Homomorphism between $\mathbb C^*$ and $S_n$

Question

I want to know that if there is any nontrivial homomorphism between $\mathbb C^*$ , the group of all non zero complex numbers and $S_n$ , the group of all permutations on the set $\{1,2, \dots , n\}$ $(n \ge 2) $ or $A_n$ , the set of all even permutations on the same set.

Any insight . Thank you.

Answer 1

Let $\phi:\mathbb C^* \longmapsto S_n$ be a homomorphism. Then $\mathbb C^*/ker\phi \cong H\le S_n$. i.e. $ker\phi $ is a finite index subgroup of $\mathbb C^*$. Let $\mid \mathbb C^*:ker\phi \mid=n\Rightarrow x^n\in ker\phi $ $\forall x\in \mathbb C^*$.

Now the equation $x^n=c$ has solution of $x$ for every $x\in \mathbb C^*\Rightarrow \mathbb C^*\subseteq ker\phi \Rightarrow ker\phi =\mathbb C^*$. Hence the only homomorphism is the trivial homomorphism.

Answer 2

(There may be an easier way to see this.)

There are no such nontrivial homomorphisms. Since $\mathbb C^*$ is a divisible group and every quotient of a divisible group is again divisible, by the first isomorphism theorem the image of such a homomorphism is divisible too. But the only finite divisible group is the trivial group, so every homomorphism is trivial.