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.

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