Transitive permutation groups of minimal movement
type: Item
instance of: scholarly article
title: Transitive permutation groups of minimal movement
zbMATH Open document ID: 0848.20002
author: Avinoam Mann
author: Cheryl E. Praeger
published in: Journal of Algebra
publication date: 1996-07-09T00:00:00Z
review text: The ``movement'' of a permutation group refers to the largest number of points which can be moved to a disjoint set by some element of the group. It was shown by \textit{C. E. Praeger} [J. Algebra 144, No. 2, 436-442 (1991; Zbl 0744.20004)] that a transitive permutation group of movement \(m\) has degree at most \(3m\). Equality is possible here (movement \(m\) on a set of \(3m\) points) if the group has exponent three, or if it is \(S_3\) (\(m=1\)), \(A_4\) (\(m=2\)), or \(A_5\) (\(m=2\)). \textit{A. Gardiner} and \textit{C. E. Praeger} [J. Algebra 168, No. 3, 798-803 (1994; Zbl 0816.20005)] conjectured that these are the only cases of equality, and they proved that a minimal counterexample is either a 3-group or a simple group acting primitively. The present paper completes the proof of this conjecture by considering first 3-groups and then simple groups. In treating 3-groups, the authors prove that a transitive \(p\)-group whose fixed-point-free elements all have order \(p\) must have exponent \(p\) if \(p=2\) or 3; they ask as an open question whether this holds for larger primes \(p\). For simple groups the authors use the classification of finite simple groups and prove a series of lemmas to eliminate the various families of simple groups.
Mathematics Subject Classification ID: 20B05
Mathematics Subject Classification ID: 20B20
Mathematics Subject Classification ID: 20B15
zbMATH DE Number: 883792
zbMATH Keywords: transitive permutation groups
zbMATH Keywords: movement
zbMATH Keywords: degree
zbMATH Keywords: exponent
zbMATH Keywords: minimal counterexamples
zbMATH Keywords: 3-groups
zbMATH Keywords: fixed-point-free elements
zbMATH Keywords: classification of finite simple groups
reviewed by: Albert J. Goodman
MaRDI profile type: Publication
full work available at URL: jabr.1996.0152
OpenAlex ID: W2001099423
Wikidata QID: Q56988241
DOI: 10.1006/JABR.1996.0152
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description: scientific article; zbMATH DE number 883792