# Group in which every element is order-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

### Symbol-free definition

A **group in which every element is order-conjugate** is a group with the following equivalent properties:

- Any two elements of the same order are conjugate elements (note that this also means that all the elements that have infinite order are conjugate).
- Whenever the group occurs as a subgroup of some group, it is a conjugacy-closed subgroup.

There are only three finite groups with this property: the symmetric groups on elements respectively. The proof of this depends on the classification of finite simple groups. `Further information: classification of finite groups in which every element is order-conjugate`

### Equivalence of definitions

The equivalence of definitions relies on a general result in the theory of HNN extensions, which says that any group can be embedded as a subgroup in some group where any two elements that had equal order in the original group become conjugate in the big group. `Further information: Equivalence of definitions of group in which every element is order-conjugate`

## Relation with other properties

### Stronger properties

- Group with two conjugacy classes: In such a group, all the non-identity elements are conjugate.