Profinite completion

Profinite completion

Given an arbitrary group ${\displaystyle G}$, there is a related profinite group ${\displaystyle {\widehat {G}},}$ the profinite completion of ${\displaystyle G}$. It is defined as the inverse limit of the groups ${\displaystyle G/N}$, where ${\displaystyle N}$ runs through the normal subgroups in ${\displaystyle G}$ of finite index (these normal subgroups are partially ordered by inclusion, which translates into an inverse system of natural homomorphisms between the quotients).