Groups decomposable into quasicentralizer products
type: Item
instance of: scholarly article
title: Groups decomposable into quasicentralizer products
zbMATH Open document ID: 0591.20047
DOI: 10.1007/BF01061182
published in: Ukrainian Mathematical Journal
publication date: 1985-01-01T00:00:00Z
review text: The quasicentralizer of a subgroup A in a group G is, by definition, the intersection Q(A) of the normalizers in G of all subgroups of A. A group G is said to be a quasicentralizer product of a subgroup A on a subgroup B if \(G=AB\) and \(B\subset Q(A)\). Periodic groups decomposable into a quasicentralizer product with locally nilpotent or abelian factors are studied in this article. It is proved that if G is a quasicentralizer product of A on B where A and B are periodic locally nilpotent subgroups then either G is locally nilpotent or the locally nilpotent coradical of G is a non-trivial abelian involution-free subgroup which is complementable in G and decomposable into the direct product of all Sylow subgroups of A which have trivial intersections with the center of G (Theorem 1). A periodic group G is decomposable into a quasi-centralizer product with abelian factors if and only if G is a semi-direct product of an abelian subgroup N all of whose subgroups are normal in G on a hypercentral subgroup M which is a quasicentralizer product of an abelian subgroup C on an abelian subgroup Q such that C is normal in G and elements in C and N have coprime orders (Theorem 3).
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 20E34
zbMATH DE Number: 3948598
zbMATH Keywords: quasicentralizer product
zbMATH Keywords: periodic locally nilpotent subgroups
zbMATH Keywords: direct product
zbMATH Keywords: Sylow subgroups
author: Igor Subbotin
reviewed by: Ju. I. Merzljakov
MaRDI profile type: Publication
cites work: Q4740729
cites work: Q5736170
cites work: Power automorphisms of a group
cites work: Zur Sylowstruktur auflösbarer Gruppen
cites work: Some questions on the general theory of representations of groups
full work available at URL: https://doi.org/10.1007/bf01061182
OpenAlex ID: W2008158966
description: scientific article; zbMATH DE number 3948598