Rank of an abelian group

Properties

• The rank of an abelian group A coincides with the dimension of the Q-vector space A ⊗ Q. If A is torsion-free then the canonical map A → A ⊗ Q is injective and the rank of A is the minimum dimension of Q-vector space containing A as an abelian subgroup. In particular, any intermediate group Zⁿ < A < Qⁿ has rank n.
• Abelian groups of rank 0 are exactly the periodic abelian groups.
• The group Q of rational numbers has rank 1. Torsion-free abelian groups of rank 1 are realized as subgroups of Q and there is a satisfactory classification of them up to isomorphism. By contrast, there is no satisfactory classification of torsion-free abelian groups of rank 2.
• Rank is additive over short exact sequences: if

${\displaystyle 0\to A\to B\to C\to 0}$

is a short exact sequence of abelian groups then rk B = rk A + rk C. This follows from the flatness of Q and the corresponding fact for vector spaces.

• Rank is additive over arbitrary direct sums:

${\displaystyle \mathrm{rank}\left(\underset{j\in J}{⨁}{A}_j\right)=\sum _{j\in J}\mathrm{rank}(A_j),}$

where the sum in the right hand side uses cardinal arithmetic.

Groups of higher rank

Abelian groups of rank greater than 1 are sources of interesting examples. For instance, for every cardinal d there exist torsion-free abelian groups of rank d that are indecomposable, i.e. cannot be expressed as a direct sum of a pair of their proper subgroups. These examples demonstrate that torsion-free abelian group of rank greater than 1 cannot be simply built by direct sums from torsion-free abelian groups of rank 1, whose theory is well understood. Moreover, for every integer ${\displaystyle n\ge 3}$, there is a torsion-free abelian group of rank ${\displaystyle 2n-2}$ that is simultaneously a sum of two indecomposable groups, and a sum of n indecomposable groups. Hence even the number of indecomposable summands of a group of an even rank greater or equal than 4 is not well-defined.

Another result about non-uniqueness of direct sum decompositions is due to A.L.S. Corner: given integers ${\displaystyle n\ge k\ge 1}$, there exists a torsion-free abelian group A of rank n such that for any partition ${\displaystyle n=r_1+\cdots +r_k}$ into k natural summands, the group A is the direct sum of k indecomposable subgroups of ranks ${\displaystyle r_1,r_2,\dots ,r_k}$. Thus the sequence of ranks of indecomposable summands in a certain direct sum decomposition of a torsion-free abelian group of finite rank is very far from being an invariant of A.

Other surprising examples include torsion-free rank 2 groups A_n,m and B_n,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

• K is indecomposable;
• K is generated by G and a single other element; and
• Every nonzero direct summand of G is decomposable.

Generalization

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:

${\displaystyle \mathrm{rank}(M)={\dim }_{R_0}M{\otimes }_R{R}_0}$

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 every 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,

${\displaystyle x{\otimes }_{\mathbf{Z}}q=0.}$