# 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

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## 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 |