If $xy = yx$ and $\lvert x\rvert$ and $\lvert y\rvert$ are coprime, then $\lvert xy\rvert = lcm(\lvert x\rvert, \lvert y\rvert)$

Question

I would like to prove the following: If $G$ is a finite group (not necessarily Abelian) and if $xy = yx$ and $\lvert x\rvert$ and $\lvert y\rvert$ are coprime, then $\lvert xy\rvert = lcm(\lvert x\rvert, \lvert y\rvert)$.

I have a part of this because $$ (xy)^{lcm(\lvert x\rvert, \lvert y\rvert)} = x^{lcm(\lvert x\rvert, \lvert y\rvert)}y^{lcm(\lvert x\rvert, \lvert y\rvert)} = e $$ so $\lvert xy\rvert$ divides $lcm(\lvert x\rvert, \lvert y\rvert)$.

But I can't quite figure out the other direction. I had thought about saying something like if $(xy)^i = x^iy^i = e$, then somehow get to what I want but I am not sure.

Answer 1

Suppose $x^ay^b=e$.
Then $y^{-b\cdot |x|} =e$ so $b\cdot |x|$ is a multiple of $|y|$, hence - by being coprimes - so is $b$.
Similarly we get that $a$ must be a multiple of $|x|$.