Let G be a group. Prove that G is abelian if and only if (ab)^-1 = a^-1 * b^-1 for all a,b in G.
First note we always have $\displaystyle (ab)^{-1}=b^{-1}a^{-1}$ See why?
so if G is abelian, the forward direction is clear (just move them past eachother). Conversely, if we have $\displaystyle a^{-1}b^{-1}=(ab)^{-1}=b^{-1}a^{-1}$ for all a and b in G.
Just multiply on the left by (ba) and on the right by (ab)