On the Erdos-Ko-Rado property for finite Groups

Abstract

Let a finite group $G$ act transitively on a finite set $X$. A subset $S\subseteq G$ is said to be {\it intersecting} if for any $s_1,s_2\in S$, the element $s_1^{-1}s_2$ has a fixed point. The action is said to have the {\it weak Erd\H{o}s-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 {\it strong Erd\H{o}s-Ko-Rado} property. In this paper we will investigate the weak and strong Erd\H{o}s-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 Erd\H{o}s-Ko-Rado property is solvable and that a nilpotent group with the strong Erd\H{o}s-Ko-Rado property is product of a $2$-group and an abelian group of odd order.