let $\mathbb{C}^*$ be a multiplicative group of non-zero complex numbers.
Suppose that $H$ be a subgroup of finite index of $\mathbb{C}^*$.Prove that $H=\mathbb{C}^*$
Attempt
If $[\mathbb{C}^*.H]=n$ then if $xH \in \frac{\mathbb{C}^*}{H}$ then $\left(xH\right)^n=H$ implies $x^n \in H$.
Because $x$ is random we have $\left(\mathbb{C}^*\right)^n=\mathbb{C}^* \in H$
Note that $\left(\mathbb{C}^*\right)^n=\mathbb{C}^*$