Minimal covers of \(S_ n\) by Abelian subgroups and maximal subsets of pairwise noncommuting elements
type: Item
instance of: scholarly article
title: Minimal covers of \(S_ n\) by Abelian subgroups and maximal subsets of pairwise noncommuting elements
zbMATH Open document ID: 0666.20012
DOI: 10.1016/0097-3165(88)90057-X
author: S. H. Smith
published in: Journal of Combinatorial Theory. Series A
publication date: 1988-01-01T00:00:00Z
review text: Let \({\mathcal G}\) be a finite group, \(\alpha\) (\({\mathcal G})\) be the maximal cardinality of a set of pairwise noncommuting elements of \({\mathcal G}\), and \(\beta\) (\({\mathcal G})\) be the minimal cardinality of a cover of \({\mathcal G}\) by abelian subgroups. These numbers were studied recently in particular by \textit{D. R. Mason} [Math. Proc. Camb. Philos. Soc. 83, 205-209 (1978; Zbl 0374.20037)] and \textit{L. Pyber} [J. Lond. Math. Soc., II. Ser. 35, 287-295 (1987; Zbl 0588.20016)]. The present author now concentrates on the case when \({\mathcal G}={\mathcal S}_ n\), the symmetric group on n letters. By lengthy and non-trivial arguments, he shows that the average values of \(\alpha\) (\({\mathcal S}_ n)/(n-2)!\) and \(\beta (S_ n)/(n-2)!\) are bounded above and below by positive constants. It is also proved that the sequence \(\{\) \(\beta\) (\({\mathcal S}_ n)/\alpha ({\mathcal S}_ n)\}\) is bounded, and that its convergence would imply \(\alpha\) (\({\mathcal S}_ n)=\beta ({\mathcal S}_ n)\) for \(n\geq 1\).
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20B35
Mathematics Subject Classification ID: 11N45
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 4089791
zbMATH Keywords: cover by abelian subgroups
zbMATH Keywords: finite group
zbMATH Keywords: noncommuting elements
zbMATH Keywords: symmetric group
reviewed by: John Knopfmacher
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4089791