Transitive permutation groups of minimal movement

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    Abstract

    For a permutation group G on a set S, the movement of G is defined as the maximum cardinality of subsets T of S for which there exists an element x is an element of G such that T is disjoint from its translate TX (that is, when such subsets have bounded cardinality). It was shown by the second author that, if G has bounded movement m and if G has no fixed points in S, then S is finite, and ISI is bounded above by a function of m. In particular, if G is transitive, then \S\ less than or equal to 3m. This paper completes the proof of a conjecture of Gardiner and Praeger that the only transitive groups on a set of size 3m which have movement m are transitive permutation groups of exponent 3 (when m is a power of 3), the symmetric group S-3 in its natural representation on a set of three points, and the alternating groups A(4) and A(5), in their transitive representations on six points. (C) 1996 Academic Press, Inc.
    Original languageEnglish
    Pages (from-to)903-911
    JournalJournal of Algebra
    Volume181
    DOIs
    Publication statusPublished - 1996

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