Let $(G, ·)$ be a group, and let $a, x$ ∈ $G$. Show that the orders of $x$ and $axa^{−1}$ are equal.
My attempt:
In order to show that the orders of $x$ and $axa^{−1}$ are equal, I have to show that $axa^{−1}$ = $x$.
Since $G$ is a group, we have that $axa^{−1}$ = $a.(x.a^{-1}) = (a.x).a^{-1}$ (associativity) = $a^{-1}.(a.x)$ (commutativity) = $(a^{-1}.a).x$ (associativity) = $e.x = x.e = x$ (property of the identity element).
Hence, I've shown that $axa^{−1}$ = $x$, and so their orders are equal.
Is my proof correct? Did I make a mistake somewhere? Do you have other suggestions?