## 8

## Structure Theory of Groups

It is well known that any cyclic group is abelian and the product of any class of abelian groups is abelian. In this chapter, we prove the celebrated theorem known as the Fundamental Theorem of finitely generated abelian groups which states that any finitely generated abelian group is a product of finite number of cyclic groups. This amounts to saying that the cyclic groups are like ‘building blocks’ for the finite or finitely generated abelian groups. Since any cyclic group is isomorphic to the group ℤ of integers or the group ℤ_{n} of integers modulo *n* for some positive integers, the Fundamental ...