Suppose $\phi$ is a homomorphism from $S_4$ to $S_3$. It is given that $\phi (12)=(13)$ and $\phi (234)=(132)$. Compute the entire mapping $\phi$.
$S_4= \{ (), (12), (13), (14), (23), (24), (34), (12)(34), (13)(24), (14)(23), (123), (124), (134), (234), (143), (243), (132), (142), (1234), (1324), (1423), (1432), (1342), (1243) \}$
You can do $\phi (12) \phi (234) =\phi (1342)=(13)(132)=(12) $ and maybe do similar calculations but it might take a while.
I feel like once I calculate the mapping of $(12), (13), (14), (23), (24), (34) $ then I can compute everything easily and not miss anything but I how do I find them?
The permutations $(1 2)$ and $(2 3 4)$ form a set of generators of $S_4$. To find the image of an arbitrary element of $S_4$, first write that element as a product of the generators (and their inverses).
Example, using the convention that composition goes from the left to the right:
$(1 3) = (2 3 4)^{-1} (1 2) (2 3 4)$
$\phi(1 3) = (1 3 2)^{-1} (1 3) (1 3 2) = (2 3)$