Group in which every element is order-conjugate

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 defining ingredient::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.

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.

Stronger properties

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

Weaker properties

 * Stronger than::Group in which every element is automorph-conjugate