TITLE: Proving this group is Abelian
QUESTION [7 upvotes]: 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

REPLY [4 votes]: Hint: Let $a,b\in G$. Therefore, $(ab)^3=gabg=gag^2bg=a^3b^3$.