Finitely generated group
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 element
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.
Examples
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 groups
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 + ...
Begin with the source’s own compact description: “Finitely generated group” is 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 element. The dossier treats that line as a proposition to test through Finitely, generated and group, not as a finished interpretation.
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Named sources, stable identifiers and responsible institutions provide the strongest route from overview to verifiable evidence. The source revision retrieved here is dated Mar 11, 2026. The linked authority identifier is Q21083858.
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This entry incorporates text from “Finitely generated group” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.