On the point-stabiliser in a transitive permutation group

Abstract

If V is a (possibly infinite) set, G a permutation group on ${V, v\in V}$, and Ω is an orbit of the stabiliser G _v , let ${G_v^{\Omega}}$ denote the permutation group induced by the action of G _v 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 _v and ${G_v^\Omega}$. If G is primitive and G _v 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 _v is also a composition factor of the group ${G_v^{\Omega(v)}}$. 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 ${\Omega=u^{G_v}}$ 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 ${\{(v^g,u^g) \mid u\in \Omega, g\in N\}}$ is locally finite and strongly connected (when viewed as a digraph on V), then every closed simple section of G _v is also a section of ${G_v^\Omega}$. To demonstrate that the topological assumptions on G and the simple sections of G _v 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 _v is isomorphic to the modular group ${{\rm PSL}(2,\mathbb{Z}) \cong C_2*C_3}$, which is known to have infinitely many finite simple groups among its sections.