# Fundamental Theorem of Finitely Generated Abelian Groups

## Primary decomposition

Every finitely generated abelian group is isomorphic to a group of the form where and are powers of (not necessarily distinct) prime numbers. The values of are (up to rearrangement) uniquely determined by .

## Invariant factor decomposition

We can also write as a direct sum of the form where . Again the rank and the invariant factors are uniquely determined by .