# Group in which every element is order-conjugate

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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## Definition

### Symbol-free definition

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

1. 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).
2. 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 $1,2,3$ 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