Binary Operation question - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-20T17:21:07Z http://mathoverflow.net/feeds/question/79673 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/79673/binary-operation-question Binary Operation question Hardy 2011-11-01T03:06:55Z 2011-11-01T04:01:20Z <p>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. </p> <p>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.</p> <p>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.</p> <p>Any help would be highly appreciated.</p> <p>An answer was provided on math.stackexchange sit. i am just adding to this post as this was closed.</p> <p>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.</p>