Kernels of covered groups
type: Item
instance of: scholarly article
title: Kernels of covered groups
zbMATH Open document ID: 0596.16031
DOI: 10.1007/BF03322350
published in: Results in Mathematics
publication date: 1986-01-01T00:00:00Z
review text: Consider an additively expressed group \(G\) with identity element 0. A collection \(\mathcal C\) of subgroups of \(G\) is a cover for \(G\) if (1) \(\cup\, \mathcal C=G\) and (2) for \(H,K\in\mathcal C\), \(H\ne \{0\}\) or \(G\), if \(K\ne H\), then \(H\not\subseteq K\). The pair \((G,\mathcal C)\) is called a covered group. If (2) is replaced by (3), for each \(H,K\in\mathcal C\) either \(H=K\) or \(H\cap K=\{0\}\), then \(\mathcal C\) is a fibration and \((G,\mathcal C)\) is a fibered group. The near-ring d.g. \(E(G,\mathcal C)=\{f\in \text{End}(G)\mid f(H)\subseteq H\) for each \(H\in\mathcal C\}\) is called the kernel of \((G, \mathcal C)\). This article is a continuation of the authors' study on relating the algebraic properties of the kernel with the geometric structure derived from \((G,\mathcal C)\). In previous investigations, attention was restricted to covers that were fibrations. That restriction is lifted here. The objective is to obtain structural information about d.g. \(E(G,\mathcal C)\) from properties of \((G,\mathcal C)\). Attention is focused on finite abelian groups (in particular elementary abelian \(p\)-groups) and the questions on whether or not \(E(G,\mathcal C)\) is a semisimple ring, a local ring, or a field. After establishing several general properties, conditions become more restrictive. The examples are excellent and relate to interesting questions on rings of matrices over a field of \(p\) elements.
Mathematics Subject Classification ID: 16Y30
Mathematics Subject Classification ID: 20D30
zbMATH DE Number: 3959709
zbMATH Keywords: covered group
zbMATH Keywords: fibration
zbMATH Keywords: fibered group
zbMATH Keywords: near-ring
zbMATH Keywords: kernel
zbMATH Keywords: covers
author: Helmut Karzel
author: C. J. Maxson
author: Günter F. Pilz
reviewed by: Homer F. Bechtell
MaRDI profile type: Publication
cites work: Über Parallelstrukturen. I: Grundbegriffe. II: Translationsstrukturen. III: Zentrale \(t\)-Strukturen
cites work: Kinematic spaces with dilatations
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full work available at URL: https://doi.org/10.1007/bf03322350
OpenAlex ID: W2143766260
description: scientific article; zbMATH DE number 3959709