Normal coverings of solvable groups.
type: Item
instance of: scholarly article
title: Normal coverings of solvable groups.
zbMATH Open document ID: 1255.20027
author: Eleonora Crestani
author: Andrea Lucchini
published in: Archiv der Mathematik
publication date: 2012-02-14T00:00:00Z
review text: If \(G\) is a noncyclic finite group, then \(G\) can be obtained as the union of some proper subgroups. The smallest integer \(n\) such that \(G\) is the union of \(n\) proper subgroups is denoted by \(\sigma(G)\). This number has been investigated by several authors (see references of the paper). It is clear that there are some restrictions for \(\sigma(G)\) and, in the solvable case, these are quite tight. For instance, \(\sigma(G)=n\) for a solvable group \(G\) if and only if \(n-1\) is a prime power. In this paper, the authors consider a variation on this problem. For noncyclic groups \(G\), the authors define \(\gamma(G)\) to be the smallest integer \(n\) for which there exist \(n\) proper subgroups such that \(G\) is the union of all conjugates of those subgroups. It is not hard to see that \(\gamma(G)\) cannot be 1, and it is shown in this paper that every other positive integer occurs as \(\gamma(G)\), and in fact, \(G\) can be taken to be solvable.
reviewed by: Adolfo Ballester-Bolinches
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 6006283
zbMATH Keywords: finite groups
zbMATH Keywords: solvable groups
zbMATH Keywords: coverings of groups
zbMATH Keywords: subgroup covers
zbMATH Keywords: unions of subgroups
zbMATH Keywords: unions of conjugates of subgroups
MaRDI profile type: Publication
full work available at URL: s00013-011-0341-3
OpenAlex ID: W2058481132
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DOI: 10.1007/S00013-011-0341-3
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description: scientific article; zbMATH DE number 6006283