Prove that a group G is Abelian if and only if (ab)^-1 =a^-1*b^-1 for all a and b in G.
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The Attempt at a Solution
Suppose (ab)^-1 =a^-1*b^-1 . Let (ab)^-1 *e = a^-1*e *b^-1*e. The multiply (ba)^-1 on the left side of the equation and b^-1 *a^-1 on the right side of the equation.
then (ab)^-1=(ba)^-1 and a^-1 *b^-1 =b^-1 *a^-1
then ((ab)^-1 *(ba)^-1)*e= (a^1*b^-1 *e)*(b^-1*a^-1*e
therefore, by method of direct proof, (ab)^-1 =a^-1 *b^-1