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 (R, +) and real numbers under multiplication (R, ×) are also not finitely generated.^[1][2]

Classification

The fundamental theorem of finitely generated abelian groups (which is a special case of the structure theorem for finitely generated modules over a principal ideal domain) can be stated two ways (analogously with principal ideal domains):

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.

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.

See also

• The Jordan–Hölder theorem is a non-abelian generalization

Notes

1. Silverman & Tate (1992), p. 102
2. La Harpe (2000), p. 46

References

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