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!
2026-04-20 15:22:39.1776698559
Finite subgroups which are normal in a matrix Lie group
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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?