Kernels of covered groups with operators
type: Item
instance of: scholarly article
title: Kernels of covered groups with operators
zbMATH Open document ID: 0643.16023
DOI: 10.1016/0021-8693(88)90213-X
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: A set \({\mathcal C}=\{C_ i\); \(i\in I\}\), of proper non-trivial subgroups of a group G is a geometric cover if \(G=\cup {\mathcal C}\) and \(C_ i\nleq C_ j\) for \(i\neq j\). A monoid S of endomorphisms of G is a semigroup of operators for (G,\({\mathcal C})\) if for each \(\sigma\in S\), \(C_ i\in {\mathcal C}\) there exists some \(C_ j\in {\mathcal C}\) such that \(\sigma (C_ i)\leq C_ j\). Then (G,\({\mathcal C},S)\) is a generalized translation space with operators. All structures are finite in this paper, which studies the near-rings \(M_ S(C,{\mathcal C})\) consisting of all mappings \(G\to G\), which respect each \(C_ i\in {\mathcal C}\) and whose actions commute with the elements of S. The authors obtain necessary and sufficient conditions for \(M_ S(G,{\mathcal C})\) to be semisimple or simple when S is a group of automorphisms. The case in which S is a cyclic semigroup is also considered and reduced to the case when the endomorphism is nilpotent. Again necessary and sufficient conditions for the simplicity of \(M_ S(G,{\mathcal C})\) are established. There are several examples and some further results along these lines. This paper generalizes the work of \textit{C. J. Maxson} and \textit{A. Oswald} [Arch. Math. 48, 353-368 (1987; Zbl 0598.16038)].
Mathematics Subject Classification ID: 16Y30
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 51A40
zbMATH DE Number: 4047904
zbMATH Keywords: nilpotent endomorphism
zbMATH Keywords: geometric cover
zbMATH Keywords: semigroup of operators
zbMATH Keywords: generalized translation space
zbMATH Keywords: near-rings
zbMATH Keywords: semisimple
zbMATH Keywords: group of automorphisms
zbMATH Keywords: simplicity
author: P. R. Fuchs
author: C. J. Maxson
MaRDI profile type: Publication
cites work: Über Parallelstrukturen. I: Grundbegriffe. II: Translationsstrukturen. III: Zentrale \(t\)-Strukturen
cites work: Near-rings associated with generalized translation structures
cites work: The centralizer of a group automorphism
cites work: The centralizer of a set of group automorphisms
cites work: Kernels of fibered groups with operators
cites work: Q3807388
cites work: Q4740705
full work available at URL: 0021-8693(88)90213-x
OpenAlex ID: W1978611153
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description: scientific article; zbMATH DE number 4047904