Group Problem

Let G be a group.
∀x,y ∈ G
Prove that xy and yx have the same order n ?
ie. $latex (xy)^{n}= (yx)^{n} = e$

Proof:
Let
$latex (xy)^{n}=e$
(xy)(xy).....(xy)(xy) = e
[(xy) n times]
By associativity,
x.(yx)(yx)...(yx).y=e
$latex (yx)(yx)...(yx).y = x^{-1}$
$latex (yx)^{n-1}= x^{-1}.y{^-1}$
$latex (yx)^{n-1}= (y.x)^{-1}$
$latex (yx) ^{n-1}.(yx) = e$
$latex (yx)^{n} = e$
ie yx has order n.
[QED]