Finitely generated group
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 definition
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This article defines a group property that is pivotal (i.e., important) among existing group properties
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|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
Conjunction with other 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|
- 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
A slender group, or Noetherian group, is a group such that all its subgroups are finitely generated.
This group property can be tested using built-in functionality of Groups, Algorithms, Programming (GAP).
The GAP command for this group property is:IsFinitelyGeneratedGroup
View GAP-testable group properties