# Group in which every class-preserving 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 class-preserving automorphism is inner** is a group with the property that any class-preserving automorphism of the group (i.e., any automorphism that sends every element to within its conjugacy class) is an inner automorphism.

Note that, in a general group, a class-preserving automorphism need not be inner, although every inner automorphism is class-preserving. `For full proof, refer: Class-preserving not implies inner, inner implies class-preserving`

## Formalisms

### 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 (class-preserving automorphism) satisfies the second property (inner automorphism), and vice versa.

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### 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 (conjugacy-closed 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