Characterization of Weak EKR Groups and Intersection Densities with Prescribed Point Stabilizers

Abstract

A finite group has the weak Erdos-Ko-Rado property if all of its transitive permutation actions have the EKR property. We characterize this property in terms of normal subgroups and chief factors. More precisely, we introduce a local intersection density and establish a normal-extension criterion which reduces the weak EKR property to difference-set conditions on the elementary abelian chief factors and the linear groups induced on them. For chief factors of rank one the condition is automatically satisfied, and for chief factors of rank two, this condition is equivalent to the induced linear group being intransitive on the one-dimensional subspaces. In the second part of the paper, we solve an open problem by determining the possible intersection densities of transitive permutation groups with a prescribed point stabilizer. We prove that, for every finite group $H$ of order $m\geq 4$ and every integer $n\geq m$, there exists a faithful transitive permutation group with point stabilizer isomorphic to $H$ and intersection density $n/m$.