of a free abelian group is itself free abelian; this fact allows a general abelian group to be understood as a quotient of a free abelian group by "relations", or as a cokernel of an injective homomorphism between free abelian groups. The only free abelian groups that are free groups are the trivial group and the infinite cyclic group. Definition and examples A free abelian group is an abelian group that has a basis. Here, being an abelian group means that it is described by a set of its elements and a binary operation conventionally denoted as an additive group by the symbol (although it need not be the usual addition of numbers) that obey the following properties: The operation is commutative and associative, meaning for all elements and Therefore, when combining two or more elements of using this operation, the ordering and grouping of the elements does not affect the result. contains an identity element (conventionally denoted with the property that, for every Every element in has an inverse element such that A basis is a subset of the elements of with the property that every element of may be formed in a unique way by choosing finitely many basis elements choosing a nonzero integer for each of the chosen basis elements, and adding together copies of the basis elements for which is positive, and copies of for each basis element for which is negative. As a special case, the identity element can always be formed in this way as the combination of zero basis elements, according to the usual convention for an empty sum, and it must not be possible to find any other combination that represents the identity. The under the usual addition operation, form a free abelian group with the The integers are commutative and associative, with 0 as the additive identity and with each integer having an additive inverse, its negation. Each non-negative is the sum of copies and each negative integer is the sum of copies so the basis property is also satisfied. An example where the group operation is different from the usual addition of numbers is given by the positive rational numbers which form a free abelian group with the usual multiplication operation on numbers and with the prime numbers as their basis. Multiplication is commutative and associative, with the number as its identity and with as the inverse element for each