In abstract algebra, an abelian group ${\displaystyle (G,+)}$ is called finitely generated if there exist finitely many elements ${\displaystyle x_{1},\dots ,x_{s}}$ in ${\displaystyle G}$ such that every ${\displaystyle x}$ in ${\displaystyle G}$ can be written in the form ${\displaystyle x=n_{1}x_{1}+n_{2}x_{2}+\cdots +n_{s}x_{s}}$ for some integers ${\displaystyle n_{1},\dots ,n_{s}}$. In this case, we say that the set ${\displaystyle \{x_{1},\dots ,x_{s}\}}$ is a generating set of ${\displaystyle G}$ or that ${\displaystyle x_{1},\dots ,x_{s}}$ generate ${\displaystyle G}$.

Every finite abelian group is finitely generated. The finitely generated abelian groups can be completely classified.

Examples

• The integers, ${\displaystyle \left(\mathbb {Z} ,+\right)}$, are a finitely generated abelian group.
• The integers modulo ${\displaystyle n}$, ${\displaystyle \left(\mathbb {Z} /n\mathbb {Z} ,+\right)}$, are a finite (hence 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 ${\displaystyle \left(\mathbb {Q} ,+\right)}$ 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 ${\displaystyle \left(\mathbb {R} ,+\right)}$ and non-zero real numbers under multiplication ${\displaystyle \left(\mathbb {R} ^{*},\cdot \right)}$ are also not finitely generated.^[1][2]

Classification

The fundamental theorem of finitely generated abelian groups can be stated two ways, generalizing the two forms of the fundamental theorem of finite abelian groups. The theorem, in both forms, in turn generalizes to the structure theorem for finitely generated modules over a principal ideal domain, which in turn admits further generalizations.

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.

See also

• The composition series in the Jordan–Hölder theorem is a non-abelian generalization.

Notes

References