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On finitely generated profinite groups I: strong completeness and uniform bounds

群论 2007-05-23 v1

摘要

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 ww be a `locally finite' group word and dNd\in\mathbb{N}. Then there exists f=f(w,d)f=f(w,d) such that in every dd-generator finite group GG, every element of the verbal subgroup w(G)w(G) is equal to a product of ff ww-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.

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引用

@article{arxiv.math/0604399,
  title  = {On finitely generated profinite groups I: strong completeness and uniform bounds},
  author = {Nikolay Nikolov and Dan Segal},
  journal= {arXiv preprint arXiv:math/0604399},
  year   = {2007}
}

备注

66 pages