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 combination (under the group operation) of finitely many elements of S and of inverses of such elements.

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, with integers α1, ..., αn.

[3] This upper bound was then significantly improved by Hanna Neumann to

A group such that all its subgroups are finitely generated is called Noetherian.

Conversely, every periodic abelian group is locally finite.

[4] Finitely generated groups arise in diverse mathematical and scientific contexts.

A frequent way they do so is by the Švarc-Milnor lemma, or more generally thanks to an action through which a group inherits some finiteness property of a space.

The word problem for a finitely generated group is the decision problem of whether two words in the generators of the group represent the same element.