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.


  • Group homomorphism

    A group ho­mo­mor­phism is a “func­tion be­tween groups” that “re­spects the group struc­ture”.