Groupprops, The Group Properties Wiki (pre-alpha)

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From Groupprops

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

Contents 1 Definition 1.1 Symbol-free definition 1.2 Equivalence of definitions 2 Examples 3 Relation with other properties 3.1 Stronger properties 3.2 Weaker 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

Examples

VIEW: groups satisfying this property | groups dissatisfying property finite group | groups dissatisfying property abelian group
VIEW: |

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