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