# Question:Normal subgroup endomorphisms

A: A normal subgroup is defined as a subgroup expressible as the kernel of a homomorphism of groups. A subgroup $H$ of a group $G$ is normal if there is a homomorphism $\varphi:G \to K$ of groups whose kernel (i.e., the inverse image of the identity element) is precisely $H$.
The target group of the homomorphism need not be $G$, and hence, the homomorphism need not be an endomorphism.
An endomorphism kernel is a subgroup arising as the kernel of an endomorphism, i.e., a (not necessarily surjective) homomorphism from $G$ to itself.
Clearly, by definition, any endomorphism kernel is normal. However, the converse is not true: normal not implies endomorphism kernel. The smallest counterexample occurs where $G$ is the quaternion group. It does turn out, however, that if $G$ is a finite abelian group, then every normal subgroup does arise as the kernel of an endomorphism, because subgroup lattice and quotient lattice of finite abelian group are isomorphic.