In J. Silverman's book "Arithmetic of Elliptic Curves" on page 40 in the beginning of the proof of Lemma 5.8.1 I have found this statement. Let $K$ be a number field and $\overline{K}$ its algebraic closure. Let $V$ be a $\overline{K}−$vector space and let the absolute Galois group $G_{\overline{K}/K}$ act continuously on $V$. Then, "The fact that $G_{\overline{K}/K}$ acts continuously on $V$ in a compatible way with its action on $\overline{K}$ means precisely that for any $v\in V$ the stabilizer of $v$ as finite index in $G_{\overline{K}/K}$". I do not understand why this has to be true. If $V=\overline{K}$ and more generally if the dimension of $V$ over $\overline{K}$ is finite, then the orbits are always finite (as any algebraic number has only finitely many conjugates), there is no relation with continuity or not. What is the correct meaning of this statement and why should it be true?
This question was asked in math stackexchange with no reaction, see https://math.stackexchange.com/q/1932912.