Finitely generated abelian group

In abstract algebra, an abelian group (G, +) is called finitely generated if there exist finitely many elements x₁, ..., x_s in G such that every x in G can be written in the form

x = n₁x₁ + n₂x₂ + ... + n_sx_s

with integers n₁, ..., n_s. In this case, we say that the set {x₁, ..., x_s} is a generating set of G or that x₁, ..., x_s generate G.

Clearly, every finite abelian group is finitely generated. The finitely generated abelian groups are of a rather simple structure and can be completely classified, as will be explained below.

Examples

The integers $\left(\mathbb {Z} ,+\right)$ are a finitely generated abelian group.
The integers modulo $n$ , $\left(\mathbb {Z} _{n},+\right)$ are a 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 $\left(\mathbb {Q} ,+\right)$ of rational numbers is not finitely generated:^[1] if $x_{1},\ldots ,x_{n}$ are rational numbers, pick a natural number $k$ coprime to all the denominators; then $1/k$ cannot be generated by $x_{1},\ldots ,x_{n}$ . The group $\left(\mathbb {Q} ^{*},\cdot \right)$ of non-zero rational numbers is also not finitely generated. The groups of real numbers under addition $\left(\mathbb {R} ,+\right)$ and non-zero real numbers under multiplication $\left(\mathbb {R} ^{*},\cdot \right)$ are also not finitely generated.^[1]^[2]

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

\mathbb {Z} ^{n}\oplus \mathbb {Z} _{q_{1}}\oplus \cdots \oplus \mathbb {Z} _{q_{t}},

where the rank n ≥ 0, and the numbers q₁, ..., q_t are powers of (not necessarily distinct) prime numbers. In particular, G is finite if and only if n = 0. The values of n, q₁, ..., q_t are (up to rearranging the indices) uniquely determined by G.

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}}\oplus \cdots \oplus \mathbb {Z} _{k_{u}},

where k₁ divides k₂, which divides k₃ and so on up to k_u. Again, the rank n and the invariant factors k₁, ..., k_u are uniquely determined by G (here with a unique order).

Equivalence

These statements are equivalent because of the Chinese remainder theorem, which here states that $\mathbb {Z} _{m}\simeq \mathbb {Z} _{j}\oplus \mathbb {Z} _{k}$ if and only if j and k are coprime and m = jk.

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 in (Gauss 1801), the finite case was proven in (Kronecker 1870), and stated in group-theoretic terms in (Frobenius & Stickelberger 1878), and the finitely generated case was proven in (Poincaré 1900); details follow.

The fundamental theorem for finite abelian groups was proven by Leopold Kronecker in (Kronecker 1870), using a group-theoretic proof,^[3] though without stating it in group-theoretic terms;^[4] a modern presentation of Kronecker's proof is given in (Stillwell 2012), 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.^[5]^[6] Another group-theoretic formulation was given by Kronecker's student Eugen Netto in 1882.^[7]^[8]

The fundamental theorem for finitely generated abelian groups was proven by Henri Poincaré in (Poincaré 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.^[3]

Kronecker's proof was generalized to finitely generated abelian groups by Emmy Noether in (Noether 1926).^[3]

Corollaries

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: $\mathbb {Q}$ is torsion-free but not free abelian.

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.

Non-finitely generated abelian groups

Note that not every abelian group of finite rank is finitely generated; the rank 1 group $\mathbb {Q}$ is one counterexample, and the rank-0 group given by a direct sum of countably infinitely many copies of $\mathbb {Z} _{2}$ is another one.

Notes

1 2 Silverman & Tate (1992), p. 102
↑ de la Harpe (2000), p. 46
1 2 3 Stillwell, John (2012). "5.2 The Structure Theorem for Finitely Generated". Classical Topology and Combinatorial Group Theory. p. 175.
↑ Wussing, Hans (2007) [1969]. Die Genesis des abstrackten Gruppenbegriffes. Ein Beitrag zur Entstehungsgeschichte der abstrakten Gruppentheorie. [The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory.]. p. 67.
↑ G. Frobenius, L. Stickelberger, Uber Grubben von vertauschbaren Elementen, J. reine u. angew. Math., 86 (1878), 217-262.
↑ Wussing (2007), pp. 234–235
↑ Substitutionentheorie und ihre Anwendung auf die Algebra, Eugen Netto, 1882
↑ Wussing (2007), pp. 234–235

References

Silverman, Joseph H.; Tate, John Torrence (1992). Rational points on elliptic curves. Undergraduate Texts in Mathematics. Springer. ISBN 978-0-387-97825-3.
de la Harpe, Pierre (2000). Topics in geometric group theory. Chicago lectures in mathematics. University of Chicago Press. ISBN 978-0-226-31721-2.

Fundamental mathematical theorems
Arithmetic Algebra Calculus Geometry Curves Projective Riemannian Groups Cyclic Finitely generated Abelian Linear algebra Linear programming Calculus of variations Vector analysis Homomorphisms Galois theory

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