Semi group $S$, $x^2y=y=yx^2$ for all $x$, $y$ show that $S$ is abelian

Question

I can not understand few steps please just give me hint what actually done in second part .

Answer 1

Because $$xyxyx^3y=x(yxyx^3y)=x(yxyxx^2y)=x(yx)^2x^2y=x(yx)^2x(xy).$$

Answer 2

$ xyxyxy=xyxyx^3y $ because of the identity $X^3=X$ valid for all $X\in S$. Now, we can bracket $$xyxyxy=x(yx)(yx)x^2y=x(yx)^2x(xy)$$ OK ?

Answer 3

In the step you pointed at, $x=x^3\,\forall x\in S$ is used repeatedly; then there's just a little rearranging using the associative property: $xy=(xy)^3=(xy)(xy)(xy)=(xy)(xy)(x^3y)=x(yx)(yx)xxy=x(yx)^2x(xy)$.