The purpose of this paper is to display in some detail the theory of groups with operators leading to the Jordan-Holder Theorem on composition series. A number of examples are given to illustrate the theory. The author hopes that the material which follows is more easily read than the well known sources on group theory. In 1832, Evariste Galois introduced the concept of group in a letter written to a friend on the eve of his tragic death in a duel at the age of twenty-one. The Norwegian mathematician Neils Hendrick Abel, among others, has contributed much to the development of group theory; Wolfgang Krull and Emmy Noether in particular made notable contributions to the theory of groups with operators. The theory of groups has found widespread application in many branches of mathematics. / Advisor: Nickolas Heerema, Professor Directing Paper. / Typescript. / "January, 1954." / "Submitted to the Graduate Council of Florida State University in partial fulfillment of the requirements for the degree of Master of Science." / Includes bibliographical references (leaf 44).