# 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.

Parents:

• Group homomorphism

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