How does a group homomorphism treat inverse elements?

In modern algebra, a group homomorphism is a structure-preserving map between two groups, $G$ and $H$. Because these mappings preserve the underlying algebraic structure, they ensure that the properties of elements in the domain are maintained in the codomain.

Specifically, a homomorphism maps the inverse of an element $g$ in group $G$ to the inverse of the corresponding element $f(g)$ in group $H$. This relationship is a direct consequence of the homomorphism's requirement to preserve the group operation: since the identity element of $G$ is mapped to the identity element of $H$, the product of an element and its inverse—which equals the identity—must be preserved under the mapping. Consequently, the image of the inverse is the inverse of the image, reinforcing why homomorphisms are referred to as structure-preserving maps.1