The embedding of CAP-subgroups in finite groups.
type: Item
instance of: scholarly article
title: The embedding of CAP-subgroups in finite groups.
zbMATH Open document ID: 1209.20016
author: Joseph Petrillo
published in: Ricerche di Matematica
publication date: 2010-09-20T00:00:00Z
review text: Let \(G\) be a finite group. A subgroup \(U\) of \(G\) is called a CAP-subgroup of \(G\) if each chief factor of \(G\) is either covered or avoided by \(U\). A subgroup \(U\) of \(G\) is called a subCAP-subgroup of \(G\) if there exists a chain of subgroups \(U=U_0\leq U_1\leq\cdots\leq U_r=G\) such that \(U_{i-1}\) is a CAP-subgroup of \(U_i\) for \(i=1,\dots,r\). The present paper is devoted to these subgroups and their influence on the structure of \(G\) and culminates in Proposition 28 stating that the following are equivalent: (1) \(G\) is soluble; (2) every Sylow subgroup of \(G\) is a CAP-subgroup of \(G\); (3) every maximal subgroup of \(G\) is a CAP-subgroup of \(G\); (4) if \(U\leq G\), then there exists a chain \(U=U_0\leq U_1\leq\cdots\leq U_r= G\) such that \(U_{i-1}\) is a CAP-subgroup of \(U_i\) and \([U_i:U_{i-1}]\) is a prime power for \(i=1,\dots,r\); (5) if \(U<G\), then there exists \(V\leq G\) such that \(U<V\) and \(U\) is a CAP-subgroup of \(V\); (6) every subgroup of \(G\) is a subCAP-subgroup of \(G\). -- Note that if, in Proposition 28, ``soluble'' is replaced by ``nilpotent'', ``CAP'' by ``normal'' and ``prime power'' by ``prime'', one obtains the classical characterisation of nilpotent groups.
reviewed by: Hans Lausch
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20E28
zbMATH DE Number: 5788846
zbMATH Keywords: finite groups
zbMATH Keywords: cover-avoidance property
zbMATH Keywords: CAP-subgroups
zbMATH Keywords: subCAP-subgroups
zbMATH Keywords: chief factors
MaRDI profile type: Publication
full work available at URL: s11587-010-0073-x
OpenAlex ID: W1989281189
cites work: Finite soluble groups
cites work: A contribution to the theory of finite supersoluble groups
cites work: Some permutability properties related to \(\mathfrak F\)-hypercentrally embedded subgroups of finite groups
cites work: Cover-avoid subgroups in finite solvable groups
cites work: Cover-avoidance properties and the structure of finite groups
cites work: CAP-subgroups in a direct product of finite groups.
cites work: Subgroup lattices of groups
cites work: Cover-Avoidance Properties in Finite Soluble Groups
DOI: 10.1007/S11587-010-0073-X
description: scientific article; zbMATH DE number 5788846