Finitely generated nilpotent group: Difference between revisions
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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 | ||
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| [[Stronger than::Slender group]] || every subgroup is finitely generated || || || {{intermediate notions short|slender group|finitely generated nilpotent group}} | |||
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| [[Stronger than::Finitely generated group]] || || || || {{intermediate notions short|finitely generated group|finitely generated nilpotent group}} | | [[Stronger than::Finitely generated group]] || || || || {{intermediate notions short|finitely generated group|finitely generated nilpotent group}} | ||
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| [[Stronger than::Nilpotent group]] || || || || {{intermediate notions short|nilpotent group|finitely generated nilpotent group}} | | [[Stronger than::Nilpotent group]] || || || || {{intermediate notions short|nilpotent group|finitely generated nilpotent group}} | ||
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| [[Stronger than::Supersolvable group]] || has a [[normal series]] where all successive quotients are cyclic || || || {{intermediate notions short|supersolvable group|finitely generated nilpotent group}} | |||
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| [[Stronger than::Polycyclic group]] || has a [[subnormal series]] where all successive quotients are cyclic || || || {{intermediate notions short|polycyclic group|finitely generated nilpotent group}} | |||
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| [[Stronger than::Finitely generated solvable group]] || || || || {{intermediate notions short|finitely generated solvable group|finitely generated nilpotent group}} | | [[Stronger than::Finitely generated solvable group]] || || || || {{intermediate notions short|finitely generated solvable group|finitely generated nilpotent group}} | ||
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Revision as of 16:45, 25 May 2010
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
Definition
Symbol-free definition
A finitely generated nilpotent group is a group satisfying the following equivalent conditions:
- It is finitely generated 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 |
|---|---|---|---|---|
| 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 |