Locally graded groups with certain minimal conditions for subgroups. II
type: Item
instance of: scholarly article
title: Locally graded groups with certain minimal conditions for subgroups. II
zbMATH Open document ID: 0662.20028
DOI: 10.5565/PUBLMAT_32288_01
author: J. M. Peña
author: Javier Otal
published in: Publicacions Matemàtiques
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/41052
review text: [For Part I cf. the preceding item Zbl 0662.20027.] These two papers deal with one of the ways of studying infinite groups many of whose subgroups have a prescribed property, namely the consideration of minimal conditions in the direction described by \textit{R. E. Phillips} and \textit{J. Wilson} [J. Algebra 51, 41-68 (1978; Zbl 0374.20042)]. The second paper extends the main result of the first one, and these results are the following. Theorem. Let P be one of the following properties: (i) abelian; (ii) normal; (iii) normal or abelian; (iv) metahamiltonian. If G is a locally graded group which satisfies the minimal condition for subgroups not having P then either G is a Chernikov group or every subgroup of G satisfies P. - Theorem. Let c be a nonnegative integer and let P be one of the following properties: (i) normal; (ii) locally nilpotent; (iii) normal or nilpotent of class at most c; (iv) normal or locally nilpotent; (v) normal or nilpotent of class at most c; (vi) c-Hamiltonian (that is, a group G is said to be c-Hamiltonian if every subgroup of G is either normal in G or nilpotent of class at most c). If G is a locally graded group, then we have: 1) A minimal non-P subgroup of G is finite. 2) G satisfies the minimal condition for subgroups not having P if and only if either G is a Chernikov group or every subgroup of G satisfies P.
Mathematics Subject Classification ID: 20F22
Mathematics Subject Classification ID: 20E15
zbMATH DE Number: 4081829
zbMATH Keywords: infinite groups
zbMATH Keywords: minimal conditions
zbMATH Keywords: locally graded group
zbMATH Keywords: Chernikov group
zbMATH Keywords: locally nilpotent
zbMATH Keywords: minimal non-P subgroup
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4081829