On graph-restrictive permutation groups

Abstract

Let Γ be a connected G-vertex-transitive graph, let v be a vertex of Γ and let L=GvΓ(v) be the permutation group induced by the action of the vertex-stabiliser Gv on the neighbourhood Γ(v). Then (Γ,G) is said to be locally-L. A transitive permutation group L is graph-restrictive if there exists a constant c(L) such that, for every locally-L pair (Γ,G) and an arc (u,v) of Γ, the inequality |Guv|⩽c(L) holds.Using this terminology, the Weiss Conjecture says that primitive groups are graph-restrictive. We propose a very strong generalisation of this conjecture: a group is graph-restrictive if and only if it is semiprimitive. (A transitive permutation group is said to be semiprimitive if each of its normal subgroups is either transitive or semiregular.) Our main result is a proof of one of the two implications of this conjecture, namely that graph-restrictive groups are semiprimitive. We also collect the known results and prove some new ones regarding the other implication.