$G$ is a subgroup of finite index in $SL(n,\mathbb{Z})$, $n\ge 3$, $N$ is finite normal subgroup of $SL(n,\mathbb{Z})$, then I want to know why $N$ is a normal subgroup of $SL(n,\mathbb{Z})$.
More generally, $A$ is an arithmetic subgroup of a simple algebraic $\mathbb{Q}$-group $G$ with $\mathbb{Q}$-$\operatorname{rank}(G)\ge 2$ and $N$ is a finite normal subgroup of $A$. Then $N$ is a normal subgroup of $G$. But I don't know why.