Gruppenklassen, deren kritische Gruppen minimal nicht nilpotent sind

Abstract

This article is concerned with classes of groups, whose critical groups are minimal non-nilpotent, i.e. all proper subgroups are nilpotent, but the group itself is not. Since the structure of these groups is well known, the structure of the critical groups is determined as well. Thus, in a first part two theorems, which show that the critical groups for certain classes are mini=mal non-nilpotent, are proved. In the second part applications are given, concerning for example groups all of whose second maximal subgroups belong to a given class or describing the structure of groups if you have informations about certain subgroups. Also a number-theoretical characterization of all those ordinary numbers, to which belong only groups with a given ab=stract group-theoretical property, is given.