a group embeds into its profinite completion if and only if it is residually finite
Let be a group.
First suppose that is residually finite, that is,
(where denotes that is a normal subgroup of finite index in ). Consider the natural mapping of into its profinite completion given by . It is clear that the kernel of this map is precisely , so that it is a monomorphism when is residually finite.
(where denotes that is an open (http://planetmath.org/TopologicalSpace) normal subgroup of ) and since open subgroups of a profinite group have finite index, we have that
so is residually finite. Then is a subgroup of a residually finite group, so is itself residually finite, as required.
- 1 J. D. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal, Analytic pro- 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|
|Date of creation||2013-03-22 15:15:56|
|Last modified on||2013-03-22 15:15:56|
|Last modified by||yark (2760)|