Are kernels unique to homomorphisms?

Question

I would like to ask if two different homomorphisms can share the same kernel.

For instance for the kernel $n \mathbb{Z} $, is it possible to come up with homomorphisms other than the function mapping integers to residue classes modulo $n$? Thanks.

Answer 1

No, a homomorphism is not uniquely determined by its kernel. Consider the following two homomorphisms from $\mathbb{Z}_2$ to $\mathbb{Z}_2\times\mathbb{Z}_2$: one sending $1$ to $(0,1)$ and the other sending $1$ to $(1,0)$. They're both homomorphisms with the same kernel to the same group, but they are different homomorphisms.

Answer 2

Unless $G$ has a "simple" structure, there are many isomorphisms $:G \to G$ and they all have a kernel of $\{e_G\}$.

Answer 3

The identity homomorphism and the homomorphism $n\mapsto -n$ from $\mathbb{Z}$ to $\mathbb{Z}$ have ...?... kernel.

Answer 4

Think of $k\mapsto\exp(\frac{2k\pi}n\mathbf i):\Bbb Z\to\Bbb C^\times$ which has kernel $n\Bbb Z$.