Proving that a group is abelian! Supposing we have a group $(G,*)$ (Which we don't know the elements of).
It is given that for each $x, y \in G$, exists $x*y*x=y$.
That's the only information we receive, How can we prove that group G is an abelian group?

If $xyx=y$ for each pair $x,y\in G$ then in particular this is true if we take $y=e$, where $e\in G$ is the netural element. We obtain $x^2=e$ for any $x\in G$, i.e. every element of $G$ is of order $2$. And such groups are abelian as you can read here.
