Finite index subgroup of $SL_n$ over an algebraic extension of $\mathbb Z$

Question

If $\mathbb{\hat Z}$ is an algebraic extension of $\mathbb Z$ and $H$ is a finite index subgroup of $SL_n(\mathbb{\hat Z})$, does $H\cap SL_n(\mathbb Z)$ has a finite index in $SL_n(\mathbb Z)$?

Answer 1

OK, so this is simply from the second isomorphism theorem where $G=SL_n(\mathbb {\hat Z}), S=SL_n(\mathbb Z)$ and let $N\subset H$ be a normal subgroup of $G$ with a finite index. So $[S:H\cap S]\leq [S:N\cap S]\leq($from the second isomorphism theorem$)\leq [G:N]\lt \infty$