Ultrasoluble coverings of some nilpotent groups by a cyclic group over number fields and related questions

摘要

Let F be a finite nilpotent group of odd order. For every finite cyclic subgroup A of odd order we find necessary and sufficient conditions for a class h is an element of H-2 (F, A) to determine an ultrasoluble extension (under the additional assumption of minimality of all p-Sylow subextensions to the extension with class h for all non-Abelian p-Sylow subgroups F-p of F), that is, for the existence of a Galois extension of number fields K/k with group F such that the corresponding embedding problem is ultrasoluble (it has solutions and all such solutions are fields). We also establish a number of related results.