Rank of An Abelian Group - Properties

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

is a s.e.s. 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:

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