#### Thesis Title

Finite Groups Acting on Sets With Applications

Summer 1975

#### Degree

Master of Science in Mathematics

Mathematics

James Downing

#### Abstract

In the May-June 1973 issue of the Mathematics Magazine appears an article entitled "Finite Groups Acting on Sets With Applications," by Louis W. Shapiro. The article poses a series of problems in a step-by-step manner involving finite group action on sets, the Polya-Burnside Theorem, and the First and Second Sylow Theorems. The purpose or aim of this thesis is to give the mathematical proofs and solutions to the selected propositions. However, it will be assumed that the reader has had a first course in modern algebra. Finite groups acting on sets is a different viewpoint of group theory. Instead of using the usual definition of a group as a set G together with a binary operation satisfying the associative, identity, and inverse axioms, G will be used as a group of functions and the so called binary operation as composition of functions acting on a set S. In this way, a different viewpoint of group theory may be obtained in solving a variety of problems. Chaper 1 and Chapter 2 lay the basic foundation of finite groups acting on sets with proofs and solutions to propositions. In Chapter 3, the Polya-Burnside Theorem is discussed with related problems and solutions involving group action. Chapter 4 deals with the Strong Cayley Theorem and the First and Second Sylow Theorem. This thesis has given the proofs and solutions to selected problems from the article, "Finite Groups Acting on Sets With Applications," by Louis W. Shapiro. Thus, the basic concept of a group acting on a set is a generalization of a permutation group. Therefore, a different viewpoint of group theory has been demonstrated in solving a variety of problems.

Mathematics