Kronecker Decomposition Theorem

Every Finite Abelian Group can be written as a Direct Product of Cyclic Groups of Prime Power Orders. In fact, the number of nonisomorphic Abelian Finite Groups $a(n)$ of any given Order $n$ is given by writing $n$ as

$$n=\prod_i {p_i}^{\alpha_i},$$

where the $p_i$ are distinct Prime Factors, then

$$a(n)=\prod_i P(\alpha_i),$$

where $P(n)$ is the Partition Function. This gives 1, 1, 1, 2, 1, 1, 1, 3, 2, ... (Sloane's A000688).

See also Abelian Group, Finite Group, Order (Group), Partition Function P

References

Sloane, N. J. A. Sequence A000688/M0064 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.