# Group in which every subgroup is automorph-conjugate

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 subgroup is automorph-conjugate** is a group satisfying the following equivalent conditions:

- Every subgroup of the group is an automorph-conjugate subgroup.
- Any two automorphic subgroups of the group are conjugate subgroups.
- Every automorphism of the group is a subgroup-conjugating automorphism.

## Formalisms

### In terms of the subgroup property collapse operator

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

View other group properties obtained in this way