Lattice and group complementability in periodic locally solvable groups
type: Item
instance of: scholarly article
title: Lattice and group complementability in periodic locally solvable groups
zbMATH Open document ID: 0582.20021
DOI: 10.1007/BF01979708
author: G. N. Titov
published in: Algebra and Logic
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/187265
review text: A subgroup B of a group G is called a lattice complement of a subgroup A if \(G=<A,B>\) and \(A\cap B=1\). The subgroup A is said to be \(G\Phi\)- complemented if there exists a subgroup H of G such that \(G=AH\) and \(A\cap H\subseteq \Phi (G)\), where \(\Phi\) (G) denotes the Frattini subgroup of G. A group G is said to be a \(\Phi_{\omega}C\)-group if the following conditions on a subgroup A of G imply that A is \(G\Phi\)- complemented: A is a p-subgroup of G for some prime p and A contains a Sylow p-subgroup of \(\Phi\) (G); A has a lattice complement B which contains a Sylow p'-subgroup of \(\Phi\) (G). Let G be a periodic locally solvable groups which is locally normal or has finite Sylow p-subgroups for all primes p. Theorem: if G is a \(\Phi_{\omega}C\)-group then G is locally supersolvable. There are some further related results.
Mathematics Subject Classification ID: 20E25
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 20F24
zbMATH DE Number: 3931253
zbMATH Keywords: lattice complement
zbMATH Keywords: Frattini subgroup
zbMATH Keywords: G\(\Phi \) -complemented
zbMATH Keywords: periodic locally solvable groups
zbMATH Keywords: locally normal
zbMATH Keywords: Sylow p-subgroups
reviewed by: Roger M. Bryant
MaRDI profile type: Publication
cites work: Q3232305
cites work: Q4745979
cites work: Q3961010
OpenAlex ID: W2056408400
description: scientific article; zbMATH DE number 3931253