Finitely generated group: Difference between revisions

Latest revision as of 04:32, 16 January 2022

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 definition
VIEW: Definitions built on this | Facts about this: (facts closely related to Finitely generated group, all facts related to Finitely generated group) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
View a complete list of semi-basic definitions on this wiki

This article defines a group property that is pivotal (i.e., important) among existing group properties
View a list of pivotal group properties | View a complete list of group properties [SHOW MORE]

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	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

@@ Line 1: / Line 1: @@
-{{group property}}
+[[importance rank::2| ]]
+==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 subgroups|join]] of finitely many cyclic subgroups.
+===Equivalence of definitions===
-{{variationof|finiteness (groups)}}
+{{further|[[equivalence of definitions of finitely generated group]]}}
 {{semibasicdef}}
+{{pivotal group property}}
+{{variation of|finite group}}
-==Definition==
+==Metaproperties==
-===Symbol-free definition===
-A [[group]] is said to be '''finitely generated''' if it has a finite [[generating set]].
+{| class="sortable" border="1"
+! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
+|-
+| [[dissatisfies metaproperty::subgroup-closed group property]] || No || [[finite generation is not subgroup-closed]] || It is possible to have a finitely generated group <math>G</math> and a subgroup <math>H</math> of <math>G</math> such that <math>H</math> is not finitely generated.
+|-
+| [[satisfies metaproperty::quotient-closed group property]] || Yes || [[finite generation is quotient-closed]] || If <math>G</math> is a finitely generated group and <math>H</math> is a [[normal subgroup]] of <math>G</math>, then the [[quotient group]] <math>G/H</math> is a finitely generated group.
+|-
+| [[satisfies metaproperty::extension-closed group property]] || Yes || [[finite generation is extension-closed]] || If <math>G</math> is a group and <math>H</math> is a [[normal subgroup]] of <math>G</math> such that both <math>H</math> and the [[quotient group]] <math>G/H</matH> are finitely generated, then <math>G</math> is also finitely generated.
+|-
+| [[satisfies metaproperty::finite direct product-closed group property]] || Yes || (via extension-closed || If <math>G_1, G_2, \dots, G_n</math> are all finitely generated groups, so is the [[external direct product]] <math>G_1 \times G_2 \times \dots \times G_n</math>.
+|-
+| [[satisfies metaproperty::finite index-closed group property]] || Yes || [[finite generation is finite index-closed]] || If <math>G</math> is a finitely generated group and <math>H</math> is a [[subgroup of finite index]] in <math>G</math>, then <math>H</math> is also a finitely generated group.
+|}
 ==Relation with other properties==
@@ Line 15: / Line 38: @@
 ===Stronger properties===
-* [[Finite group]]
+{| class="sortable" border="1"
-* [[Finitely presented group]]: {{proofofstrictimplicationat|[[finitely presented implies finitely generated]]|[[finitely generated not implies finitely presented]]}}
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Finitely generated free group]]
+|-
+| [[Weaker than::finite group]] || || || || {{intermediate notions short|finitely generated group|finite group}}
+|-
+| [[Weaker than::Noetherian group]] (also called '''slender group''') || every subgroup is finitely generated || (by definition) || [[finitely generated not implies Noetherian]] || {{intermediate notions short|finitely generated group|slender group}}
+|-
+| [[Weaker than::finitely presented group]] || has a [[presentation of a group|presentation]] with finitely many generators and finitely many relations || [[finitely presented implies finitely generated]] || [[finitely generated not implies finitely presented]] || {{intermediate notions short|finitely generated group|finitely presentable group}}
+|}
+===Conjunction with other properties===
+{| class="sortable" border="1"
+! Conjunction !! Other component of conjunction !! Intermediate notions between finitely generated group and conjunction !! Intermediate notions between other component and conjunction !! Additional comments
+|-
+| [[Weaker than::finitely generated free group]] || [[free group]] || {{intermediate notions short|finitely generated group|finitely generated free group}} || {{intermediate notions short|free group|finitely generated free group}} || A finitely generated free group is a group with finite freely generating set
+|-
+| [[Weaker than::finitely generated abelian group]] || [[abelian group]] || {{intermediate notions short|finitely generated group|finitely generated abelian group}} || {{intermediate notions short|abelian group|finitely generated abelian group}} || turns out to be a direct product of finitely many cyclic groups by the [[structure theorem for finitely generated abelian groups]]
+|-
+| [[Weaker than::finitely generated residually finite group]] || [[residually finite group]] || {{intermediate notions short|finitely generated group|finitely generated residually finite group}} || {{intermediate notions short|residually finite group|finitely generated residually finite group}} ||
+|-
+| [[Weaker than::finitely generated Hopfian group]] || [[Hopfian group]] || {{intermediate notions short|finitely generated group|finitely generated Hopfian group}} || {{intermediate notions short|Hopfian group|finitely generated Hopfian group}} ||
+|-
+| [[Weaker than::finitely generated nilpotent group]] || [[nilpotent group]] || {{intermediate notions short|finitely generated group|finitely generated nilpotent group}} || {{intermediate notions short|nilpotent group|finitely generated nilpotent group}} || equivalent to abelianization being finitely generated
+|-
+| [[Weaker than::finitely generated solvable group]] || [[solvable group]] || {{intermediate notions short|finitely generated group|finitely generated solvable group}} || {{intermediate notions short|solvable group|finitely generated solvable group}} ||
+|-
+| [[Weaker than::finitely generated periodic group]] || [[periodic group]] || {{intermediate notions short|finitely generated group|finitely generated periodic group}} || {{intermediate notions short|periodic group|finitely generated periodic group}} ||
+|}
 ===Weaker properties===
-* [[Countable group]]
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Stronger than::Countable group]] || cardinality of underlying set is either finite or countably infinite|| || || {{intermediate notions short|countable group|finitely generated group}}
+|-
+| [[Stronger than::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 || || || {{intermediate notions short|group with finitely many homomorphisms to any finite group|finitely generated group}}
+|-
+| [[Stronger than::Group with no infinite minimal generating set]] || any [[minimal generating set]] is finite || || || {{intermediate notions short|group with no infinite minimal generating set|finitely generated group}}
+|-
+| [[Stronger than::Normally finitely generated group]] || [[normal closure]] of a finitely generated subgroup || || || {{intermediate notions short|normally finitely generated group|finitely generated group}}
+|}
+===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==
+{{applyingoperatorgives|hereditarily operator|slender group}}
+A slender group, or Noetherian group, is a group such that all its subgroups are finitely generated.
+==Testing==
+{{GAP command for gp|
+test = IsFinitelyGeneratedGroup}}