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 $\GL_2(\Z/n\Z)$ as $n$ varies over positive integers, order them by divisibility of $n$ 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 $$ \lim_{\overset{\longleftarrow}{n}} \GL_2(\Z/n\Z) \simeq \GL_2(\widehat \Z) $$ 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 $\Z/n\Z$ equipped with the discrete topology.

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- Review status: reviewed
- Last edited by John Cremona on 2022-12-12 07:19:20

**History:**(expand/hide all)

- 2022-12-12 07:19:20 by John Cremona (Reviewed)
- 2021-09-18 13:56:00 by Andrew Sutherland
- 2021-07-17 14:54:23 by Andrew Sutherland

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