Transitive permutation groups of prime-squared degree

We explicitly determine all of the transitive groups of degree p(2), p a prime, whose Sylow p-subgroup is not isomorphic to the wreath product Z(p) Z(p). Furthermore, we provide a general description of the transitive groups of degree p(2) whose Sylow p-subgroup is isomorphic to Z(p) Z(p), and explicitly determine most of them. As applications, we solve the Cayley Isomorphism problem for Cayley objects of an abelian group of order p(2), explicitly determine the full automorphism group of Cayley graphs of abelian groups of order p(2), and find all nonnormal Cayley graphs of order p(2).