Equal Order Elements may not be Conjugate

Theorem
Let $G$ be a group

Let $x, y \in G$ be elements of $G$ such that:
 * $\order x = \order y$

where $\order x$ denotes the order of $x$.

Then it is not necessarily the case that $x$ and $y$ are conjugates.

Proof
Consider the dihedral group $D_4$, whose group presentation is:

We have that:
 * $\order {a^2} = 2$

and:
 * $\order b = 2$

but $a^2$ and $b$ are not conjugate to each other.

Also see

 * Order of Conjugate Element equals Order of Element: the converse of this.