a group embeds into its profinite completion if and only if it is residually finite

Let $G$ be a group.

First suppose that $G$ is residually finite, that is,

$$\mathrm{R}(G):=\bigcap _{N{⊴}_{\mathrm{f}}G}N=1$$

(where $N{⊴}_{\mathrm{f}}G$ denotes that $N$ is a normal subgroup of finite index in $G$).Consider the natural mapping of $G$ into its profinite completion $\widehat{G}$given by $g\mapsto {(Ng)}_{N{⊴}_{\mathrm{f}}G}$.It is clear that the kernel of this map is precisely $\mathrm{R}(G)$,so that it is a monomorphism when $G$ is residually finite.

Now suppose that $G$ embeds into its profinite completion $\widehat{G}$and identify $G$ with a subgroup of $\widehat{G}$. Now, a theorem onprofinite groups tells us that

$$\bigcap _{N{⊴}_{\mathrm{o}}\widehat{G}}N=1,$$

(where $N{⊴}_{\mathrm{o}}G$ denotes that $N$ is an open (http://planetmath.org/TopologicalSpace) normal subgroup of $G$) and since open subgroups of a profinite group have finite index, wehave that

$$\mathrm{R}(\widehat{G})=1,$$

so $\widehat{G}$ is residually finite. Then $G$ is a subgroup of aresidually finite group, so is itself residually finite, as required.