# 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”.