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Prove that a group G is Abelian if and only if (ab)^-1 = (a^-1)(b^-1) for all a and b in G.
I know how to do the first part of the proof.
If G is Abelian then (ab)^-1 = (a^-1)(b^-1).
(ab)^-1 = (b^-1)(a^-1)=(a^-1)(b^-1), since G is abelian.
The second part is where I am snagged.
If (ab)^-1 = (a^-1)(b^-1) then G is abelian.
(ab)^-1 = (b^-1)(a^-1) = (ab)^-1 = (a^-1)(b^-1) Therefore (b^-1)(a^-1)=(a^-1)(b^-1). So G must be abelian.
I'm not sure if i can make these statements. It seems like i'm assuming stuff.
Any help is appreciated.
