Free abelian group
In abstract algebra, the free abelian group on a set X may be constructed as the abelian group of functions on X, taking integer values all but finitely many of which are zero. One can verify directly that this has the appropriate universal property in relation to arbitrary functions on X with values in some abelian group A: namely unique extension to a homomorphism of the free group.When X is finite of cardinality n the free abelian group on X is the same up to isomorphism as the product of n copies of the infinite cyclic group. This breaks down for infinite X, though.
This construction is a special case of the construction of free modules.
