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An open subgroup $H$ of a profinite group $G$ is a subgroup that is open in the topology of $G$, which implies that it is equal to the inverse image of its projection to a a finite quotient of $G$.

Open subgroups of $G$ necessarily have finite index (since $G$ is compact), but not every finite index subgroup of $G$ is necessarily open.

When the profinite group $G$ is a matrix group over a ring $R$ that is equipped with canonical projections to finite rings of the form $\Z/n\Z$ (take $R=\Z_\ell$ or $R=\widehat \Z$, for example), we use $G(n)$ to denote the image of $G$ under the group homomorphism induced by the projection $R\to\Z/n\Z$. In this situation we may identify $H$ with its projection to $G(N)$, where $N$ is the least positive integer for which $H$ is the inverse image of its projection to $G(N)$ (this $N$ is the level of $H$).

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  • Last edited by Andrew Sutherland on 2021-09-18 14:44:54
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