Finitely generated group

Definition

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

1. It has a finite generating set.
2. Every generating set of the group has a subset that is finite and is also a generating set.
3. The group has at least one minimal generating set and every minimal generating set of the group is finite.
4. The minimum size of generating set of the group is finite.
5. 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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VIEW RELATED: Group property implications | Group property non-implications | Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions |Group property dissatisfactions<random> @@@
RANDOM GROUP PROPERTY: <random>Simple group: A nontrivial group having no proper nontrivial normal subgroup.@@@Noetherian group: A group where every subgroup is finitely generated.@@@Complete group: A group for which the natural homomorphism to its automorphism group is an isomorphism, i.e., a centerless group for which every automorphism is inner.@@@SQ-universal group: A group for which every finitely generated group can be realized as a subquotient.@@@T-group: A group where any subnormal subgroup is normal, i.e., where normality is transitive. In general, normality is not transitive.@@@Locally finite group: A group in which every finitely generated subgroup is finite.@@@Group satisfying normalizer condition: A group with no proper self-normalizing subgroup.</random></random>

This is a variation of finite group|Find other variations of finite group |

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 $G$ and a subgroup $H$ of $G$ such that $H$ is not finitely generated.
quotient-closed group property	Yes	finite generation is quotient-closed	If $G$ is a finitely generated group and $H$ is a normal subgroup of $G$, then the quotient group $G/H$ is a finitely generated group.
extension-closed group property	Yes	finite generation is extension-closed	If $G$ is a group and $H$ is a normal subgroup of $G$ such that both $H$ and the quotient group $G/H$ are finitely generated, then $G$ is also finitely generated.
finite direct product-closed group property	Yes	(via extension-closed	If $G_{1},G_{2},\dots ,G_{n}$ are all finitely generated groups, so is the external direct product $G_{1}\times G_{2}\times \dots \times G_{n}$.
finite index-closed group property	Yes	finite generation is finite index-closed	If $G$ is a finitely generated group and $H$ is a subgroup of finite index in $G$, then $H$ is also a finitely generated group.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
finite group				|FULL LIST, MORE INFO
Noetherian group (also called slender group)	every subgroup is finitely generated	(by definition)	finitely generated not implies Noetherian	|FULL LIST, MORE INFO
finitely presented group	has a presentation with finitely many generators and finitely many relations	finitely presented implies finitely generated	finitely generated not implies finitely presented	|FULL LIST, MORE INFO

Conjunction with other properties

Conjunction	Other component of conjunction	Intermediate notions between finitely generated group and conjunction	Intermediate notions between other component and conjunction	Additional comments
finitely generated free group	free group	|FULL LIST, MORE INFO	|FULL LIST, MORE INFO	A finitely generated free group is a group with finite freely generating set
finitely generated abelian group	abelian group	|FULL LIST, MORE INFO	|FULL LIST, MORE INFO	turns out to be a direct product of finitely many cyclic groups by the structure theorem for finitely generated abelian groups
finitely generated residually finite group	residually finite group	|FULL LIST, MORE INFO	|FULL LIST, MORE INFO
finitely generated Hopfian group	Hopfian group	|FULL LIST, MORE INFO	|FULL LIST, MORE INFO
finitely generated nilpotent group	nilpotent group	|FULL LIST, MORE INFO	|FULL LIST, MORE INFO	equivalent to abelianization being finitely generated
finitely generated solvable group	solvable group	|FULL LIST, MORE INFO	|FULL LIST, MORE INFO
finitely generated periodic group	periodic group	|FULL LIST, MORE INFO	|FULL LIST, MORE INFO

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 of Groups, Algorithms, Programming (GAP).
The GAP command for this group property is:IsFinitelyGeneratedGroup
View GAP-testable group properties