Let $n \in \mathbb{Z}$ such that $n \geq 2$. Let $\mathcal{S}_n$ be the permutation group on $n$ letters and $\mathcal{A}_n$ be the alternating group. We also denote $\mathbb{C}^*$ the group of non zero complex numbers under multiplication.
Which of the following are correct statements?
- For every integer $n \geq 2$ there is a non trivial homomorphism $\chi: \mathcal{S}_n \rightarrow \mathbb{C}^*$.
- For every integer $n \geq 2$ there is a unique non trivial homomorphism $\chi:\mathcal{S}_n \rightarrow \mathbb{C}^*$.
- For every integer $n \geq 3$ there is a non trivial homomorphism $\chi:\mathcal{A}_n \rightarrow \mathbb{C}^*$.
- For every integer $n \geq 5$ there is non trivial homomorphism $\chi:\mathcal{A}_n \rightarrow \mathbb{C}^*$.
Justify your answers.