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
Canonical name	AGroupEmbedsIntoItsProfiniteCompletionIfAndOnlyIfItIsResiduallyFinite
Date of creation	2013-03-22 15:15:56
Last modified on	2013-03-22 15:15:56
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	13
Author	yark (2760)
Entry type	Theorem
Classification	msc 20E18
Related topic	ProfiniteCompletion
Related topic	ProfiniteGroup
Related topic	ResiduallyCalP