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.

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 is finitely generated. A subgroup of a finitely generated group need not be finitely generated.

See also

• Element (mathematics)
• Finitely generated module
• Generating set of a group
• Group (mathematics)
• Presentation of a group

External links

• Finitely generated group @ Wikpedia