Orbital digraphs of infinite primitive permutation groups

Smith, Simon (2007) Orbital digraphs of infinite primitive permutation groups. Journal of Group Theory, 10 (6). ISSN 1433-5883

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Abstract

If G is a group acting on a set Ω, and α, β ∈ Ω, the digraph whose vertex set is Ω and whose arc set is the orbit (α, β)G is called an orbital digraph of G. Each orbit of the stabilizer G α acting on Ω is called a suborbit of G.

A digraph is locally finite if each vertex is adjacent to at most finitely many other vertices. A locally finite digraph Γ has more than one end if there exists a finite set of vertices X such that the induced digraph Γ\X contains at least two infinite connected components; if there exists such a set containing precisely one element, then Γ has connectivity one.

In this paper we show that if G is a primitive permutation group whose suborbits are all finite, possessing an orbital digraph with more than one end, then G has a primitive connectivityone orbital digraph, and this digraph is essentially unique. Such digraphs resemble trees in many respects, and have been fully characterized in a previous paper by the author.

Keywords:Infinite permutation groups, Orbital graphs
Subjects:G Mathematical and Computer Sciences > G110 Pure Mathematics
Divisions:College of Science > School of Mathematics and Physics
ID Code:27495
Deposited On:12 May 2017 10:08

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