profinite completion

The profinite completion of a group $G$is defined to be the profinite group

$$\widehat{G}={\underleftarrow{\lim }}_{N{⊴}_{\mathrm{f}}G}G/N,$$

where $N{⊴}_{\mathrm{f}}G$ means that $N$ is a normal subgroup of finite index in $G$.

A group embeds into its profinite completionif and only if it is residually finite.