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?
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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... 

