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
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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				|FULL LIST, MORE INFO
Finite group that is 1-isomorphic to an abelian group				|FULL LIST, MORE INFO
Finite group that is order statistics-equivalent to an abelian group				|FULL LIST, MORE INFO