Same order iff potentially conjugate

Statement

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 $L$.

Related facts

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. Isomorphic iff potentially conjugate