On the Erdős–Ko–Rado property for finite groups

Abstract

Let a finite group $$G$$G act transitively on a finite set $$X$$X. A subset $$S\subseteq G$$S⊆G is said to be intersecting if for any $$s_1,s_2\in S$$s1,s2?S, the element $$s_1^{-1}s_2$$s1-1s2 has a fixed point. The action is said to have the weak Erd?s---Ko---Rado (EKR) property, if the cardinality of any intersecting set is at most $$|G|/|X|$$|G|/|X|. If, moreover, any maximum intersecting set is a coset of a point stabilizer, the action is said to have the strong EKR property. In this paper, we will investigate the weak and strong EKR property and attempt to classify groups in which all transitive actions have these properties. In particular, we show that a group with the weak EKR property is solvable and that a nilpotent group with the strong EKR property is the direct product of a $$2$$2-group and an abelian group of odd order.