The action is doubly transitive if some permutation takes any pair of elements to any other pair.
Assume the action of G on S is transitive, and |S| = p (prime). If G has a kernel that maps to the identity permutation, mod out by the kernel and call the factor group H. Now H acts effectively on S.
since the orbit has size p, |H| is divisible by p. By Macay's theorem, H contains a subgroup Zp. The only possible permutation with period p is a p cycle. Thus the action of G includes a p cycle.
If S is divisible by p, G contains an element of order p, whose action on S is one or more disjoint p cycles.