Same order iff potentially conjugate

Verbal statement
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 $$x,y \in G$$:


 * The order of $$x$$ equals the order of $$y$$.
 * There exists a group $$L$$ containing $$G$$ such that $$x,y$$ are conjugate in $$L$$.

Applications

 * Equivalence of definitions of group in which every element is order-conjugate
 * Every group is a subgroup of a group with two conjugacy classes

Facts used

 * 1) uses::Isomorphic iff potentially conjugate