Transitive permutation groups with bounded movement having maximal degree

Hassani, Akbar, Khayaty, Mehdi, 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

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Let G be a transitive permutation group on a set Ω such that G is not a 2-group and let m be a positive integer. It was shown by the fourth author that if |�g\�| � m for every subset � of Ω and all g � G, then |Ω| � �2mp/(p-1)�, where p is the least odd prime dividing |G|. If p = 3 the upper bound for |Ω| 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 � 5 satisfy one of the following: (a) G := Zp � Z2a, |Ω| = p, m = (p-1)/2, where 2a|(p-1) for some a � 1; (b) G := K � P, |Ω| = 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 Ω, and P = Zp is fixed point free on Ω; (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). © 1999 Academic Press.

Keywords:Algebra, Groups
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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