Question

give an example of a non–trivial homomorphism or explain why none exists

φ:S₄ → S₃

Answer 1

An example could by a homomorphism that is composed by two homomorphisms:

1. There is a homomorphism "signature" $\text{sgn}: S_n \to \mathbb{Z}/2\mathbb{Z}$ that associates to a permutation its signature.
2. There is an injection $\iota: \mathbb{Z}/2\mathbb{Z} \to S_m$ for any $m\geq 2$ that associates $1$ to a permutation $\begin{pmatrix} 1 & 2 \end{pmatrix}$.

Composition of these two homomorphisms works, in particular, for $n=4, m=3$ which gives us a homomorphism $\phi: S_4 \to S_3$ with $\phi(even)=id; \phi(odd)=\begin{pmatrix} 1 & 2 \end{pmatrix}$.

There exist another non-trivial homomorphism, that is more interesting because it is surjective. We can prove that $K=\{id, (12)(34), (13)(24), (14)(23) \}$ is a normal subgroup of $S_4$ and the quotient $S_4/K \simeq S_3$, so the composition of these two surjective homomorphisms is a surjective homomorphism.