The
fundamental theorem of finitely generated abelian groups can be stated two ways, generalizing the two forms of the
fundamental theorem of finite abelian groups. The theorem, in both forms, in turn generalizes to the
structure theorem for finitely generated modules over a principal ideal domain, which in turn admits further generalizations.
Primary decomposition The primary decomposition formulation states that every finitely generated abelian group
G is isomorphic to a
direct sum of
primary cyclic groups and infinite
cyclic groups. A primary cyclic group is one whose
order is a power of a
prime. That is, every finitely generated abelian group is isomorphic to a group of the form :\mathbb{Z}^n \oplus \mathbb{Z}/q_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/q_t\mathbb{Z}, where
n ≥ 0 is the
rank, and the numbers
q1, ...,
qt are powers of (not necessarily distinct) prime numbers. In particular,
G is finite if and only if
n = 0. The values of
n,
q1, ...,
qt are (
up to rearranging the indices) uniquely determined by
G, that is, there is one and only one way to represent
G as such a decomposition. The proof of this statement uses the basis theorem for
finite abelian group: every finite abelian group is a
direct sum of
primary cyclic groups. Denote the
torsion subgroup of
G as
tG. Then,
G/tG is a
torsion-free abelian group and thus it is free abelian.
tG is a
direct summand of
G, which means there exists a subgroup
F of
G s.t. G=tG\oplus F, where F\cong G/tG. Then,
F is also free abelian. Since
tG is finitely generated and each element of
tG has finite order,
tG is finite. By the basis theorem for finite abelian group,
tG can be written as direct sum of primary cyclic groups.
Invariant factor decomposition We can also write any finitely generated abelian group
G as a direct sum of the form :\mathbb{Z}^n \oplus \mathbb{Z}/{k_1}\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/{k_u}\mathbb{Z}, where
k1
divides k2, which divides
k3 and so on up to
ku. Again, the rank
n and the
invariant factors k1, ...,
ku are uniquely determined by
G (here with a unique order). The rank and the sequence of invariant factors determine the group up to isomorphism.
Equivalence These statements are equivalent as a result of the
Chinese remainder theorem, which implies that \mathbb{Z}_{jk}\cong \mathbb{Z}_{j} \oplus \mathbb{Z}_{k} if and only if
j and
k are
coprime.
History The history and credit for the fundamental theorem is complicated by the fact that it was proven when group theory was not well-established, and thus early forms, while essentially the modern result and proof, are often stated for a specific case. Briefly, an early form of the finite case was proven by Gauss in 1801, the finite case was proven by Kronecker in 1870, and stated in group-theoretic terms by Frobenius and Stickelberger in 1878. The
finitely presented case is solved by
Smith normal form, and hence frequently credited to , The fundamental theorem for
finite abelian groups was proven by
Leopold Kronecker in 1870, using a group-theoretic proof, though without stating it in group-theoretic terms; a modern presentation of Kronecker's proof is given in , 5.2.2 Kronecker's Theorem, 176–177. This generalized an earlier result of
Carl Friedrich Gauss from
Disquisitiones Arithmeticae (1801), which classified quadratic forms; Kronecker cited this result of Gauss's. The theorem was stated and proved in the language of groups by
Ferdinand Georg Frobenius and
Ludwig Stickelberger in 1878. Another group-theoretic formulation was given by Kronecker's student
Eugen Netto in 1882. The fundamental theorem for
finitely presented abelian groups was proven by
Henry John Stephen Smith in , as integer matrices correspond to finite presentations of abelian groups (this generalizes to finitely presented modules over a principal ideal domain), and
Smith normal form corresponds to classifying finitely presented abelian groups. The fundamental theorem for
finitely generated abelian groups was proven by
Henri Poincaré in 1900, using a matrix proof (which generalizes to principal ideal domains). This was done in the context of computing the
homology of a complex, specifically the
Betti number and
torsion coefficients of a dimension of the complex, where the Betti number corresponds to the rank of the free part, and the torsion coefficients correspond to the torsion part. Kronecker's proof was generalized to
finitely generated abelian groups by Emmy Noether in 1926. ==Corollaries==