# Free abelian group

## Definition

A free abelian group is an abelian group satisfying the following equivalent conditions:

No. Shorthand A group is a free abelian group if ... A group $G$ is a free abelian group if ...
1 Direct sum of copies of integers it is isomorphic to a possibly infinite restricted direct product (also called direct sum) of copies of the group of integers $G \cong \bigoplus_{i \in I} \mathbb{Z}$, where $I$ is some set.
2 Freely generating set, in terms of unique integer combinations there exists a subset of the group such that every element of the group has a unique expression as an integer linear combination of elements of the subset. Such a subset is termed a freely generating set or a basis. there exists a subset $S$ of $G$ such that every element of $G$ can be written uniquely as $\sum_{s \in S} a_ss$, where $a_s$ is an integer and only finitely many $a_s$s are nonzero
3 Freely generating set, in terms of universal property there exists a subset of the group such that any set map from that subset to any abelian group extends uniquely to a group homomorphism from the whole group to that abelian group there exists a subset $S$ of $G$ such that any set map $f$ from $S$ to an abelian group $H$ extends uniquely to a group homomorphism from $G$ to $H$
4 Projective object in category of abelian groups any surjective homomorphism from an abelian group to it has a one-sided inverse for any surjective homomorphism $\varphi:H \to G$ with $H$ an abelian group, there exists an injective homomorphism $\alpha:G \to H$ such that $\varphi \circ \alpha$ is the identity map on $G$.
5 Abelianization of free group it occurs as the abelianization of a free group, i.e., as the quotient of a free group by its derived subgroup there is a free group $F$ such that $G \cong F/[F,F]$

The rank of a free abelian group is defined as the cardinality of a freely generating set for it. The rank of a free abelian group is fixed; in other words, any two freely generating sets of a free abelian group have the same cardinality. Further information: free abelian groups satisfy IBN

The free abelian group of rank $\alpha$ is isomorphic to a direct sum of $\alpha$ copies of the group of integers. In particular, the free abelian group of rank $n$ for a natural number $n$ is isomorphic to the group $\mathbb{Z}^n$, which is a direct product of $n$ copies of the group of integers.

## Examples

### Extreme examples

• The trivial group is a free abelian group of rank $0$.
• The group of integers is a free abelian group of rank $1$.

### Non-examples

An (unrestricted) external direct power of the group of integers need not be free abelian. In fact, it is not free abelian if it is an infinite power. The smallest counterexample is the Baer-Specker group, which arises as the external direct product of countably many copies of the group of integers.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Finitely generated free abelian group

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Torsion-free abelian group abelian, no non-identity element of finite order |FULL LIST, MORE INFO
Reduced free group quotient of a free group by a verbal subgroup |FULL LIST, MORE INFO
Residually finite group free abelian implies residually finite |FULL LIST, MORE INFO

## Metaproperties

Metaproperty name Satisfied? Proof
Subgroup-closed group property Yes free abelian is subgroup-closed
Quotient-closed group property No in fact, every abelian group is a quotient of a free abelian group
Finite-direct product-closed group property Yes