Difference between revisions of "Finitely generated nilpotent group"

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{{group property conjunction|finitely generated group|nilpotent group}}
 
{{group property conjunction|finitely generated group|nilpotent group}}
 
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{{group property conjunction|slender group|nilpotent group}}
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[[importance rank::3| ]]
 
==Definition==
 
==Definition==
 
===Symbol-free 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]]
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# It is [[finitely presented group|finitely presented]] and [[nilpotent group|nilpotent]].
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# It is [[Noetherian group|Noetherian]] (i.e., every subgroup is finitely generated -- this is also described using the adjective "slender") and [[nilpotent group|nilpotent]].
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# It is [[nilpotent group|nilpotent]] and its [[abelianization]] is [[finitely generated group|finitely generated]].
  
 
===Equivalence of definitions===
 
===Equivalence of definitions===
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{| class="sortable" border="1"
 
{| class="sortable" border="1"
 
! 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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|-
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| [[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}}
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|-
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| [[Stronger than::Finitely presented group]] || || || || {{intermediate notions short|finitely presented group|finitely generated nilpotent group}}
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|-
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| [[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::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::Nilpotent group]] || || || || {{intermediate notions short|nilpotent group|finitely generated nilpotent group}}
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|-
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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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|-
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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}}
 
|-
 
|-
 
| [[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}}
 
|}
 
|}

Latest revision as of 00:46, 17 April 2013

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:

  1. It is finitely generated and nilpotent.
  2. It is finitely presented and nilpotent.
  3. It is Noetherian (i.e., every subgroup is finitely generated -- this is also described using the adjective "slender") and nilpotent.
  4. 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 Finitely presented solvable group, Group in which every subgroup is finitely presented, Polycyclic group|FULL LIST, MORE INFO
Noetherian group (also called slender group) every subgroup is finitely generated Group in which every subgroup is finitely presented, Polycyclic group|FULL LIST, MORE INFO
Finitely generated group Finitely generated group for which all homomorphisms to any finite group can be listed in finite time, Finitely generated solvable group, Finitely presented solvable group, Group in which every subgroup is finitely presented, Group with a finite complete rewriting system, Group with polynomial-time solvable word problem, Group with solvable word problem, Noetherian group, Polycyclic 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 Supersolvable group|FULL LIST, MORE INFO
Finitely generated solvable group Finitely presented solvable group, Polycyclic group, Supersolvable group|FULL LIST, MORE INFO