finitely generated group

A finitely generated group is a group that has a finite generating setPlanetmathPlanetmath.

Every finite groupMathworldPlanetmath is obviously finitely generatedMathworldPlanetmath. Every finitely generated group is countable.

Any quotient ( of a finitely generated group is finitely generated. However, a finitely generated group may have subgroupsMathworldPlanetmathPlanetmath that are not finitely generated. (For example, the free groupMathworldPlanetmath of rank 2 is generated by just two elements, but its commutator subgroupMathworldPlanetmath is not finitely generated.) Nonetheless, a subgroup of finite index in a finitely generated group is necessarily finitely generated; a bound on the number of generatorsPlanetmathPlanetmathPlanetmathPlanetmath required for the subgroup is given by the Schreier index formula (

The finitely generated groups all of whose subgroups are also finitely generated are precisely the groups satisfying the maximal condition. This includes all finitely generated nilpotent groupsMathworldPlanetmath and, more generally, all polycyclic groupsMathworldPlanetmath.

A group that is not finitely generated is sometimes said to be infinitely generated.

Title finitely generated group
Canonical name FinitelyGeneratedGroup
Date of creation 2013-03-22 12:16:38
Last modified on 2013-03-22 12:16:38
Owner yark (2760)
Last modified by yark (2760)
Numerical id 24
Author yark (2760)
Entry type Definition
Classification msc 20A05
Related topic FundamentalTheoremOfFinitelyGeneratedAbelianGroups
Related topic AFinitelyGeneratedGroupHasOnlyFinitelyManySubgroupsOfAGivenIndex
Defines finitely generated
Defines finitely generated subgroup
Defines infinitely generated
Defines infinitely generated group
Defines infinitely generated subgroup