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Concept
Finitely generated group
Summary
In algebra, a finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the combination (under the group operation) of finitely many elements of S and of inverses of such elements.
By definition, every finite group is finitely generated, since S can be taken to be G itself. Every infinite finitely generated group must be countable but countable groups need not be finitely generated. The additive group of rational numbers Q is an example of a countable group that is not finitely generated.
Every quotient of a finitely generated group G is finitely generated; the quotient group is generated by the images of the generators of G under the canonical projection.
A group that is generated by a single element is called cyclic. Every infinite cyclic group is isomorphic to the additive group of the integers Z.
A locally cyclic group is a group in which every finitely generated subgroup is cyclic.
The free group on a finite set is finitely generated by the elements of that set (§Examples).
A fortiori, every finitely presented group (§Examples) is finitely generated.
Finitely generated abelian group
Every abelian group can be seen as a module over the ring of integers Z, and in a finitely generated abelian group with generators x1, ..., xn, every group element x can be written as a linear combination of these generators,
x = α1⋅x1 + α2⋅x2 + ... + αn⋅xn
with integers α1, ..., αn.
Subgroups of a finitely generated abelian group are themselves finitely generated.
The fundamental theorem of finitely generated abelian groups states 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 which are unique up to isomorphism.
A subgroup of a finitely generated group need not be finitely generated. The commutator subgroup of the free group on two generators is an example of a subgroup of a finitely generated group that is not finitely generated.
On the other hand, all subgroups of a finitely generated abelian group are finitely generated.
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