Let $G\subseteq GL_n(\mathbb{C})$ be a irreducible, affine, algebraic group (Zariski-Closed). Moreover let $H \subseteq G$ be a finite normal group. I want to show that $H \subseteq Z(G):=\{g \in G|hg=gh \hspace{0.3cm} \forall h \in G\}$. I am not quite sure how to start. I think that if $H$ is not central, then it should be irreducible, but I am not sure.
Thank you.
$G$ is irreducible, so for any $h\in H$ the set $G\cdot h=\{ghg^{-1}\mid g\in G\}$ is connected. But $H\unlhd G$, so $G\cdot h\subseteq H$. As a finite set $H$ is discrete, so this implies that $G\cdot h$ must be a singleton. Therefore $h\in Z(G)$.