Every

finite abelian group is the

direct sum of

cyclic groups, each of

prime power order. Additionally, two finite

abelian groups are

isomorphic iff their

representations as direct sums of cyclic groups of

prime power

order are the same (up to permutation of the cyclic groups, of course). These prime powers are called the

elementary divisors of the group.

For example, there are exactly four abelian groups of order 36 (up to isomorphism):

**Z**_2 (+) **Z**_2 (+) **Z**_3 (+) **Z**_3 = **Z**_6 (+) **Z**_6
**Z**_2 (+) **Z**_2 (+) **Z**_9 = **Z**_2 (+) **Z**_18
**Z**_4 (+) **Z**_3 (+) **Z**_3 = **Z**_3 (+) **Z**_12
**Z**_4 (+) **Z**_9 = **Z**_36

The FTFAG is a complete classification of finite abelian groups.