Finite abelian group

This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: finite group and Abelian group
View other group property conjunctions OR view all group properties

Definition

Symbol-free definition

A finite Abelian group is a group satisfying the following equivalent conditions:

1. It is both finite and Abelian.
2. It is isomorphic to a direct product of finitely many finite cyclic groups.
3. It is isomorphic to a direct product of Abelian groups of prime power order.
4. It is isomorphic to a direct product of cyclic groups of prime power order.

Equivalence of definitions

For full proof, refer: Structure theorem for finitely generated Abelian groups

Relation with other properties

Stronger properties

• Abelian group of prime power order
• Finite cyclic group
• Odd-order Abelian group

Weaker properties

• Finitely generated Abelian group