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.