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,

 $\mathrm{R}(G):=\bigcap_{N\trianglelefteqslant_{\mathrm{f}}G}N=1$

(where $N\trianglelefteqslant_{\mathrm{f}}G$ denotes that $N$ is a normal subgroup of finite index in $G$). Consider the natural mapping of $G$ into its profinite completion $\hat{G}$ given by $g\mapsto(Ng)_{N\trianglelefteqslant_{\mathrm{f}}G}$. 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 $\hat{G}$ and identify $G$ with a subgroup of $\hat{G}$. Now, a theorem on profinite groups tells us that

 $\bigcap_{N\trianglelefteqslant_{\mathrm{o}}\hat{G}}N=1,$

(where $N\trianglelefteqslant_{\mathrm{o}}G$ 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, we have that

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

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

References

• 1 J. D. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal, Analytic pro-$p$ groups, 2nd ed., Cambridge studies in advanced mathematics, Cambridge University Press, 1999.
Title a group embeds into its profinite completion if and only if it is residually finite AGroupEmbedsIntoItsProfiniteCompletionIfAndOnlyIfItIsResiduallyFinite 2013-03-22 15:15:56 2013-03-22 15:15:56 yark (2760) yark (2760) 13 yark (2760) Theorem msc 20E18 ProfiniteCompletion ProfiniteGroup ResiduallyCalP