4.1. Absence of the Weak EKR Property for non-Solvable Groups 8

Abstract. Let a finite group G act transitively on a finite set X. A subset S ⊆ G is said to be intersecting if for any s1, s2 ∈ S, the element s−11 s2 has a fixed point. The action is said to have the weak Erdős-Ko-Rado property, if the cardinality of any intersecting set is at most |G|/|X|. If, moreover, any maximal intersecting set is a coset of a point stabilizer, the action is said to have the strong Erdős-Ko-Rado property. In this paper we will investigate the weak and strong Erdős-Ko-Rado property and attempt to classify the groups whose all transitive actions have these properties. In particular, we show that a group with the weak Erdős-Ko-Rado property is solvable and that a nilpotent group with the strong Erdős-Ko-Rado property is product of a 2-group and an abelia