normal subgroup of Classical Groups

https://math.stackexchange.com/questions/3474827/normal-subgroups-of-gln-mathbbr-and-sln-mathbbr

Proposition

Let $G=\mathrm{SL}(n,q)$ with $\mathrm{PSL}(n,q)$ simple, and let $N⊲G$ be a normal subgroup of $G$. Then $N⩽Z(G)$ or $N=\mathrm{SL}(n,q)$.

\begin{proof} Consider $\pi :G\to G/Z(G)$. Then $\pi (N)$ is a normal subgroup of $\mathrm{PSL}(n,q)$, and so $N⩽Z(G)$ or $NZ(G)=\mathrm{SL}(n,q)$. If $NZ(G)=G$, then

$$G/N=NZ/N\cong Z/Z\cap N$$

is abelian and so $N⩾G^{\prime }=G$. \end{proof}

• in fact, for each non-solvable group in ^bcmv8d, their normal subgroups are center or SL …