In abstract algebra, an abelian group <math>(G,+)</math> is called finitely generated if there exist finitely many elements <math>x_1,\dots,x_s</math> in <math>G</math> such that every <math>x</math> in <math>G</math> can be written in the form <math>x = n_1x_1 + n_2x_2 + \cdots + n_sx_s</math> for some integers <math>n_1,\dots, n_s</math>. In this case, we say that the set <math>\{x_1,\dots, x_s\}</math> is a generating set of <math>G</math> or that <math>x_1,\dots, x_s</math> generate <math>G</math>. So, finitely generated abelian groups can be thought of as a generalization of cyclic groups.

Every finite abelian group is finitely generated. The finitely generated abelian groups can be completely classified.

Examples

  • The integers, <math>\left(\mathbb{Z},+\right)</math>, are a finitely generated abelian group.
  • The integers modulo <math>n</math>, <math>\left(\mathbb{Z}/n\mathbb{Z},+\right)</math>, are a finite (hence finitely generated) abelian group.
  • Any direct sum of finitely many finitely generated abelian groups is again a finitely generated abelian group.
  • Every lattice forms a finitely generated free abelian group.

There are no other examples (up to isomorphism). In particular, the group <math>\left(\mathbb{Q},+\right)</math> of rational numbers is not finitely generated: if <math>x_1,\ldots,x_n</math> are rational numbers, pick a natural number <math>k</math> coprime to all the denominators; then <math>1/k</math> cannot be generated by <math>x_1,\ldots,x_n</math>. The group <math>\left(\mathbb{Q}^*,\cdot\right)</math> of non-zero rational numbers is also not finitely generated. The groups of real numbers under addition <math> \left(\mathbb{R},+\right)</math> and non-zero real numbers under multiplication <math>\left(\mathbb{R}^*,\cdot\right)</math> are also not finitely generated.

Classification

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

:<math>\mathbb{Z}^n \oplus \mathbb{Z}/q_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/q_t\mathbb{Z},</math>

where n ≥ 0 is the rank, and the numbers q<sub>1</sub>, ..., q<sub>t</sub> are powers of (not necessarily distinct) prime numbers. In particular, G is finite if and only if n = 0. The values of n, q<sub>1</sub>, ..., q<sub>t</sub> 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. <math>G=tG\oplus F</math>, where <math>F\cong G/tG</math>. 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

:<math>\mathbb{Z}^n \oplus \mathbb{Z}/{k_1}\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/{k_u}\mathbb{Z},</math>

where k<sub>1</sub> divides k<sub>2</sub>, which divides k<sub>3</sub> and so on up to k<sub>u</sub>. Again, the rank n and the invariant factors k<sub>1</sub>, ..., k<sub>u</sub> 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 <math>\mathbb{Z}_{jk}\cong \mathbb{Z}_{j} \oplus \mathbb{Z}_{k}</math> 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 ,