# 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

## Examples

VIEW: groups satisfying this property | groups dissatisfying property finite group | groups dissatisfying property abelian group
VIEW: 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 $G$ is a finite abelian group and $H$ is a subgroup of $G$, then $H$ is also a finite abelian group.
quotient-closed group property Yes follows from abelianness is quotient-closed If $G$ is a finite abelian group and $H$ is a normal subgroup of $G$, then the quotient group $G/H$ is also a finite abelian group.
finite direct product-closed group property Yes follows from abelianness is direct product-closed If $G_1, G_2, \dots, G_n$ are finite abelian groups, so is the external direct product $G_1 \times G_2 \times \dots \times G_n$.
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 $G_1, G_2$ with isomorphic lattices of subgroups such that $G_1$ is finite abelian and $G_2$ 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

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Finitely generated abelian group |FULL LIST, MORE INFO
Periodic abelian group |FULL LIST, MORE INFO
Finite nilpotent group Finite Lazard Lie group, Finite group that is 1-isomorphic to an abelian group, Finite group that is order statistics-equivalent to an abelian group|FULL LIST, MORE INFO
Finite group that is 1-isomorphic to an abelian group Finite group admitting a bijective quasihomomorphism to an abelian group|FULL LIST, MORE INFO
Finite group that is order statistics-equivalent to an abelian group Finite group admitting a bijective quasihomomorphism to an abelian group|FULL LIST, MORE INFO