On graph-restrictive permutation groups
Let $Γ$ be a connected $G$-vertex-transitive graph, let $v$ be a vertex of $Γ$ and let $L=G_v^{Γ(v)}$ be the permutation group induced by the action of the vertex-stabiliser $G_v$ on the neighbourhood $Γ(v)$. Then $(Γ,G)$ is said to be \emph{locally-$L$}. A transitive permutation group $L$ is \emph{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 $|G_{uv}|\leq c(L)$ 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 \emph{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.