Finitely generated group: Difference between revisions

Revision as of 00:39, 23 May 2010

Definition

Symbol-free definition

A group is said to be finitely generated if it has a finite generating set.

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
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 |

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
Slender group	every subgroup is finitely generated	(by definition)	finitely generated not implies slender	\|FULL LIST, MORE INFO
Finitely presentable group	has a presentation with finitely many generators and finitely many relations	finitely presentable implies finitely generated	finitely generated not implies finitely presentable	\|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		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				\|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

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: @@
-{{pivotal group property}}
-{{variationof|finiteness (groups)}}
-{{semibasicdef}}
 ==Definition==
@@ Line 10: / Line 4: @@
 A [[group]] is said to be '''finitely generated''' if it has a finite [[generating set]].
+{{semibasicdef}}
+{{pivotal group property}}
+{{variation of|finite group}}
 ==Relation with other properties==
@@ Line 15: / Line 13: @@
 ===Stronger properties===
-* [[Weaker than::Finite group]]
+{| class="sortable" border="1"
-* [[Weaker than::Slender group]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Weaker than::Finitely presented group]]: {{proofofstrictimplicationat|[[finitely presented implies finitely generated]]|[[finitely generated not implies finitely presented]]}}
+|-
-* [[Weaker than::Finitely generated free group]]
+| [[Weaker than::Finite group]] || || || || {{intermediate notions short|finitely generated group|finite group}}
+|-
+| [[Weaker than::Slender group]] || every subgroup is finitely generated || (by definition) || [[finitely generated not implies slender]] || {{intermediate notions short|finitely generated group|slender group}}
+|-
+| [[Weaker than::Finitely presentable group]] || has a [[presentation of a group|presentation]] with finitely many generators and finitely many relations || [[finitely presentable implies finitely generated]] || [[finitely generated not implies finitely presentable]] || {{intermediate notions short|finitely generated group|finitely presentable group}}
+|}
 ===Conjunction with other properties===
-* [[Weaker than::Finitely generated free group]] is a group that is both free and finitely generated; equivalently, it is freely generated by a finite set.
+{| class="sortable" border="1"
-* [[Weaker than::Finitely generated Abelian group]] is a group that is both finitely generated and [[Abelian group|Abelian]] -- such groups are classified by the [[structure theorem for finitely generated Abelian groups]].
+! 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 nilpotent group]] is a group that is both finitely generated and [[nilpotent group|nilpotent]].
+|-
-* [[Weaker than::Finitely generated periodic group]] is a group that is both finitely generated and [[periodic group|periodic]]: every element has finite order.
+| [[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|
+| [[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]] || || || || {{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}}
+|}
-===Incomparable properties===
+===Opposite properties===
-* [[Periodic group]]: This is a group where every element has finite order. A finitely generated group need not be periodic, and a periodic group need not be finitely generated.
+* [[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==