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

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.


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
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