# If $x * (y * z) = (x * z) * y$ for all $x, y, z \in S$, then $*$ is both associative and commutative

This is not an assignment question. I have been self teaching myself abstract algebra from this book and this is one of the exercise questions which I could n't solve.

Suppose $e$ is the identity element for a binary operation $*$ defined on S. If $*$ satisfies the identity $x * (y * z) = (x * z) * y$ where $x,y,z$ are elements of $S$, then show that $*$ is both commutative and associative.

In all my attempts I started with the left hand side of the identity but kept getting stuck at how to can $z$ jump across $y$. The only other known piece of information I was considering was that $e * a = a * e = a$, where $a$ belongs to $S$ and this holds for all elements of $S$.

Any help would be highly appreciated.

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Set $x=e$ and you have commutativity. Once you have that, you can commute $y$ and $z$ to get $$x*(z*y)=(x*z)*y$$ and there's your associativity.