Group whose automorphism group is nilpotent: Difference between revisions

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==Definition==
==Definition==


A [[group]] is termed an '''aut-nilpotent group''' if its [[automorphism group]] is a [[nilpotent group]].
A [[group]] is termed an '''group whose automorphism group is nilpotent''' (or '''group with nilpotent automorphism group''') if its [[automorphism group]] is a [[nilpotent group]].


==Relation with other properties==
==Relation with other properties==
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===Stronger properties===
===Stronger properties===


{| class="wikitable" 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
|-
|-
| [[Weaker than::Aut-abelian group]] || [[automorphism group]] is [[abelian group|abelian]] || follows from [[abelian implies nilpotent]] || [[aut-nilpotent not implies aut-abelian]] ||  
| [[Weaker than::group whose automorphism group is abelian]] || [[automorphism group]] is [[abelian group|abelian]] || follows from [[abelian implies nilpotent]] || [[nilpotent automorphism group not implies abelian automorphism group]] ||  
|-
|-
| [[Weaker than::Cyclic group]] || || [[cyclic implies aut-abelian|via aut-abelian]] || || {{intermediate notions short|cyclic group|aut-nilpotent group}}
| [[Weaker than::cyclic group]] || || [[cyclic implies abelian automorphism group|via automorphism group being abelian]] || || {{intermediate notions short|cyclic group|group whose automorphism group is nilpotent}}
|-
|-
| [[Weaker than::Locally cyclic group]] || || [[locally cyclic implies aut-abelian|via aut-abelian]] || || {{intermediate notions short|locally cyclic group|aut-nilpotent group}}
| [[Weaker than::locally cyclic group]] || || [[locally cyclic implies abelian automorphism group|via automorphism group being abelian]] || || {{intermediate notions short|locally cyclic group|group whose automorphism group is nilpotent}}
|}
|}


===Weaker properties===
===Weaker properties===


{| class="wikitable" 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
|-
|-
| [[Stronger than::Nilpotent group]] || || [[aut-nilpotent implies nilpotent of class at most one more]] || [[nilpotent not implies aut-nilpotent]] || {{intermediate notions short|nilpotent group|aut-nilpotent group}}
| [[Stronger than::nilpotent group]] || || [[nilpotent automorphism group implies nilpotent of class at most one more]] || [[nilpotent not implies nilpotent automorphism group]] || {{intermediate notions short|nilpotent group|group whose automorhism group is nilpotent}}
|}
|}

Latest revision as of 19:02, 20 June 2013

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

Definition

A group is termed an group whose automorphism group is nilpotent (or group with nilpotent automorphism group) if its automorphism group is a nilpotent group.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
group whose automorphism group is abelian automorphism group is abelian follows from abelian implies nilpotent nilpotent automorphism group not implies abelian automorphism group
cyclic group via automorphism group being abelian |FULL LIST, MORE INFO
locally cyclic group via automorphism group being abelian |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
nilpotent group nilpotent automorphism group implies nilpotent of class at most one more nilpotent not implies nilpotent automorphism group |FULL LIST, MORE INFO
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