Groups with 3-Abelian normal closures
type: Item
instance of: scholarly article
title: Groups with 3-Abelian normal closures
zbMATH Open document ID: 0653.20037
DOI: 10.1007/BF01206466
author: Luise-Charlotte Kappe
author: Robert Fitzgerald Morse
published in: Archiv der Mathematik
publication date: 1988-01-01T00:00:00Z
review text: By a theorem of \textit{F. W. Levi} [J. Indian Math. Soc., New Ser. 6, 87-97 (1942; Zbl 0061.026)] a group G satisfies the second Engel condition if and only if every element of G is contained in an abelian normal subgroup of G. Call G n-abelian (n-Levi resp. n-Bell) if it satisfies the law \((XY)^ n=X^ nY^ n\) \(([X^ n,Y]=[X,Y]^ n\) resp. \([X^ n,Y]=[X,Y^ n]).\) The following generalization of Levi's result is proved: Theorem 1. For any group G the following are equivalent: i) every element of G is contained in a 3-abelian normal subgroup, ii) G is a 3-Levi group, iii) G is a 3-Bell group, iv) G is a 3-Engel group and \([X,Y,Y]^ 3=1\) is a law in G. - In Theorem 2, some further properties of 3-Bell groups G are given. For example, \((G_ 3)^ 9=(G_ 4)^ 3=G_ 5=1\).
Mathematics Subject Classification ID: 20F45
Mathematics Subject Classification ID: 20F12
Mathematics Subject Classification ID: 20F14
Mathematics Subject Classification ID: 20E10
zbMATH DE Number: 4065281
zbMATH Keywords: n-abelian groups
zbMATH Keywords: n-Levi groups
zbMATH Keywords: n-Bell groups
zbMATH Keywords: Engel condition
zbMATH Keywords: 3- abelian normal subgroup
reviewed by: Rolf Brandl
MaRDI profile type: Publication
cites work: Factorization of n-Soluble and n-Nilpotent Groups
cites work: ON n—bell groups
cites work: Engelsche Elemente der Länge drei. (Engel elements of length three.)
cites work: A class of three-Engel groups
cites work: On the H p -Problem for Finite p-Groups
cites work: On Levi-formations
cites work: On n-Levi groups
cites work: On three-Engel groups
cites work: Q5840069
cites work: Q5840072
full work available at URL: https://doi.org/10.1007/bf01206466
OpenAlex ID: W2015459036
description: scientific article; zbMATH DE number 4065281