Given the semigroup $(S,*)$ with at least two elements such that for every $x$ and $y$ in $S$ it holds that $x*y^{2}*x=x^2$ let us look at structure $(S,×) $ where operation "$×$" is defined as $x×y=y^2*x$. Prove that $(S,×)$ is a semigroup and check whether it can be a group.
I'm kind of stuck with this problem, it's obvious that closure holds, but I'm stuck to prove that it's associative, I got that $(x×y)×z = z^2*y^2*x$ and $x×(y×z) = z^4*y^2*x$. I don't know what I can do to apply that given condition here. And since I didn't even prove it is associative I have no idea if it can be a group.