Group whose automorphism group is nilpotent: Difference between revisions
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===Weaker properties=== | ===Weaker properties=== | ||
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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::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}} | | [[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}} | ||
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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 |