# finitely generated group

A *finitely generated group* is a group that has a finite generating set^{}.

Every finite group^{} is obviously finitely generated^{}.
Every finitely generated group is countable.

Any quotient (http://planetmath.org/QuotientGroup)
of a finitely generated group is finitely generated.
However, a finitely generated group may have subgroups^{}
that are not finitely generated.
(For example, the free group^{} of rank $2$ is generated by just two elements,
but its commutator subgroup^{} 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 generators^{} required for the subgroup is given by
the Schreier index formula (http://planetmath.org/ScheierIndexFormula).

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 groups^{} and,
more generally, all polycyclic groups^{}.

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 |