Abstract Algebra Groups??

Question

Let G be a group. Prove that G is abelian if and only if (ab)^-1=a^-1*b^-1 for all a, b is in G.

Answer 1

For the "If" part, you need to show that

AB = BA

If it is, then

(AB)^-1 * AB = (AB)^-1 * BA

But we can now substitute and get

1 = A^-1 * B^-1 * B * A = A^-1 * (B^-1 * B) * A - 1*-1 * 1 * A = 1

Since AB and BA have the same inverse, they must be the same.

Going the other way is trivial, because you know that the commutative property hold.