Finitely generated nilpotent group: Difference between revisions
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{{group property conjunction|finitely generated group|nilpotent group}} | {{group property conjunction|finitely generated group|nilpotent group}} | ||
{{group property conjunction|slender group|nilpotent group}} | |||
[[importance rank::3| ]] | |||
==Definition== | ==Definition== | ||
A '''finitely generated nilpotent group''' is a group satisfying the following equivalent conditions: | A '''finitely generated nilpotent group''' is a group satisfying the following equivalent conditions: | ||
# It is [[finitely generated group|finitely generated]] and [[nilpotent group|nilpotent]] | # It is [[finitely generated group|finitely generated]] and [[nilpotent group|nilpotent]]. | ||
# It is [[nilpotent group|nilpotent]] and its [[abelianization]] is [[finitely generated group|finitely generated]] | # It is [[finitely presented group|finitely presented]] and [[nilpotent group|nilpotent]]. | ||
# It is [[Noetherian group|Noetherian]] (i.e., every subgroup is finitely generated -- this is also described using the adjective "slender") and [[nilpotent group|nilpotent]]. | |||
# It is [[nilpotent group|nilpotent]] and its [[abelianization]] is [[finitely generated group|finitely generated]]. | |||
===Equivalence of definitions=== | ===Equivalence of definitions=== | ||
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ||
|- | |- | ||
| [[Weaker than:: | | [[Weaker than::finitely generated abelian group]] || both a [[finitely generated group]] and an [[abelian group]] || || || {{intermediate notions short|finitely generated nilpotent group|finitely generated abelian group}} | ||
|- | |- | ||
| [[Weaker than:: | | [[Weaker than::finite nilpotent group]] || both a [[finite group]] and a [[nilpotent group]] || || || {{intermediate notions short|finitely generated nilpotent group|finite nilpotent group}} | ||
|} | |} | ||
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ||
|- | |- | ||
| [[Stronger than:: | | [[Stronger than::group in which every subgroup is finitely presented]] || || || || {{intermediate notions short|group in which every subgroup is finitely presented|finitely generated nilpotent group}} | ||
|- | |||
| [[Stronger than::finitely presented group]] || || || || {{intermediate notions short|finitely presented group|finitely generated nilpotent group}} | |||
|- | |||
| [[Stronger than::Noetherian group]] (also called '''slender group''') || every subgroup is finitely generated || || || {{intermediate notions short|Noetherian group|finitely generated nilpotent group}} | |||
|- | |||
| [[Stronger than::finitely generated group]] || || || || {{intermediate notions short|finitely generated group|finitely generated nilpotent group}} | |||
|- | |||
| [[Stronger than::nilpotent group]] || || || || {{intermediate notions short|nilpotent group|finitely generated nilpotent group}} | |||
|- | |||
| [[Stronger than::supersolvable group]] || has a [[normal series]] where all successive quotients are cyclic || || || {{intermediate notions short|supersolvable group|finitely generated nilpotent group}} | |||
|- | |- | ||
| [[Stronger than:: | | [[Stronger than::polycyclic group]] || has a [[subnormal series]] where all successive quotients are cyclic || || || {{intermediate notions short|polycyclic group|finitely generated nilpotent group}} | ||
|- | |- | ||
| [[Stronger than:: | | [[Stronger than::finitely generated solvable group]] || || || || {{intermediate notions short|finitely generated solvable group|finitely generated nilpotent group}} | ||
|} | |} | ||
Latest revision as of 05:51, 18 April 2024
This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: finitely generated group and nilpotent group
View other group property conjunctions OR view all group properties
This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: slender group and nilpotent group
View other group property conjunctions OR view all group properties
Definition
A finitely generated nilpotent group is a group satisfying the following equivalent conditions:
- It is finitely generated and nilpotent.
- It is finitely presented and nilpotent.
- It is Noetherian (i.e., every subgroup is finitely generated -- this is also described using the adjective "slender") and nilpotent.
- It is nilpotent and its abelianization is finitely generated.
Equivalence of definitions
For full proof, refer: Equivalence of definitions of finitely generated nilpotent group
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| finitely generated abelian group | both a finitely generated group and an abelian group | |FULL LIST, MORE INFO | ||
| finite nilpotent group | both a finite group and a nilpotent group | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| group in which every subgroup is finitely presented | |FULL LIST, MORE INFO | |||
| finitely presented group | |FULL LIST, MORE INFO | |||
| Noetherian group (also called slender group) | every subgroup is finitely generated | |FULL LIST, MORE INFO | ||
| finitely generated group | |FULL LIST, MORE INFO | |||
| nilpotent group | |FULL LIST, MORE INFO | |||
| supersolvable group | has a normal series where all successive quotients are cyclic | |FULL LIST, MORE INFO | ||
| polycyclic group | has a subnormal series where all successive quotients are cyclic | |FULL LIST, MORE INFO | ||
| finitely generated solvable group | |FULL LIST, MORE INFO |