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.
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 seriesReferences:
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