Hassani, Akbar and Khayaty, Mehdi and Khukhro, E. I. and Praeger, Cheryl E.
(1999)
*Transitive permutation groups with bounded movement having maximal degree.*
Journal of Algebra, 214
(1).
pp. 317-337.
ISSN 0021-8693

Full content URL: http://dx.doi.org/10.1006/jabr.1998.7681

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Item Type: | Article |
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Item Status: | Live Archive |

## Abstract

Let G be a transitive permutation group on a set Î© such that G is not a 2-group and let m be a positive integer. It was shown by the fourth author that if |Î�g\Î�| â�¤ m for every subset Î� of Î© and all g â�� G, then |Î©| â�¤ â��2mp/(p-1)â��, where p is the least odd prime dividing |G|. If p = 3 the upper bound for |Î©| is 3m, and the groups G attaining this bound were classified in the work of Gardiner, Mann, and the fourth author. Here we show that the groups G attaining the bound for p â�¥ 5 satisfy one of the following: (a) G := Zp â�� Z2a, |Î©| = p, m = (p-1)/2, where 2a|(p-1) for some a â�¥ 1; (b) G := K â�� P, |Î©| = 2sp, m = 2s-1(p-1), where 1 < 2s < p, K is a 2-group with p-orbits of length 2s, each element of K moves at most 2s(p-1) points of Î©, and P = Zp is fixed point free on Î©; (c) G is a p-group. All groups in case (a) are examples. In case (b), there exist examples for every p with s = 1. In case (c), where G is a p-group, we also prove that the exponent of G is bounded in terms of p only. Each transitive group of exponent p is an example, and it may be that these are the only examples in case (c). Â© 1999 Academic Press.

Keywords: | Algebra, Groups |
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Subjects: | G Mathematical and Computer Sciences > G100 Mathematics |

Divisions: | College of Science > School of Mathematics and Physics |

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ID Code: | 15744 |

Deposited On: | 14 Nov 2014 13:14 |

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