Problem: What are the possible kernels of homomorphisms $\mathbb{Z} \to \mathbb{Z}$?
I understand why the possible kernels are $0$ and $\mathbb{Z}$ if the binary operation is addition for both domain and codomain of the homomorphism.
However, would the answer be same if binary operation for domain and codomain did not have to be addition?
I cannot think of any binary operation other than addition now, but I want to know if we can generally find the possible kernels of homomorphisms when we do not restrict the binary operation on both domain and codomain to be addition.
Suppose that you define $x\oplus y=x+y+1$. It's a binary operation and $(\mathbb Z,\oplus)$ is a group. And the kernel of the identity function from $(\mathbb Z,\oplus)$ into itself will be $\{-1\}$ then. By a similar construction, for any $a\in\mathbb Z$, there is a group operation defined on $\mathbb Z$ such that $\ker\operatorname{Id}=\{a\}$.
More generaly, if $K\subset\mathbb Z$ is non-empty and $K^\complement$ is an infinite set, that, for some group operation on $\mathbb Z$, $K=\ker\operatorname{Id}$.