The Erd\H{o}s-Ko-Rado property for some permutation groups

Abstract

A subset in a group $G \leq Sym(n)$ is intersecting if for any pair of permutations $\pi,\sigma$ in the subset there is an $i \in \{1,2,\dots,n\}$ such that $\pi(i) = \sigma(i)$. If the stabilizer of a point is the largest intersecting set in a group, we say that the group has the Erd\H{o}s-Ko-Rado (EKR) property. Moreover, the group has the strict EKR property if every intersecting set of maximum size in the group is either the stabilizer of a point or the coset of the stabilizer of a point. In this paper we look at several families of permutation groups and determine if the groups have either the EKR property or the strict EKR property. First, we prove that all cyclic groups have the strict EKR property. Next we show that all dihedral and Frobenius groups have the EKR property and we characterize which ones have the strict EKR property. Further, we show that if all the groups in an external direct sum or an internal direct sum have the EKR (or strict EKR) property, then the product does as well. Finally, we show that the wreath product of two groups with EKR property also has the EKR property.