# Understanding the hint of a question to show that $G$ is Abelian.

Here is the question I want to answer:

Let $$G$$ be a finite group such that 3 does not divide $$|G|$$ and such that the identity $$(xy)^3 = x^3 y^3$$ holds for all $$x,y in G.$$ Show that $$G$$ is abelian.

And here is the hint I got for the question:

First show that the map $$G rightarrow G$$ given by $$x mapsto x^3$$ is bijective. Then show that $$x^2 in Z(G)$$ for all $$x in G.$$

My questions are:

1- Could anyone explain for me why we should show that the map $$G rightarrow G$$ given by $$x mapsto x^3$$ is bijective?

2- Also, why we should show that $$x^2 in Z(G)$$ for all $$x in G$$?

3- And how proving the hint will show that $$G$$ is Abelian?

Mathematics Asked on January 2, 2021

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