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

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 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.