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.