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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En algèbre, dans un groupe, un élément est dit de torsion s'il est d'ordre fini, c'est-à-dire si l'une de ses puissances non nulle est l'élément neutre. La torsion d'un groupe est l'ensemble de ses éléments de torsion. Un groupe est dit sans torsion si sa torsion ne contient que le neutre, c'est-à-dire si tout élément différent du neutre est d'ordre infini. Si le groupe est abélien, sa torsion est un sous-groupe. Par exemple, le sous-groupe de torsion du groupe abélien est .
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.
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