Permutation groups and close-knit families of sets
type: Item
instance of: scholarly article
title: Permutation groups and close-knit families of sets
zbMATH Open document ID: 0865.20004
DOI: 10.1007/BF01197589
author: Peter M. Neumann
published in: Archiv der Mathematik
publication date: 1997-07-14T00:00:00Z
review text: This work is motivated by the study of \(m\)-quasi-invariant subsets of a set \(\Omega\) with respect to a group \(G\) of permutations of \(\Omega\). This notion has been introduced by \textit{L. Brailovsky, D. V. Pasechnik} and \textit{C. E. Praeger} [Proc. Am. Math. Soc. 123, No. 8, 2283-2295 (1995; Zbl 0843.20005)]: a subset \(\Gamma\subseteq\Omega\) is called \(m\)-quasi-invariant with respect to \(G\) if \(|\Gamma^g\cap\Gamma|\leq m\) for all \(g\in G\), where \(m\in\mathbb{N}\) is given. One main purpose is to improve a result due to Brailovsky, Pasechnik and Praeger [loc.cit.]: it is shown in Theorem 1.1 that if \(\Gamma\) is an \(m\)-quasi-invariant set for a permutation group \(G\) acting on \(\Omega\) then there exists a \(G\)-invariant subset \(\Delta\) of \(\Omega\) such that the distance \(d(\Gamma,\Delta)\leq 2m-1\), where the (Hamming) distance is the cardinality of the symmetric difference. (Brailovsky, Pasechnik and Praeger had shown in the paper quoted above that \(d(\Gamma,\Delta)\leq f(m)\) for some \(G\)-invariant set \(\Delta\) where \(f\) is a function such that \(f(m)=O(m\log m)\).) The bound obtained in Theorem 1.1 is best possible. It is obtained via considering the more general concept of close-knit families of subsets. A family \(\mathcal F\) of subsets of \(\Omega\) is said to be close-knit (with bound \(m\)) if there is a natural number \(m\) such that \(|\Gamma_1\setminus\Gamma_2|\leq m\) for all \(\Gamma_1,\Gamma_2\in{\mathcal F}\). A structure theorem for arbitrary infinite close-knit families of subsets is the second main purpose of the paper. It is proved in Theorem 3.4 that for such an infinite close-knit family \(\mathcal F\) with bound \(m\) there exists a set \(\Sigma\) such that \(d(\Gamma,\Sigma)\leq 2m-1\) for all \(\Gamma\in{\mathcal F}\) and there is a finite decomposition \({\mathcal F}={\mathcal F}_0\cup{\mathcal F}_1\cup\dots\cup{\mathcal F}_t\) where \({\mathcal F}_0\) is finite and all the other families \({\mathcal F…
Mathematics Subject Classification ID: 20B07
Mathematics Subject Classification ID: 03E05
zbMATH DE Number: 938602
zbMATH Keywords: quasi-invariant subsets
zbMATH Keywords: permutation groups
zbMATH Keywords: close-knit families of subsets
zbMATH Keywords: nuclear families
MaRDI profile type: Publication
cites work: Subgroups close to normal subgroups
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description: scientific article; zbMATH DE number 938602