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A profinite group is a compact totally disconnected topological group. Equivalently, it is the inverse limit of a system of finite groups equipped with the discrete topology.

For example, if we take the finite groups $\GL(2,\Z/m\Z)$ as $m$ varies over positive integers, order them by divisibility of $m$ and consider the inverse system equipped with reduction maps $\GL(2,\Z/n\Z)\to \GL(2,\Z/m\Z)$ for all positive integers $m|n$, then the inverse limit $$\lim_{\overset{\longleftarrow}{m}} \GL(2,\Z/m\Z) \simeq \GL(2,\widehat \Z)$$ is a profinite group which is isomorphic to the group of invertible $2\times 2$ matrices over the topological ring $\widehat \Z$, which is the inverse limit of the finite rings $\Z/m\Z$ equipped with the discrete topology.

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• Review status: beta
• Last edited by Andrew Sutherland on 2021-07-17 14:54:23
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