Show that a complex, compact, abelian linear group is finite (using Liouville's theorem)

Question

I'm trying to solve the following problem from the second chapter of Rossmann's book on Lie groups:

This seems like a very interesting result (and a cool application of Liouville's theorem), but I'm not sure on how to approach this.

I thought about maybe looking at the matrix norm/determinant as analytic functions from the matrices to $\mathbb C$ (which should be bounded since the group is compact), but the version I know of Liouville's theorem requires an analytic function who's domain is all of $\mathbb C$ (and here I'm defining it on a group of matrices). So I'm not at all sure what function I should use, and what set to define it on.

Does anyone have an idea on how one should approach this problem? Where should the finiteness come from?

Thanks in advance.

Answer 1

If complex linear group means a complex submanifold of $GL_n(\Bbb{C})$, then the maximum modulus principle is more natural: let $M$ be a compact complex submanifold of $\Bbb{C}^k$ then $\phi:M\to \Bbb{C},\phi(z)=z_j$ is analytic and $|\phi|$ attains its maximum at some $z\in M$ thus $\phi$ is constant near $z$, doing so with each connected component shows that $\phi$ is everywhere locally constant so $M$ is discrete thus finite.