I have seen the following statment somewhere, for example in Appendix B2 on Silverman's book "The Arithmetic of Elliptic Curves" : Let $M$ be an abelian group with discrete topology and $G$ be a profinite group. Then an linear action ( which means that $\sigma(m_1+m_2)=\sigma(m_1)+\sigma(m_2)$, i.e it is a $G$-module) $ \phi : G \times M \rightarrow M$ is continuous if and only if the stabilizer $ \sigma \in G | \sigma(m)=m $ has finite index in $G$ for all $m \in M$. But what we need is that this stabilizer is open in $G$. I also saw that in a profinite group, not every subgroup of finite index is open. So is this statement correct? Or how to see that this stabilizer is open if it has finite rank?
Not every finite index subgroup is open, but closed subgroups of finite index are open.
So if the stabilizer is closed, that would be sufficient...