Suppose G is a group. We say that G has "middle cancelation" if, for any a, b, c, d, xe G, arb = crd implies ab = cd. (a) Prove that if G is abelian then it has middle cancelation. (b) Prove the opposite direction, i.e. if G has middle cancelation, then G is abelian. (Hint: Use middle cancelation on the expression aa-¹b = ba¹a.) Therefore G is abelian if and only if G has middle cancelation.

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