Free Abelian Group

In abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Hence, free abelian groups over a basis B are also known as formal sums over B. Informally, free abelian groups or formal sums may also be seen as signed multisets with elements in B.

Free abelian groups have very nice properties which make them similar to vector spaces and allow a general abelian group to be understood as a quotient of a free abelian group by "relations". Every free abelian group has a rank defined as the cardinality of a basis. The rank determines the group up to isomorphism, and the elements of such a group can be written as finite formal sums of the basis elements. Every subgroup of a free abelian group is itself free abelian, which is important for the description of a general abelian group as a cokernel of a homomorphism between free abelian groups.