Normal Subgroups of $O(n)$

Question

In general, what are the normal subgroups of the orthogonal group? I know $SO(n)$, $\{I\}$, $\{I,-I\}$ are examples, are there others?

Answer 1

If $G$ is a connected Lie group and $N$ a closed normal subgroup, then its Lie algebra $\mathfrak{n}$ is an ideal of the Lie algebra $\mathfrak{g}$ of $G$. Because the Lie algebra $\mathfrak{so}(n)$ is simple, $\mathfrak{n}=0$ or $\mathfrak{n}=\mathfrak{so}(n)$. We can apply this to the connected components of $O(n)$. This shows that $N$ is $SO(n)$, or $N$ is discrete (if $\mathfrak{n}=0$). If $N$ is discrete, then $N$ must be contained in the center $\{I,-I\}$.