Free Group - Free Abelian Group

Free Abelian Group

The free abelian group on a set S is defined via its universal property in the analogous way, with obvious modifications: Consider a pair (F, φ), where F is an abelian group and φ: S → F is a function. F is said to be the free abelian group on S with respect to φ if for any abelian group G and any function ψ: S → G, there exists a unique homomorphism f: F → G such that

f(φ(s)) = ψ(s), for all s in S.

The free abelian group on S can be explicitly identified as the free group F(S) modulo the subgroup generated by its commutators, i.e. its abelianisation. In other words, the free abelian group on S is the set of words that are distinguished only up to the order of letters. The rank of a free group can therefore also be defined as the rank of its abelianisation as a free abelian group.