I'm trying to show that there is no subgroup of $GL_2(\mathbb{C})$ which is isomorphic to $S_4$. I'm doing a proof by contradiction; Let $G$ be a subgroup of $GL_2(\mathbb{C})$ which is isomorphic to $S_4$. I saw that we can conclude then that $G$ is a subgroup of $U(2)$. Why is that right? How can I show that?
I'll be thankful for having answers.
Let $G$ be a finite subgroup of $\mathrm{GL}_n(\Bbb C)$. Define a new inner product on $\Bbb C^n$ via $(v,w) := \sum_{g \in G}\langle gv,gw\rangle$, where $\langle-,-\rangle$ is the standard inner product (check that this is actually an inner product). By construction, $(v,w)$ is $G$-invariant, i.e. $(v,w)=(gv,gw)$ for all $v,w \in \Bbb C^n, g \in G$. This implies that if we choose an orthonormal basis for this inner product, $G$ will be contained in $U(n)$.
(This is a standard argument in representation theory.)