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 of any given
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, ... (Sloane's A000688).

**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.

© 1996-9

1999-05-26