Finitely generated group
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 product of finitely many elements of the finite set S and of inverses of such elements.
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.
Quotients and subgroups
Every quotient of a finitely generated group is finitely generated. However, a subgroup of a finitely generated group need not be finitely generated. For example, the commutator subgroup of the free group F2 on two generators is not finitely generated. However, a subgroup of finite index in a finitely generated group is always finitely generated, and the Schreier index formula gives a bound on the number of generators required.
Finitely generated abelian groups
In a finitely generated abelian group with generators x1, ..., xn, every group element x can be written in the form
- x = α1⋅x1 + α2⋅x2 + ... + αn⋅xs
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.
Geometric group theory
The lattice of subgroups of a group satisfies the ascending chain condition if and only if all subgroups of the group are finitely generated. A group such that all its subgroups are finitely generated is called Noetherian.
The word problem for a finitely generated group is the decision problem whether two words in the generators of the group represent the same element. The word problem for a given finitely generated group is solvable if and only if the group can be embedded in every algebraically closed group.
A group is locally finite if every finitely generated subgroup is finite. Every locally finite group is periodic, i.e., every element has finite order. Conversely, every periodic abelian group is locally finite.
A group is locally cyclic if every finitely generated subgroup is cyclic. The additive group of the rational numbers Q is an example of a non-cyclic locally cyclic group. Every locally cyclic group is abelian. Every finitely-generated locally cyclic group is cyclic.