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.

Examples

There are no other examples (up to isomorphism). In particular, the group ${\displaystyle \left(\mathbb {Q} ,+\right)}$ of rational numbers is not finitely generated:[1] if ${\displaystyle x_{1},\ldots ,x_{n}}$ are rational numbers, pick a natural number ${\displaystyle k}$ coprime to all the denominators; then ${\displaystyle 1/k}$ cannot be generated by ${\displaystyle x_{1},\ldots ,x_{n}}$. The group ${\displaystyle \left(\mathbb {Q} ^{*},\cdot \right)}$ 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

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

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

${\displaystyle \mathbb {Z} ^{n}\oplus \mathbb {Z} _{k_{1}}\oplus \cdots \oplus \mathbb {Z} _{k_{u}},}$

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 ${\displaystyle \mathbb {Z} _{m}\simeq \mathbb {Z} _{j}\oplus \mathbb {Z} _{k}}$ 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: ${\displaystyle \mathbb {Q} }$ 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 ${\displaystyle \mathbb {Q} }$ is one counterexample, and the rank-0 group given by a direct sum of countably infinitely many copies of ${\displaystyle \mathbb {Z} _{2}}$ is another one.