[Math] Center of an irreducible faithful representation is cyclic.

Question

I read on many references that : given $G$ a finite group which has an irreducible faithful representation, then its center $Z(G)$ is cyclic. I want to prove it over $\mathbb{C}$. Any help ?

Answer 1

The $A$ group of automorphisms of the representation $f:G\rightarrow Gl(V)$ is contained in the group of homothetic map $(h(x)=cx, x\in\mathbb{C})$ since the representation is irreducible (Schur's lemma), you deduce that for every $g\in Z(G)$, $g(x)=c_gx$, $|c_g|=1$ since $G$ is finite. A finite subgroup of $S^1$ is cyclic.