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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.

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
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