rank. A free abelian group is finitely generated if and only if its rank is a finite number , in which case the group is isomorphic to . This notion of rank can be generalized, from free abelian groups to abelian groups that are not necessarily free. The rank of an abelian group is defined as the rank of a free abelian subgroup of for which the quotient group is a torsion group. Equivalently, it is the cardinality of a maximal subset of that generates a free subgroup. The rank is a group invariant: it does not depend on the choice of the subgroup. Subgroups Every subgroup of a free abelian group is itself a free abelian group. This result of Richard Dedekind was a precursor to the analogous Nielsen–Schreier theorem that every subgroup of a free group is free, and is a generalization of the fact that every nontrivial subgroup of the infinite cyclic group is infinite cyclic. The proof needs the axiom of choice. A proof using Zorn's lemma (one of many equivalent assumptions to the axiom of choice) can be found in Serge Lang's Algebra. Solomon Lefschetz and Irving Kaplansky argue that using the well-ordering principle in place of Zorn's lemma leads to a more intuitive proof. In the case of finitely generated free abelian groups, the proof is easier, does not need the axiom of choice, and leads to a more precise result. If is a subgroup of a finitely generated free abelian group , then is free and there exists a basis of and positive integers (that is, each one divides the next one) such that is a basis of Moreover, the sequence depends only on and and not on the basis. A constructive proof of the existence part of the theorem is provided by any algorithm computing the Smith normal form of a matrix of integers. Uniqueness follows from the fact that, for any , the greatest common divisor of the minors of rank of the matrix is not changed during the Smith normal form computation and is the product at the end of the computation. Torsion and divisibility All free abelian groups are torsion-free, meaning that there is no non-identity group element and nonzero integer such that . Conversely, all finitely generated torsion-free abelian groups are free abelian. The additive group of rational numbers provides an example of a torsion-free (but not finitely