##
**On finitely generated profinite groups. I: Strong completeness and uniform bounds. II: Products in quasisimple groups.**
*(English)*
Zbl 1126.20018

The result proved in these two papers is the most impressive result on profinite groups within several decades. It states that every subgroup of finite index of a finitely generated profinite group is open. This means that the profinite completion of a finitely generated profinite group coincides with the group itself, i.e., the profinite completion of a finitely generated group can not be completed any further.

The result also implies that every homomorphism of a finitely generated profinite group to any profinite group is continuous. This can be used to deduce that the continuous (Galois) first cohomology coincides with usual first cohomology of \(G\) when a module of coefficients is finite.

The result follows from a ‘uniformity theorem’ about finite groups proved by the authors: given a group word \(w\) that defines a locally finite variety and a natural number \(d\), there exists \(f=f_w(d)\) such that in every finite \(d\)-generator group \(G\), each element of the verbal subgroup \(w(G)\) is a product of \(f\) \(w\)-values. Similar methods show that in a finite \(d\)-generator group, each element of the derived group is a product of \(g(d)\) commutators; this implies that the (abstract) commutator subgroup in any finitely generated profinite group is closed.

The result also implies that every homomorphism of a finitely generated profinite group to any profinite group is continuous. This can be used to deduce that the continuous (Galois) first cohomology coincides with usual first cohomology of \(G\) when a module of coefficients is finite.

The result follows from a ‘uniformity theorem’ about finite groups proved by the authors: given a group word \(w\) that defines a locally finite variety and a natural number \(d\), there exists \(f=f_w(d)\) such that in every finite \(d\)-generator group \(G\), each element of the verbal subgroup \(w(G)\) is a product of \(f\) \(w\)-values. Similar methods show that in a finite \(d\)-generator group, each element of the derived group is a product of \(g(d)\) commutators; this implies that the (abstract) commutator subgroup in any finitely generated profinite group is closed.

Reviewer: Pavel Zalesskij (Brasília)

### MSC:

20E18 | Limits, profinite groups |

20F05 | Generators, relations, and presentations of groups |

20E07 | Subgroup theorems; subgroup growth |

20F12 | Commutator calculus |

20D05 | Finite simple groups and their classification |