# Finitely generated group

From Groupprops

## Definition

A group is said to be **finitely generated** if it satisfies the following equivalent conditions:

- It has a finite generating set.
- Every generating set of the group has a subset that is finite and is also a generating set.
- The group has at least one minimal generating set and every minimal generating set of the group is finite.
- The minimum size of generating set of the group is finite.
- The group is a join of finitely many cyclic subgroups.

### Equivalence of definitions

`Further information: equivalence of definitions of finitely generated group`

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definitionVIEW: Definitions built on this | Facts about this: (factscloselyrelated to Finitely generated group, all facts related to Finitely generated group) |Survey articles about this | Survey articles about definitions built on this

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This article defines a group property that is pivotal (i.e., important) among existing group properties

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## Metaproperties

Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|

subgroup-closed group property | No | finite generation is not subgroup-closed | It is possible to have a finitely generated group and a subgroup of such that is not finitely generated. |

quotient-closed group property | Yes | finite generation is quotient-closed | If is a finitely generated group and is a normal subgroup of , then the quotient group is a finitely generated group. |

extension-closed group property | Yes | finite generation is extension-closed | If is a group and is a normal subgroup of such that both and the quotient group are finitely generated, then is also finitely generated. |

finite direct product-closed group property | Yes | (via extension-closed | If are all finitely generated groups, so is the external direct product . |

finite index-closed group property | Yes | finite generation is finite index-closed | If is a finitely generated group and is a subgroup of finite index in , then is also a finitely generated group. |

## Relation with other properties

### Stronger properties

### Conjunction with other properties

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Countable group | cardinality of underlying set is either finite or countably infinite | |FULL LIST, MORE INFO | ||

Group with finitely many homomorphisms to any finite group | for any fixed finite group, there are finitely many homomorphisms from the given group to that group | |FULL LIST, MORE INFO | ||

Group with no infinite minimal generating set | any minimal generating set is finite | |FULL LIST, MORE INFO | ||

Normally finitely generated group | normal closure of a finitely generated subgroup | |FULL LIST, MORE INFO |

### Opposite properties

- Locally finite group is a group where every finitely generated subgroup is finite. A group is locally finite and finitely generated if and only if it is finite.

## Effect of property operators

### The hereditarily operator

*Applying the hereditarily operator to this property gives*: slender group

A slender group, or Noetherian group, is a group such that all its subgroups are finitely generated.

## Testing

### GAP command

This group property can be tested using built-in functionality ofGroups, Algorithms, Programming(GAP).

The GAP command for this group property is:IsFinitelyGeneratedGroup

View GAP-testable group properties