Rank of an abelian group

In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset.[1] 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.

The term rank has a different meaning in the context of elementary abelian groups.

where all but finitely many coefficients nα are zero (so that the sum is, in effect, finite), then all summands are 0. Any two maximal linearly independent sets in A have the same cardinality, which is called the rank of A.

The rank of an abelian group is analogous to the dimension of a vector space. The main difference with the case of vector space is a presence of torsion. An element of an abelian group A is classified as torsion if its order is finite. The set of all torsion elements is a subgroup, called the torsion subgroup and denoted T(A). A group is called torsion-free if it has no non-trivial torsion elements. The factor-group A/T(A) is the unique maximal torsion-free quotient of A and its rank coincides with the rank of A.

The notion of rank with analogous properties can be defined for modules over any integral domain, the case of abelian groups corresponding to modules over Z. For this, see finitely generated module#Generic rank.

Other surprising examples include torsion-free rank 2 groups An,m and Bn,m such that An is isomorphic to Bn if and only if n is divisible by m.

For abelian groups of infinite rank, there is an example of a group K and a subgroup G such that

The notion of rank can be generalized for any module M over an integral domain R, as the dimension over R0, the quotient field, of the tensor product of the module with the field:

It makes sense, since R0 is a field, and thus any module (or, to be more specific, vector space) over it is free.

It is a generalization, since any abelian group is a module over the integers. It easily follows that the dimension of the product over Q is the cardinality of maximal linearly independent subset, since for any torsion element x and any rational q