Show the stabilizer is an open subgroup

Question

Let $V$ be a $l-$adic representation of a Galois group $G$ where $V$ is equipped with the $l-$adic topology. Let $T_0$ be a lattice in $V$ and $H=\{g \in G \mid g(T_0)=T_0\}$. Show $H$ is an open subgroup of $G$. Thanks for any help.

Answer 1

Let $v_1,\dots,v_n$ be a $\mathbb{Z}_\ell$-basis for $T_0$. Considering the map $G\to V^n$ given by $g\mapsto (gv_1,\dots,gv_n)$, we see that $U=\{g\in G\mid g(T_0)\subset T_0\}$ is an open subset of $G$. Since $g(T_0)=T_0$ holds if and only if both $g(T_0)\subset T_0$ and $g^{-1}(T_0)\subset T_0$ holds, we get that $$ \{g\in G\mid g(T_0)=T_0\}=U\cap U^{-1} $$ is also open.