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Published online by Cambridge University Press: 05 April 2013
ABSTRACT
We investigate geometric groups and sets of permutations of an infinite set. (These are a generalisation of sharply t–transitive groups and sets). We prove non–existence of groups, and give constructions of sets, for certain parameters. This work was done while the authors were visiting the Ohio State University, to whom we express our gratitude.
INTRODUCTION
It is known that sharply t–transitive groups of permutations of an infinite set exist only for t ≤ 3 (Tits (1952)), while sharply t–transitive sets exist for al l t (Barlotti & Strambach 1984).
Geometric groups and sets of permutations have been proposed as a natural generalisation of sharply t–transitive groups and sets (Cameron & Deza (1979)). Our purpose i s to investigate such objects on infinite sets. Not surprisingly, we give nonexistence results for groups, and constructions for sets.
Let L = {ℓ0, ℓt …, ℓs–1) be a finit e set of natural numbers, with ℓ0 < … < ℓs–1. The permutation group G on the set X is a geometric group of type L if there exist points x1, …, xs ϵ X such that
(i) the stabiliser of x1 …, xs is the identity;
(ii) for i < s, the stabiliser of x1 …, xi fixes ℓi points and acts transitively on its non–fixed points.
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