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 ![]() |
---|---|---|---|
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 | ![]() ![]() |
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 ![]() ![]() ![]() ![]() ![]() ![]() |
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 ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 ![]() ![]() ![]() ![]() ![]() |
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 ![]() ![]() |
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 |