ASL STEM

Definition

If G is abelian and finitely generated, then there is a fundamental theorem to the effect that G is the direct sum of two subgroups, H and T. H is a free abelian of finite rank, and T is a subgroup of G consisting of all elements of finite order. The rank of H is uniquely determined by G, since it equals the rank of the quotient G by its torsion subgroup. This number is often called the betti number of G.

Source: Topology (second edition) by James R. Munkres