For any Group homomorphism , we have .
Indeed, , and similarly for multiplication on the left.
https://arbital.com/p/group_homomorphism_image_of_inverse
by Patrick Stevens Jun 14 2016 updated Jun 14 2016
The operations of "taking inverses" and "applying a group homomorphism" commute: it does not matter in which order we do them.
For any Group homomorphism , we have .
Indeed, , and similarly for multiplication on the left.