tailieunhanh - Đề tài " On finitely generated profinite groups, I: strong completeness and uniform bounds "

We prove that in every finitely generated profinite group, every subgroup of finite index is open; this implies that the topology on such groups is determined by the algebraic structure. This is deduced from the main result about finite groups: let w be a ‘locally finite’ group word and d ∈ N. Then there exists f = f (w, d) such that in every d-generator finite group G, every element of the verbal subgroup w(G) is equal to a product of f w-values. | Annals of Mathematics On finitely generated profinite groups I strong completeness and uniform bounds By Nikolay Nikolov and Dan Segal Annals of Mathematics 165 2007 171 238 On finitely generated profinite groups I strong completeness and uniform bounds By NiKOLAy Nikolov and Dan Segal Abstract We prove that in every finitely generated profinite group every subgroup of finite index is open this implies that the topology on such groups is determined by the algebraic structure. This is deduced from the main result about finite groups let w be a locally finite group word and d E N. Then there exists f f w d such that in every d-generator finite group G every element of the verbal subgroup w G is equal to a product of f w-values. An analogous theorem is proved for commutators this implies that in every finitely generated profinite group each term of the lower central series is closed. The proofs rely on some properties of the finite simple groups to be established in Part II. Contents 1. Introduction 2. The Key Theorem 3. Variations on a theme 4. Proof of the Key Theorem 5. The first inequality lifting generators 6. Exterior squares and quadratic maps 7. The second inequality soluble case 8. Word combinatorics 9. Equations in semisimple groups 1 the second inequality 10. Equations in semisimple groups 2 powers 11. Equations in semisimple groups 3 twisted commutators Work done while the first author held a Golda-Meir Fellowship at the Hebrew University of Jerusalem. 172 NIKOLAY NIKOLOV AND DAN SEGAL 1. Introduction A profinite group G is the inverse limit of some inverse system of finite groups. Thus it is a compact totally disconnected topological group properties of the original system of finite groups are reflected in properties of the topological group G. An algebraist may ask does this remain true if one forgets the topology Now a base for the neighbourhoods of 1 in G is given by the family of all open subgroups of G and each such subgroup has finite index so if .

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