Finitely-generated abelian group

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.

Contents

Examples

There are no other examples (up to isomorphism). The group (Q,+) of rational numbers is not finitely generated:[1] if x1,...,xs are rational numbers, pick a natural number w coprime to all the denominators; then 1/w cannot be generated by x1,...,xs. The group (Q*,*) 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 PIDs):

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 Zm is isomorphic to the direct product of Zj and Zk 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: 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 Q is one counterexample, and the rank-0 group given by a direct sum of countably infinitely many copies of Z2 is another one.

See also

Notes

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

References

Geometric
of Groups