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.     

Surface characters of internal waves generated by Rankine ovoid

A linear theory on the internal waves generated in the stratified fluid with a pycnocline is presented in this paper. The internal wave fields such as the velocity fields in the stratified fluid and velocity gradient fields at the free surface are also investigated by means of the theoretical and numerical method. From the numerical results, it is shown that the internal wave generated by horizontally moving Rankine ovoid is a sort of trapped wave which propagates in a wave guide, and its waveform is a kind of Mach front-type internal wave in the pycnocline. Influence of the internal wave on the flow fields at the free surface is represented by the velocity gradient fields resulted from the internal waves generated by motion of the Rankine ovoid. At the same time, it is also shown that under the hypothesis of inviscid fluid, the synchronism between the surface velocity gradient fields at the free surface and the internal wave fields in the fluid is retained. This theory opens a possibility to study further the modulated spectrum of the Bragg waves at the free surface.The project supported by the National Natural Science Foundation of China (40576010). The English text was polished by Keren Wang.