Fundamental theorem of finite abelian groups

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):

groups of order 8	proof of Sylow's theorem	cyclic group	direct sum
Sylow's theorem	classification theorem of finite simple groups	group	Taylor polynomial
finite field	Abelian	abelian group	minimal criminal
wreath product	isomorphic	Classification	Infinite monkeys theorem
taylor	Fundamental group	functor	Fundamental theorem of arithmetic
Fundamental theorem of calculus	The math Project	consistent	Pascal's Triangle