The , $\left(\mathbb {Z} ,+\right)$ , are a finitely generated abelian group.

integers

The , $\left(\mathbb {Z} /n\mathbb {Z} ,+\right)$ , are a finite (hence finitely generated) abelian group.

integers modulo $n$

Any of finitely many finitely generated abelian groups is again a finitely generated abelian group.

direct sum

Every forms a finitely generated free abelian group.

lattice

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]

Corollaries[edit]

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[edit]

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.

The composition series in the is a non-abelian generalization.