Is the following statement is True/false regarding a non-trivial homomorphism

abstract-algebra

Is the following statement True ?

For every integer $n \geq 2$, there is a unique non-trivial homomorphism
$\phi:S_n \rightarrow \mathbb{C}^*$. where $\mathbb{C}^*$,denotes the multiplicative group of non-zero complex number.

My attempt : This is false, because this only true for $n \le 4$ because cyclic groups map to cyclic group, as as every finite subgroup of $ \mathbb{C}^*$ is cyclic

Is it correct ?

Best Answer

What about this ? $$g:\sigma \mapsto \begin{cases} 1&\text{if}\;\sigma \;\text{is even}\\-1 &\text{otherwise}\end{cases} $$

Uniqueness :

Suppose there is another non trivial homomorphism, say, $f$. Then $\ker f$ is a normal subgroup of $G$. But for $n \geq 5$, $A_n$ is the only proper normal subgroup of $S_n$. So $$\ker f\in\Big\{\{e\},A_n,S_n\Big\}$$

  • $\ker f=S_n$ implies $f$ is trivial, which is not under consideration
  • $\ker f=\{e\}$ implies $f(S_n)$ is a subgroup of order $6$ in $C^*$ and so cyclic, which is not true

Hence $\ker f=A_n$ and so $f=g$

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