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

The first part of the paper establishes results about products of commutators in a d-generator finite group G, for example: if H?G=??g ₁,??,g _r ?? then every element of the subgroup H,G] is a product of f(r) factors of the form $h_{1},g_{1}]h_{1}^{\prime},g_{1}^{-1}]\ldots\lbrack h_{r},g_{r}]h_{r}^{\prime },g_{r}^{-1}]$ with $h_{1},h_{1}^{\prime},\ldots,\allowbreak h_{r},h_{r}^{\prime }\in H$ . Under certain conditions on H, a similar conclusion holds with the significantly weaker hypothesis that G=H??g ₁,??,g _r ??, where f(r) is replaced by f ₁(d,r). The results are applied in the second part of the paper to the study of normal subgroups in finitely generated profinite groups, and in more general compact groups. Results include the characterization of (topologically) finitely generated compact groups which have a countably infinite image, and of those which have a virtually dense normal subgroup of infinite index. As a corollary it is deduced that a compact group cannot have a finitely generated infinite abstract quotient.