2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/221988We explicitly determine all of the transitive groups of degree p-squared, p a prime, whose Sylow p-subgroup is not the wreath product of two cyclic groups of order p. Furthermore, we provide a general description of the transitive groups of degree p-squared whose Sylow p-subgroup is such a wreath product, and explicitly determine most of them. As applications, we solve the Cayley Isomorphism problem for Cayley objects of an abelian group of order p-squared, explicitly determine the full automorphism group of Cayley graphs of abelian groups of order p-squared, and find all nonnormal Cayley graphs of order p-squared.29 pages, no figures. This version corrects an error in the statement of Theorem 12Group TheoryCombinatorics20B35; 05C25, 05E20text