I get the equation $a^{-1} + b^{-1} = (a + b)^{-1}$ from ordinary + operation. For ordinary + operation I mean $a^{-1} = -a$. It is also true for * of rational numbers $3^{-1}*4^{-1} = \frac{1}{3} * \frac{1}{4} = \frac{1}{12}=(3*4)^{-1}$.
I would like to know whether it is true for any Abelian group? If it is true I would like to know why?