Free Abelian Group
Further information: free abelian groupThe 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.
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