Finite index subgroups in profinite groups. (English. Abridged French version) Zbl 1033.20029

Nikolov, Nikolay; Segal, Dan

C. R., Math., Acad. Sci. Paris 337, No. 5, 303-308 (2003).

The authors announce the following important result: in a finitely-generated profinite group all subgroups of finite index are open. This extends Serre’s theorem for finitely-generated pro-\(p\) groups from the 1970s and is an extension which has long been sought for. It follows that for a finitely-generated profinite group the topology is uniquely determined by its group structure.
The authors also prove that the terms of the (algebraic) lower central series of a finitely generated profinite group are closed.
The proofs depend on properties of certain verbal subgroups and are outlined in this announcement.

Reviewer: R. D. Camina (Cambridge)

MSC:

20E18	Limits, profinite groups
20E07	Subgroup theorems; subgroup growth
22A05	Structure of general topological groups
20F05	Generators, relations, and presentations of groups
20F14	Derived series, central series, and generalizations for groups

Keywords:

finitely generated profinite groups; strongly complete groups; subgroups of finite index; open subgroups; lower central series

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