# Free abelian group

## Contents

## 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 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 | , where 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 of such that every element of can be written uniquely as , where is an integer and only finitely many 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 of such that any set map from to an abelian group extends uniquely to a group homomorphism from to |

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 with an abelian group, there exists an injective homomorphism such that is the identity map on . |

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 such that |

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 is isomorphic to a direct sum of copies of the group of integers. In particular, the free abelian group of rank for a natural number is isomorphic to the group , which is a direct product of copies of the group of integers.

## Examples

### Extreme examples

- The trivial group is a free abelian group of rank .
- The group of integers is a free abelian group of rank .

### 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 |