywords: locally solvable
zbMATH Keywords: semidirect product
zbMATH Keywords: Dedekind group
reviewed by: Vyacheslav A. Artamonov
MaRDI profile type: Publication
cites work: Q3961010
cites work: Q5637070
cites work: Q3867970
cites work: Q5736170
cites work: Finite Groups with Pro-Normal Subgroups
cites work: Groups with only normal and abnormal subgroups
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description: scientific article; zbMATH DE number 4046011