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

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

2011-02-15

arxiv.org/abs/1102.3037v6

Mathematics

Group Theory

This paper supersedes the preprint arXiv:0901.0244v2 by the first author and answers the questions raised there. Latest versio

Scientific paper

10.1007/s00222-012-0383-6

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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