On the point-stabiliser in a transitive permutation group

P. Potočnik, Steve Wilson

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Abstract

If V is a (possibly infinite) set, G a permutation group on V, ν ∈ V, and Ω is an orbit of the stabiliser G ν, let G Ω ν denote the permutation group induced by the action of G ν on Ω, and let N be the normaliser of G in Sym(V). In this article, we discuss a relationship between the structures of G ν and G Ω ν. If G is primitive and G ν is finite, then by a theorem of Betten et al. (J Group Theory 6:415-420, 2003) we can conclude that every composition factor of the group G ν is also a composition factor of the group G Ω(ν) ν. In this paper we generalize this result to possibly imprimitive permutation groups G with infinite vertex-stabilisers, subject to certain restrictions that can be expressed in terms of the natural permutation topology on Sym(V). In particular, we show the following: If Ω = u is a suborbit of a transitive closed subgroup G of Sym(V) with a normalizing overgroup N ≤ N Sym(V)(G) such that the N-orbital {(ν g, u g) {pipe} u ∈ Ω, g ∈ N} is locally finite and strongly connected (when viewed as a digraph on V), then every closed simple section of G ν is also a section of G Ω ν. To demonstrate that the topological assumptions on G and the simple sections of G ν cannot be omitted in this statement, we give an example of a group G acting arc-transitively on an infinite cubic tree, such that the vertex-stabiliser G ν is isomorphic to the modular group PSL(2, ℤ)≅ C 2*C 3, which is known to have infinitely many finite simple groups among its sections.

Original languageEnglish (US)
Pages (from-to)497-504
Number of pages8
JournalMonatshefte fur Mathematik
Volume166
Issue number3-4
DOIs
StatePublished - Jun 1 2012

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Keywords

  • Locally-compact group
  • Permutation group
  • Point-stabiliser

ASJC Scopus subject areas

  • Mathematics(all)

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