Homomorphisms S4 to Z Find all homomorphisms f: S4 -> Z.
I know that Ker(f) must be a normal subgroup in S4. So, Ker(f) must be {e}, V4, A4 or S4.
However, I don't know how to use this information.
Thanks.

There is only one such homomorphism, the zero homomorphism (mapping everything to 0).
The reason is there every non-zero element of $\mathbb{Z}$ has infinite order, and the order of the image (under a homomorphism) of any element of finite order must be finite.
