Submission date: 24 October 2009.

Abstract:

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.

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

Keywords and phrases: primitive permutation group, pseudofinite field, supersimple theory

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