Group theory exercise from Judson text S=R−1 and define a binary operation on S by a×b=a+b+ab. Prove that (S,×) is an abelian group.

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Group theory exercise from Judson text  S=R−1 and define a binary operation on S by a×b=a+b+ab. Prove that (S,×) is an abelian group.

zesponderyd · Accepted Answer

Step 1  Let  f(x)=x+1  and  g(x)=x−1.  Then what we have is  x×y=(x+1)(y+1)−1=g(f(x)f(y))  Or, to write this in a way that might be easier to catch what is going on:  g(x)×g(y)=g(xy)  So S is isomorphic to (R0, ×)

jgardner33v4 · Accepted Answer

Step 1Here −1 is an obstacle, essentially, because of the following.Lemma: Let e be the identity of a group G. Then the only idempotent of G is e.Proof: Let x2=x. Then ex=x=×. Multiplying on the right by x−1 then gives e=xWe have−1×−1=−1−1+(−1)2=−1and 0×0=0+0+02=0Moreover, we have thata×a′=a+a′+aa′=0 impliesa′=−a1+awhere a′ is the inverse of a with respect to ×

Group theory exercise from Judson text S=\frac{R}{-1} and define a binary

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