Question:Normal subgroup endomorphisms

Q: Is a normal subgroup the same thing as the kernel of an endomorphism of the group?

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.