Prove that * is commutative and associative

Question

Assume that $*$ is an operation on $S$ with identity element $e$ and that $x*(y*z)=(x*z)*y$ for all $x, y, z$ in $S$. prove that $*$ is commutative and associative

Ok, I know that in order for it to be associative then $x*(y*z)=(x*y)*z$ and that $(x*z)*y=x*(z*y)$ and that $x=e$, so then $y*z=z*y$ for every $z,y$ in $S$ but I'm not sure how to actually prove this statement

Answer 1

Hint:

First prove commutativity, setting $x=e$. Then it is very easy to deduce associativity.

A small remark: to prove associativity, you have to prove a single equality, not two.

Answer 2

Let $x=e$. Then, for general $y, z\in S$, we have $$\begin{align} y\ast z &=e\ast (y\ast z) \\ &=(e\ast z)\ast y& \\ &=z\ast y.\end{align}$$ Hence $\ast$ is commutative.

Now for any $a,b,c\in S$, we have $$\begin{align}a\ast (b\ast c)&=a\ast (c\ast b) \\ &=(a\ast b)\ast c, \end{align}$$ so $\ast$ is associative.