ZS-metacyclic groups and their endomorphism near-rings
type: Item
instance of: scholarly article
title: ZS-metacyclic groups and their endomorphism near-rings
zbMATH Open document ID: 0820.16043
DOI: 10.1007/BF01301692
author: Gordon Mason
author: J. J. jun. Malone
published in: Monatshefte für Mathematik
publication date: 1995-09-11T00:00:00Z
full work available at URL: https://eudml.org/doc/178667
review text: The first author and \textit{C. G. Lyons} started the study of endomorphism near-rings [in Proc. Edinb. Math. Soc., II. Ser. 17, 71-78 (1970; Zbl 0203.336)]. This paper follows in that line and determines the endomorphism near-rings of a large class of groups, using an extension of the methods developed in the paper above. Let \(S\) be a semigroup of endomorphisms of the group \(G\). Then \(S\) generates a near-ring of mappings from \(G\) to itself. The three important cases are \(S = \text{Inn }G\), the inner automorphisms, \(S = \text{Aut }G\), all the automorphisms, \(S = \text{End }G\), all the endomorphisms. The corresponding nearrings are called \(I(G)\), \(A(G)\) and \(E(G)\) respectively. The groups considered here are ZS-metacyclic groups, which are extensions of a cyclic group by another cyclic group of order prime to the first. This is equivalent to saying that all the Sylow subgroups are cyclic. The authors determine all the endomorphisms of a ZS-metacyclic group \(G\). They use a Peirce decomposition method to obtain the structure of \(I(G)\). Then they show that \(I(G) = E(G)\), so every endomorphism of \(G\) is a sum of inner automorphisms. This is a major advance in the study of \(I\)-\(E\) groups, groups \(H\) for which \(I(H) = E(H)\). Finally the radical of \(I(G)\) is determined. As may be expected, there is a great deal of careful detailed work in this paper.
Mathematics Subject Classification ID: 16Y30
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 16N80
zbMATH DE Number: 729092
zbMATH Keywords: endomorphism near-rings
zbMATH Keywords: semigroup of endomorphisms
zbMATH Keywords: near-ring of mappings
zbMATH Keywords: inner automorphisms
zbMATH Keywords: ZS-metacyclic groups
zbMATH Keywords: sum of inner automorphisms
zbMATH Keywords: \(I\)-\(E\) groups
zbMATH Keywords: radicals
reviewed by: John D. P. Meldrum
describes a project that uses: Maple
MaRDI profile type: Publication
cites work: Distributively generated near-rings with descending chain condition
cites work: Q5770541
cites work: Q4002478
cites work: Q3855356
cites work: The Near-Ring Generated by the Inner Automorphisms of a Finite Simple Group
cites work: Radicals of endomorphism near-rings
cites work: Q3197968
cites work: Endomorphism near-rings
cites work: Q5645569
cites work: Q5682133
cites work: On the structure of morphism near-rings
cites work: Q3807388
cites work: On the structure of an endomorphism near-ring
cites work: Q3671335
cites work: Q5331572
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description: scientific article; zbMATH DE number 729092