# Group in which every automorphism is inner

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 in which every automorphism is inner** is a group satisfying the following equivalent conditions:

- Every automorphism of the group is an inner automorphism, i.e., can be expressed via conjugation by an element.
- The outer automorphism group of the group is trivial.

## Formalisms

### In terms of the supergroup property collapse operator

This group property can be defined in terms of the collapse of two subgroup properties in the following sense. Whenever the given group is embedded as a subgroup satisfying the first subgroup property (normal subgroup), in some bigger group, it also satisfies the second subgroup property (central factor), and vice versa.

View other group properties obtained in this way

### In terms of the automorphism property collapse operator

This group property can be defined in terms of the collapse of two automorphism properties. In other words, a group satisfies this group property if and only if every automorphism of it satisfying the first property (automorphism) satisfies the second property (inner automorphism), and vice versa.

View other group properties obtained in this way

## Relation with other properties

### Stronger properties

### Weaker properties

- Group in which every normal subgroup is characteristic
- Group in which every subgroup is automorph-conjugate
- Group in which every automorphism is class-preserving
- Group in which every automorphism is normal-extensible

## Facts

Group property | What can we say about which groups with the property also have the property that every automorphism is inner? |
---|---|

finite nilpotent group | The only examples are the trivial group and cyclic group:Z2. See also finite nilpotent and every automorphism is inner implies trivial or cyclic of order two. |

finite solvable group | There are many example. In fact, there exist finite solvable groups that are complete: they are centerless and every automorphism is inner. In fact, there can exist multiple non-isomorphic complete solvable groups of the same order. See complete and composition factor-equivalent not implies isomorphic. |

finitely generated nilpotent group | The only examples are the trivial group and cyclic group:Z2. See also finitely generated nilpotent and every automorphism is inner implies trivial or cyclic of order two. |

(infinite) nilpotent group | There exist infinite nilpotent groups in which every automorphism is inner |