Profinite topology

In mathematics, profinite groups are topological groups that are in a certain sense assembled from finite groups. They share many properties with their finite quotients: for example, both Lagrange's theorem and the Sylow theorems generalise well to profinite groups.

A non-compact generalization of a profinite group is a locally profinite group.

Profinite groups can be defined in one of two equivalent ways.

A profinite group is a topological group that is isomorphic to the inverse limit of an inverse system of discrete finite groups. In this context, an inverse system consists of a directed set ${\displaystyle (I,\leq )}$, a collection of finite groups ${\displaystyle {\mathcal {G}}=\{G_{i}:i\in I\}}$, each having the discrete topology, and a collection of homomorphisms ${\displaystyle \{f_{i}^{j}:G_{j}\to G_{i}\mid i,j\in I,i\leq j\}}$ such that ${\displaystyle f_{i}^{i}}$ is the identity on ${\displaystyle G_{i}}$ and the collection satisfies the composition property ${\displaystyle f_{i}^{j}\circ f_{j}^{k}=f_{i}^{k}}$. The inverse limit is the set:

...
Wikipedia