Kronecker Decomposition Theorem

Every finite Abelian group can be written as a group direct product of cyclic groups of prime power group orders. In fact, the number of nonisomorphic Abelian finite groups of any given group order is given by writing as

where the are distinct prime factors, then

where is the partition function. This gives 1, 1, 1, 2, 1, 1, 1, 3, 2, ... (OEIS A000688).

More generally, every finitely generated Abelian group is isomorphic to the group direct sum of a finite number of groups, each of which is either cyclic of prime power order or isomorphic to . This extension of Kronecker decomposition theorem is often referred to as the Kronecker basis theorem.