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arXiv · 1102.3037

Generators and commutators in finite groups; abstract quotients of compact groups

Nikolay Nikolov·Dan Segal
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Abstract

Let N be a normal subgroup of a finite group G. We prove that under certain (unavoidable) conditions the subgroup [N,G] is a product of commutators [N,y] (with prescribed values of y from a given set Y) of length bounded by a function of d(G) and |Y| only. This has several applications: 1. A new proof that G^n is closed (and hence open) in any finitely generated profinite group G. 2. A finitely generated abstract quotient of a compact Hausdorff group must be finite. 3. Let G be a topologically finitely generated compact Hausdorff group. Then G has a countably infinite abstract quotient if and only if G has an infinite virtually abelian continuous quotient.

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Nikolay Nikolov, Dan Segal. 2011-02-15. Generators and commutators in finite groups; abstract quotients of compact groups. https://doi.org/10.1007/s00222-012-0383-6

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