TITLE: Are kernels unique to homomorphisms?
QUESTION [8 upvotes]: 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.

REPLY [3 votes]: Think of $k\mapsto\exp(\frac{2k\pi}n\mathbf i):\Bbb Z\to\Bbb C^\times$ which has kernel $n\Bbb Z$.