(G,+)
x1,...,xs
G
x
G
x=n1x1+n2x2+ … +nsxs
n1,...,ns
\{x1,...,xs\}
G
x1,...,xs
G
Every finite abelian group is finitely generated. The finitely generated abelian groups can be completely classified.
\left(Z,+\right)
n
\left(Z/nZ,+\right)
There are no other examples (up to isomorphism). In particular, the group
\left(Q,+\right)
x1,\ldots,xn
k
1/k
x1,\ldots,xn
\left(Q*, ⋅ \right)
\left(R,+\right)
\left(R*, ⋅ \right)
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.
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
Zn ⊕ Z/q1Z ⊕ … ⊕ Z/qtZ,
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 ⊕ F
F\congG/tG
We can also write any finitely generated abelian group G as a direct sum of the form
Zn ⊕ Z/{k1}Z ⊕ … ⊕ Z/{ku}Z,
These statements are equivalent as a result of the Chinese remainder theorem, which implies that
Zjk\congZj ⊕ Zk
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, though the finitely generated case is sometimes instead credited to Poincaré in 1900; details follow.
Group theorist László Fuchs states:[3]
The fundamental theorem for finite abelian groups was proven by Leopold Kronecker in 1870, using a group-theoretic proof,[4] though without stating it in group-theoretic terms;[5] 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.[6] [7] Another group-theoretic formulation was given by Kronecker's student Eugen Netto in 1882.[8] [9]
The fundamental theorem for finitely presented abelian groups was proven by Henry John Stephen Smith in,[3] 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 thehomology 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.
Stated differently the fundamental theorem says that a finitely generated abelian group is the direct sum of a free abelian group of finite rank and a finite abelian group, each of those being unique up to isomorphism. The finite abelian group is just the torsion subgroup of G. The rank of G is defined as the rank of the torsion-free part of G; this is just the number n in the above formulas.
A corollary to the fundamental theorem is that every finitely generated torsion-free abelian group is free abelian. The finitely generated condition is essential here:
Q
Every subgroup and factor group of a finitely generated abelian group is again finitely generated abelian. The finitely generated abelian groups, together with the group homomorphisms, form an abelian category which is a Serre subcategory of the category of abelian groups.
Note that not every abelian group of finite rank is finitely generated; the rank 1 group
Q
Z2