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:

$D_4 = \gen {a, b: a^4 = b^2 = e, a b = b a^{-1} }$

We have that:

$\order {a^2} = 2$

and:

$\order b = 2$

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

$\blacksquare$

Also see

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

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $10$: The Orbit-Stabiliser Theorem: Proposition $10.18$: Remark $2$