On finite permutation groups with the same orbits on unordered sets
type: Item
instance of: scholarly article
title: On finite permutation groups with the same orbits on unordered sets
zbMATH Open document ID: 0582.20001
DOI: 10.1007/BF01194888
author: Johannes Siemons
author: Ascher Wagner
published in: Archiv der Mathematik
publication date: 1985-01-01T00:00:00Z
review text: Let G and H be permutation groups acting on a set \(\Omega\). Then \(H\approx^{k}G\) iff H and G have the same orbits on the unordered sets of k points. G and H are said to be orbit equivalent (H\(\approx G)\) iff \(H\approx^{k}G\) for all k. The main result of the present paper is the following Theorem A: Let K be a finite primitive permutation group on a set \(\Omega\). Let \(H<K\) and \(H\approx K\). Suppose there exists a prime r dividing the order of K but not the order of H. Then \(| \Omega |\) belongs to \(\{\) 3,5,8,9\(\}\) and the possibilities for H and K are explicitly stated. A somehow stronger version of the result, Theorem B, is proved using the classification of 3-transitive groups. Remark. The concept of orbit equivalence should be compared with \textit{H. Wielandt}'s concept of k- equivalence and k-closure, see [Permutation groups through invariant relations and invariant functions; Lecture notes, Ohio State Univ. (1969)]. Hence Theorem A corresponds in some way to a well known theorem of \textit{R. A. Beaumont} and \textit{R. P. Peterson} [Can. J. Math. 7, 35-42 (1955; Zbl 0064.025)].
Mathematics Subject Classification ID: 20B10
Mathematics Subject Classification ID: 20B05
Mathematics Subject Classification ID: 20B15
zbMATH DE Number: 3931233
zbMATH Keywords: orbits
zbMATH Keywords: finite primitive permutation group
zbMATH Keywords: classification of 3- transitive groups
zbMATH Keywords: orbit equivalence
MaRDI profile type: Publication
cites work: Set-Transitive Permutation Groups
cites work: An Interchange Property in Finite Permutation Groups
cites work: Q3253828
cites work: \(k\)-homogeneous groups
cites work: Transitivity of finite permutation groups on unordered sets
cites work: On partitions and permutation groups on unordered sets
cites work: Q5512231
full work available at URL: https://doi.org/10.1007/bf01194888
OpenAlex ID: W2012402255
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description: scientific article; zbMATH DE number 3931233