Finitely generated group: Difference between revisions
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| [[Weaker than::Finite group]] || || || || {{intermediate notions short|finitely generated group|finite group}} | | [[Weaker than::Finite group]] || || || || {{intermediate notions short|finitely generated group|finite group}} | ||
|- | |- | ||
| [[Weaker than:: | | [[Weaker than::Noetherian group]] (also called '''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}} | | [[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}} | ||
Revision as of 13:55, 26 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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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]
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 | |||
| Noetherian group (also called 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
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