Group whose automorphism group is nilpotent
From Groupprops
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Contents
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 |