Finitely-generated abelian group

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

x = n1x1 + n2x2 + ... + nsxs

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

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.[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,...,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.

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 ku. Again, the rank n and the invariant factors k1,...,ku 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.

Notes

1. ^ a b 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.