What is an example of a proper normal subgroup of the kernel of a homomorphism?

Question

I'm reading this proof by Hungerford that concerns any normal subgroup of the kernel of a homomorphism. I understand the proof well enough, but I wanted to have some concrete example to guide or ground my understanding, but I can't think of an example from among the symmetric groups, integers, rationals, or general or special linear groups that is small enough to be manageable and still can provide an example of such a thing. Anyone have a suggestion? I guess maybe I could take any group that has a normal subgroup and use the homomorphism that just maps everything to the identity, but then I feel like this wouldn't be easily generalized because then I'd want to add other elements to the domain, and at that point I wouldn't be sure that my normal subgroup was still normal in the new, bigger group.

Answer 1

$\phi:\mathbb Z\to\mathbb Z/ n\mathbb Z$ has $n\mathbb Z$ as the kernel, which has any $m\mathbb Z$ as a normal subgroup when $n\mid m$.