Let $S$ is a semi-group such that $x^2y=y=yx^2$ for all $x,y$ in $S$. Is it a group and if yes, is it abelian?
I am asked whether $S$ may be a group or not. If yes, is it abelian?
I can't see how I am supposed to solve this question, given the fact that I am not given the binary operation for it. Is it multiplication?
For it to be a group, it should be closed, associative, have an identity element and an inverse element. How should I proceed to check those properties with the given property?
We are given that $S$ is a semigroup: so it's already equipped with an associative operation.
It becomes a group if we show existence of the unit and inverse elements.
Fix an arbitrary $x\in S$, and let $e:=x^2$. By the conditions, we have $ey=y=ye$ for all $y\in S$, so $e$ is a unit element, and as such it's unique.
(Alternatively, we have $u^2 =u^2v^2 =v^2$ when applying the conditions once with $x=u, y=v^2$ and once with $x=v, y=u^2$.)
Consequently, $x^2=e$ for every $x\in S$, and thus $x^{-1}$ exists and $x^{-1}=x$.
Finally, it's also Abelian, because $xyxy=(xy)^2=e=x^2y^2 =xxyy\implies yx=xy$.