profinite group

A profinite group is a topological group that is isomorphic to the inverse limit of discrete finite groups.[

Examples:

The p-adic integers
The profinite integers

Profinite completions

Given \(G\in {\mathsf{Grp}}\) a related profinite group \(\widehat{G} \in {\mathsf{Grp}}\), the profinite completion of G. It is defined as \begin{align*} \widehat{G} := \varprojlim_{\substack{ N{~\trianglelefteq~}G \\ [G:N] < \infty }} G/N ,\end{align*} where \(N\) runs through the normal subgroups in G of finite index.

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Warning: on computing limits

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Examples

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