Let f : G → H be a group homomorphism. The kernel of f is Kerf = {a ∈ G | f(a) = eH}. It is a (normal) subgroup of G. ... A homomorphism f is a monomorphism if and only if Kerf = {e}; f is an epimorphism if and only if f(G) = H.

Is a homomorphism a group?

A group homomorphism that is bijective; i.e., injective and surjective. Its inverse is also a group homomorphism. In this case, the groups G and H are called isomorphic; they differ only in the notation of their elements and are identical for all practical purposes.

Does homomorphism preserve subgroup?

ϕ[H] denotes the image of H under ϕ ≤ denotes subgroup. That is, group homomorphism preserves subgroups.