Group whose automorphism group is nilpotent
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 |