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On the point-stabiliser in a transitive permutation group. (English) Zbl 1247.20001

Summary: If \(V\) is a (possibly infinite) set, \(G\) a permutation group on \(V\), \(v\in V\), and \(\Omega\) is an orbit of the stabiliser \(G_v\), let \(G_v^\Omega\) denote the permutation group induced by the action of \(G_v\) on \(\Omega\), and let \(N\) be the normaliser of \(G\) in \(\text{Sym}(V)\).
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 A. Betten, A. Delandtsheer, A. C. Niemeyer, and Ch. E. Praeger [J. Group Theory 6, No. 4, 415-420 (2003; Zbl 1038.20002)] 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 \(\text{Sym}(V)\).
In particular, we show the following: If \(\Omega=u^{G_v}\) is a suborbit of a transitive closed subgroup \(G\) of \(\text{Sym}(V)\) with a normalizing overgroup \(N\leq N_{\text{Sym}(V)}(G)\) 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 \(\text{PSL}(2,\mathbb Z)\cong C_2*C_3\), which is known to have infinitely many finite simple groups among its sections.

MSC:

20B22 Multiply transitive infinite groups
05E18 Group actions on combinatorial structures

Citations:

Zbl 1038.20002
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References:

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