Under a group homomorphism, the image of the inverse is the inverse of the image

For any group homomorphism \(f: G \to H\), we have \(f(g^{-1}) = f(g)^{-1}\).

Indeed, \(f(g^{-1}) f(g) = f(g^{-1} g) = f(e_G) = e_H\), and similarly for multiplication on the left.