Group whose automorphism group is nilpotent: Difference between revisions

From Groupprops
No edit summary
Line 1: Line 1:
{{wikilocal}}
 
{{group property}}
{{group property}}


==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 '''aut-nilpotent group''' for short)if its [[automorphism group]] is a [[nilpotent group]].


==Relation with other properties==
==Relation with other properties==
Line 13: Line 13:
! 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::aut-abelian group]] || [[automorphism group]] is [[abelian group|abelian]] || follows from [[abelian implies nilpotent]] || [[aut-nilpotent not implies aut-abelian]] ||  
|-
|-
| [[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 aut-abelian|via aut-abelian]] || || {{intermediate notions short|cyclic group|aut-nilpotent group}}
|-
|-
| [[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 aut-abelian|via aut-abelian]] || || {{intermediate notions short|locally cyclic group|aut-nilpotent group}}
|}
|}


Revision as of 18:21, 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 aut-nilpotent group for short)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
aut-abelian group automorphism group is abelian follows from abelian implies nilpotent aut-nilpotent not implies aut-abelian
cyclic group via aut-abelian |FULL LIST, MORE INFO
locally cyclic group via aut-abelian |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Nilpotent group aut-nilpotent implies nilpotent of class at most one more nilpotent not implies aut-nilpotent |FULL LIST, MORE INFO
Retrieved from "https://groupprops.subwiki.org/w/index.php?title=Group_whose_automorphism_group_is_nilpotent&oldid=46751"