Schur multipliers and power endomorphisms of groups.
type: Item
instance of: scholarly article
title: Schur multipliers and power endomorphisms of groups.
zbMATH Open document ID: 1120.20034
author: Primož Moravec
published in: Journal of Algebra
publication date: 2007-02-26T00:00:00Z
review text: A group \(G\) is said to be \(n\)-central if \(\exp(G/Z(G))\) divides \(n\). Clearly, if \(\exp(G)=n\), then a representation group of \(G\) is \(n\)-central. Using his results on \(n\)-central groups, the author proves some new estimates of \(\exp(M(G))\), where \(M(G)\) is the Schur multiplier of \(G\). As a consequence, he proves that if \(G\) is a metabelian group of exponent \(p\), then \(\exp(M(G))\) divides \(p\) (in fact, according to D. L. Johnson, we have \(\exp(M(G))=p\) here). Next, if \(\exp(G)=4\), then \(\exp(M(G))\) divides \(8\), and this is the best possible result.
Mathematics Subject Classification ID: 20F14
Mathematics Subject Classification ID: 20E10
Mathematics Subject Classification ID: 20E34
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 19C09
zbMATH DE Number: 5129703
zbMATH Keywords: Schur multipliers
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: power endomorphisms
zbMATH Keywords: exponents
zbMATH Keywords: exponential ranks
zbMATH Keywords: powerful \(p\)-groups
zbMATH Keywords: \(n\)-central groups
reviewed by: Yakov Berkovich
describes a project that uses: GAP
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2006.06.035
OpenAlex ID: W1999037577
cites work: Q3207176
cites work: A Classification Of n-Abelian Groups
cites work: Factorization of n-Soluble and n-Nilpotent Groups
cites work: Q4776046
cites work: Abelian groups with a vanishing homology group
cites work: On 2-generator metabelian groups of prime-power exponent
cites work: Q5426265
cites work: Q4264394
cites work: On the relation between upper central quotients and lower central series of a group
cites work: A note on centre-by-finite-exponent varieties of groups
cites work: Q3253828
cites work: Endliche Gruppen I
cites work: Some Inequalities for the Multiplicator of a Finite Group. II
cites work: On n-Levi groups
cites work: Q3755633
cites work: A bound for the exponent of the Schur multiplier of some finite \(p\)-groups
cites work: Baer-invariants, isologism, varietal laws and homology
cites work: Q5840072
cites work: Q5840075
cites work: Powerful \(p\)-groups. I: Finite groups.
cites work: On certain varieties of groups
cites work: On power endomorphisms of n-central groups
cites work: Q5654235
cites work: Q4288980
DOI: 10.1016/J.JALGEBRA.2006.06.035
Recommended article: A note on the Schur multiplier of groups of prime power order.
Recommended article: A Remark on the Schur Multiplier ofp-Groups
Recommended article: On the schur multiplier ofp-groups
Recommended article: On power endomorphisms of n-central groups
Recommended article: Schur multipliers of unitriangular groups.
Recommended article: Q4187985
Recommended article: Q4782289
Recommended article: Q3337636
Recommended article: Some properties on the Schur multiplier of a pair of groups.
Recommended article: On the exponent of the Schur multiplier of a pair of finite \(p\)-groups.
description: scientific article; zbMATH DE number 5129703