# Group in which any two elements generating the same cyclic subgroup are automorphic

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 any two elements generating the same cyclic subgroup are automorphic** is a group satisfying the following equivalent conditions:

- are such that , then there is an automorphism of sending to .
- There exists a group containing as a normal subgroup such that all elements of are rational elements of .

## Relation with other properties

### Stronger properties

- Group with two conjugacy classes
- Group whose automorphism group is transitive on non-identity elements
- Group in which every element is order-conjugate
- Group in which every element is order-automorphic
- Rational group
- Finite abelian group

### Weaker properties

## Facts

- Normal subgroup of rational group implies any two elements generating the same cyclic subgroup are automorphic
- Alternating group implies any two elements generating the same cyclic subgroup are automorphic

## Metaproperties

### Characteristic subgroups

This group property is characteristic subgroup-closed: any characteristic subgroup of a group with the property, also has the property

View characteristic subgroup-closed group properties]]

### Quotients

This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property

View a complete list of quotient-closed group properties