## Binary Operation question [closed]

Hi All, 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.

An answer was provided on math.stackexchange sit. i am just adding to this post as this was closed.

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.

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Although your question doesn't come from an assignment, it is not appropriate for MathOverflow. It would fit much better on math.stackexchange.com where you should be able to get help with it. – Yemon Choi Nov 1 2011 at 3:15
$yz=e(yz)=(ez)y=zy$, $x(yz)=x(zy)=(xy)z$. – Mark Sapir Nov 1 2011 at 3:40