Normal subgroups of GL$(n,\mathbb{R})$ and SL$(n,\mathbb{R})$

Question

I wants to compute all the normal subgroups of SL$(n,\mathbb{R})$. Please give me some idea or reference to study about this. I have seen following question and answer given for $n=2$ but I just not neede the statement I wants to go through proof as well (Normal subgroups of the Special Linear Group).

Answer 1

Let $K$ be any field and $n \ge 2$ and let $G = {\rm GL}(n,K)$ and $S = {\rm SL}(n,K)$. Let's exclude the two small exceptional cases $n=2$ and $|K| \le 3$.

Then $[G,G]=S$ with $S$ perfect, $Z(G)$ is the subgroup $Z$ of nonzero scalar matrices, and $S/(S \cap K) = {\rm PSL}(n,K)$ is simple.

It follows that, if $N$ is a normal subgroup of $G$ with $N \not\le Z$, then $[G,N] = S$, so $S \le N$.

So there are just two types of normal subgroups of $G$, those contained in $Z$, and those that contain $S$. Since $Z \cong G/S \cong (K \setminus \{0\},\times)$, the normal subgroups of both types correspond exactly to the subgroups of $(K \setminus \{0\},\times)$.