Question:Normal subgroup endomorphisms

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This question is about normal subgroup and endomorphism| See more questions about normal subgroup | See more questions about endomorphism
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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.