Finitely generated group: Difference between revisions
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===Stronger properties=== | ===Stronger properties=== | ||
* [[Finite group]] | * [[Weaker than::Finite group]] | ||
* [[Slender group]] | * [[Weaker than::Slender group]] | ||
* [[Finitely presented group]]: {{proofofstrictimplicationat|[[finitely presented implies finitely generated]]|[[finitely generated not implies finitely presented]]}} | * [[Weaker than::Finitely presented group]]: {{proofofstrictimplicationat|[[finitely presented implies finitely generated]]|[[finitely generated not implies finitely presented]]}} | ||
* [[Finitely generated free group]] | * [[Weaker than::Finitely generated free 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. | |||
* [[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]]. | |||
* [[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 properties=== | ===Weaker properties=== | ||
* [[Countable group]] | * [[Countable group]] | ||
===Incomparable 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. | |||
==Effect of property operators== | ==Effect of property operators== | ||
Revision as of 20:46, 17 August 2008
This article defines a group property that is pivotal (i.e., important) among existing group properties
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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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Definition
Symbol-free definition
A group is said to be finitely generated if it has a finite generating set.
Relation with other properties
Stronger properties
- Finite group
- Slender group
- Finitely presented group: For proof of the implication, refer finitely presented implies finitely generated and for proof of its strictness (i.e. the reverse implication being false) refer finitely generated not implies finitely presented.
- Finitely generated free group
Conjunction with other properties
- Finitely generated free group is a group that is both free and finitely generated; equivalently, it is freely generated by a finite set.
- Finitely generated Abelian group is a group that is both finitely generated and Abelian -- such groups are classified by the structure theorem for finitely generated Abelian groups.
- Finitely generated nilpotent group is a group that is both finitely generated and nilpotent.
- Finitely generated periodic group is a group that is both finitely generated and periodic: every element has finite order.
Weaker properties
Incomparable 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.
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.