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,
$$\mathrm{R}(G):=\bigcap _{N{\u22b4}_{\mathrm{f}}G}N=1$$ |
(where $N{\u22b4}_{\mathrm{f}}G$ 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 $g\mapsto {(Ng)}_{N{\u22b4}_{\mathrm{f}}G}$. 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 on profinite groups tells us that
$$\bigcap _{N{\u22b4}_{\mathrm{o}}\widehat{G}}N=1,$$ |
(where $N{\u22b4}_{\mathrm{o}}G$ 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
$$\mathrm{R}(\widehat{G})=1,$$ |
so $\widehat{G}$ 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 |
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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 |