# Finite abelian group

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

## Definition

### Symbol-free definition

A **finite abelian group** is a group satisfying the following equivalent conditions:

- It is both finite and abelian.
- It is isomorphic to a direct product of finitely many finite cyclic groups.
- It is isomorphic to a direct product of abelian groups of prime power order.
- 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 groupVIEW: Related group property satisfactions | Related group property dissatisfactions

## Metaproperties

Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|

subgroup-closed group property | Yes | follows from abelianness is subgroup-closed | If is a finite abelian group and is a subgroup of , then is also a finite abelian group. |

quotient-closed group property | Yes | follows from abelianness is quotient-closed | If is a finite abelian group and is a normal subgroup of , then the quotient group is also a finite abelian group. |

finite direct product-closed group property Yes | follows from abelianness is direct product-closed | If are finite abelian groups, so is the external direct product . | |

lattice-determined group property | No | there exists an abelian group of prime power order that is lattice-isomorphic to a non-abelian group not of prime power order | There exist groups with isomorphic lattices of subgroups such that is finite abelian and is not. |

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

abelian group of prime power order | ||||

finite cyclic group | |FULL LIST, MORE INFO | |||

odd-order abelian group | |FULL LIST, MORE INFO |