If $x,y$ are element of the finite group $G$ such that $xy=yx$, then is the equation $(xy)^n=x^ny^n$ necessarily true?

Question

If $x,y$ are element of the finite group $G$ such that $xy=yx$, then is the equation $(xy)^n=x^ny^n$ necessarily true?

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I know that if $x^ny^n=(xy)^n$ then $x$ and $y$ commute, but I am not sure about the converse though.

Any tips are welcome.

Answer 1

Hint: Show by induction on $m$ that $xy^m=y^mx$ and $x^my=yx^m$ for any $m\in\Bbb Z$.

Answer 2

$$(xy)^n=x(yx)^{n-1}y=x(xy)^{n-1}y=\ldots=x^k(xy)^{n-k}y^k=\ldots=x^ny^n$$