Finite subgroups which are normal in a matrix Lie group

Question

I have the following question：

Let $G$ be a closed subgroup of $GL(n,\mathbb{C})$. Denote $Z(G)$ by the center of $G$.

${\bf Question}:$ Is it true that every finite normal subgroup of $G$ are contained in $Z(G)$?

Thanks very much!

Answer 1

Generally one assumes that $G$ is connected. If $G$ is not connected then the answer is certainly no, because $G$ could be finite itself.

Anyway, assume $G$ is connected.

Hint: Let $H \leq G$ be finite and normal. Pick $h \in H$ and consider the map $G \to H$ defined by $g \mapsto ghg^{-1}$. What could the image be?