Every finite abelian group
is the direct sum
of cyclic group
s, each of prime power order
. Additionally, two finite abelian group
s are isomorphic iff
s as direct sums of cyclic groups of prime
are the same (up to permutation of the cyclic groups, of course). These prime powers are called the elementary divisor
s 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.