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 completion if and only if it is residually finite.

Title	profinite completion
Canonical name	ProfiniteCompletion
Date of creation	2013-03-22 15:15:54
Last modified on	2013-03-22 15:15:54
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	10
Author	yark (2760)
Entry type	Definition
Classification	msc 20E18
Related topic	AGroupsEmbedsIntoItsProfiniteCompletionIfAndOnlyIfItIsResiduallyFinite