Proving this group is Abelian

Question

Let $(G,.)$ be a group where there exists an element $g \in G$ such that for any $x \in G$ it is the case that $x^3 = gxg$.

I've been stumped on this one. All I have found is that $e = e^3 = geg = g^2$. Does anyone have advice on some starting points to solve this?

Thanks

Answer 1

Hint: Let $a,b\in G$. Therefore, $(ab)^3=gabg=gag^2bg=a^3b^3$.