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.