Same order iff potentially conjugate
The following are equivalent for two elements of a group:
- They have the same order.
- The group can be embedded in a bigger group in which the two elements are conjugate elements.
Statement with symbols
The following are equivalent for elements :
- The order of equals the order of .
- There exists a group containing such that are conjugate in .
- Equivalence of definitions of group in which every element is order-conjugate
- Every group is a subgroup of a group with two conjugacy classes