ABSTRACT
A group is arguably the most important structure in abstract mathematics. In essence, a group is any set of objects—for example, numbers, functions, vectors, etc.—combined with a binary operation—such as addition, multiplication, composition, etc.—satisfying certain fundamental properties. Abelian groups—named after the Norwegian mathematician Niels Abel—play a central role in most branches of mathematics; our goal here is to investigate some of their fascinating number theoretic properties. Other well-studied abelian groups include the set of vectors in n-dimensional space, the set of n-by-m real matrices, and the set of real polynomials. Another example of an abelian group is the set of even integers; however, the set of odd integers is not a group, since the closure and zero properties fail. The groups studied, however, are finite groups: those that have only a finite number of elements. The simplest family of finite abelian groups are the cyclic groups.
