Classification of Finite Abelian Groups

Abstract

In this chapter, we study the finite abelian groups. We show that any finite abelian group is isomorphic to the product of additive groups $${\mathbb{Z}}_{{n_{1} }} \times {\mathbb{Z}}_{{n_{2} }} \times \ldots \times {\mathbb{Z}}_{{n_{k} }}$$ for some positive integers $$n_{1} , n_{2} , \ldots , n_{k}$$ . The first section is devoted to study cyclic groups (finite and infinite). The cyclic groups form examples of abelian groups that are described in Chap 7. We shall see that, up to isomorphism, there is only one infinite cyclic group $$\left( {{\mathbb{Z}}, + } \right)$$ , and for each n∈N, the additive group $$\left( {{\mathbb{Z}}_{n} , \oplus_{n} } \right)$$ is the only cyclic group of order n. In Sect. 9.2, we define and study primary groups, and in Sect. 9.3, we study independent and spanning subsets of an abelian group. The primary groups, the independent subset, and spanning subset in abelian groups are required to prove the fundamental theorem of finite abelian groups in Sect. 9.4.