Transitive permutation groups with bounded movement
type: Item
instance of: scholarly article
title: Transitive permutation groups with bounded movement
zbMATH Open document ID: 0816.20005
author: V. Pereyra
published in: Journal of Algebra
publication date: 1994-09-28T00:00:00Z
review text: A permutation group \(G\) on a finite set \(\Omega\) is said to satisfy the ``bounded movement'' condition with respect to \(m\) (in short: \(BM(m)\)) if and only if the following implication holds: If \(g \in G\) and \(\Delta \subseteq \Omega\) satisfy \(\Delta \cap \Delta^ g = \emptyset\), then \(| \Delta| \leq m\). A previous result of the second author [J. Algebra 144, 436-442 (1991; Zbl 0744.20004)] states that if \(G\) acts transitively on \(\Omega\) and satisfies \(BM(m)\) then \(| \Omega| \leq 3m\) (Theorem 1). In the present paper the groups \(G\) meeting this bound are studied. It is conjectured that if \(G\) is transitive on \(\Omega\), satisfies \(BM(m)\) and \(| \Omega| = 3m\), one of (a) \(G = S_ 3\), \(m = 1\); (b) \(G = A_ 4, A_ 5\), \(m = 2\); (c) \(G\) is a 3- group, \(m = 3^ r\) holds. The main result is Theorem 3: Let \(m\) be the least integer for which the conjecture fails and let \(G\) be a minimal counterexample of degree \(3m\). Then \(G\) is a simple group acting primitively on \(\Omega\). Hence the proof of the conjecture is reduced to the study of simple groups acting primitively of degree \(3m\). It has been shown by the second author (loc. cit.) that in this case \(m = 2^ \varepsilon 3^ r 5^{\varepsilon'}\) where \(\varepsilon, \varepsilon' = 0\) or 1 and \(r \geq 0\) holds (Theorem 2). Proofs of Theorem 1 and Theorem 2 are included for convenience.
Mathematics Subject Classification ID: 20B10
Mathematics Subject Classification ID: 20B05
Mathematics Subject Classification ID: 20B15
Mathematics Subject Classification ID: 20B20
zbMATH DE Number: 645106
zbMATH Keywords: primitive permutation groups
zbMATH Keywords: bounded movement
zbMATH Keywords: finite 3-groups
zbMATH Keywords: finite simple groups
zbMATH Keywords: minimal counterexample
zbMATH Keywords: simple group acting primitively
MaRDI profile type: Publication
full work available at URL: jabr.1994.1254
OpenAlex ID: W1967275610
Wikidata QID: Q56988305
DOI: 10.1006/JABR.1994.1254
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description: scientific article; zbMATH DE number 645106