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Proceedings of the London Mathematical Society Advance Access published online on August 5, 2009
Proceedings of the London Mathematical Society, doi:10.1112/plms/pdp024
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© 2009 London Mathematical Society

Primitive permutation groups of bounded orbital diameter

Martin W. Liebeck

Department of Mathematics
Imperial College
London
SW7 2BZ
United Kingdom
m.liebeck@imperial.ac.uk

Dugald Macpherson

School of Mathematics
University of Leeds
Leeds
LS2 9JT
United Kingdom

Katrin Tent

Institut für Mathematische Logik
Universität Münster
Einsteinstrasse 62
48149 Münster
Germany
tent@uni-muenster.de

Received 20 June 2008. Revision received 29 April 2009.

We give a description of infinite families of finite primitive permutation groups for which there is a uniform finite upper bound on the diameter of all orbital graphs. This is equivalent to describing families of finite permutation groups such that every ultraproduct of the family is primitive. A key result is that, in the almost simple case with socle of fixed Lie rank, apart from very specific cases, there is such a diameter bound. This is proved using recent results on the model theory of pseudofinite fields and difference fields.

2000 Mathematics Subject Classification 20B15 (primary), 03C60 (secondary).

The second author was partially funded by the Marie Curie Research Training Network MODNET (MRTN-CT-2004-512234).


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