Profinite group (reviewed)

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.