Given an algebra structure $(X,*)$ s.t. $(x*y)*y = y*(y*x) = x$ , prove$x*y=y*x$.

Question

Suppose $(X,*)$ is arbitrary algebraic structure such that $\forall x,y\in X$, we have $(x*y)*y = y*(y*x) = x$, prove that $x*y=y*x$.

This question seems pretty simple but I tried and I failed.

Answer 1

From the condition, we have $$ \begin{cases} \underline{(x*(x*y))*(x*y)} = (x*y)*((x*y)*x)=\underline{x} &(1) \\ (y*x)*x = \underline{x*(x*y)=y} &(2) \\ ((x*y)*y)*y = \underline{y*(y*(x*y)) = x*y} &(3) \end{cases} $$ for all $x,y \in X.$

Therefore $$ \begin{align*} y*x &= y*((x*(x*y))*(x*y)) & \textrm{by (1)} \\ &= y*(y*(x*y)) & \textrm{by (2)}\\ &= x*y & \textrm{by (3)}\\ \end{align*} $$ Q.E.D.