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. We call T the torsion subgroup of G. T itself is a direct sum; it is the direct sum of a finite number of finite cyclic groups whose orders are powers of primes. The orders off these groups are uniquely determined by T, and hence by G, and are called the elementary divisors of G.

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