  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.
