In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group "A" is the cardinality of a maximal linearly independent subset. The rank of "A" determines the size of the largest free abelian group contained in "A". If "A" is torsion-free then it embeds into a vector space over the rational numbers of dimension rank "A". For finitely generated abelian groups, rank is a strong invariant and every such group is determined up to isomorphism by its rank and torsion subgroup. Torsion-free abelian groups of rank 1 have been completely classified. However, the theory of abelian groups of higher rank is more involved.