The effect of a CZ-group structure on certain generalized solvable and generalized nilpotent classes of groups

Abstract

This paper attempts to take known theorems about linear groups which lie in certain well known generalized solvable and generalized nilpotent classes of groups, and to generalize these theorems to a class of groups called CZ-groups. These groups are defined in terms of the possession of a topology having certain properties known to be possessed in the case of linear groups by the Zariski topology. This class is a proper generalization of the linear groups however in that all abelian groups are also CZ-groups. The first question considered is whether a locally solvable CZ-group need be solvable, or perhaps hyperabelian. It is known that locally nilpotent CZ-groups are solvable, and this fact is used to obtain several results regarding locally solvable CZ-groups which also lie in one of several well known generalized nilpotence classes of Baer and others, the most general of which are the residually central and the Baer-Nilpotent groups. Locally solvable CZ-groups satisfying finiteness conditions on chains of certain kinds of subgroups, locally solvable CZ-groups in which normality is a transitive relation, and locally solvable CZ-groups of finite rank are other areas in which affirmative results are obtained. Also considered is the question whether locally nil potent CZ-groups need be hypercentral. Finiteness conditions are dealt with exclusively in this area, with particular emphasis placed on locally nilpotent CZ-groups whose chains of certain abelian or normal subgroups are restricted to being finite. In addition to obtaining the desired results concerning the structure of locally nilpotent CZ-groups satisfying these conditions, the equivalence of several of these conditions for the class of locally nilpotent CZ-groups is also determined. Finally, the question is considered whether radical CZ-groups need be solvable, or perhaps just hyperabelian. Both of the major methods of attack of the previous two questions are utilized, with results obtained by the addition of certain additional conditions which force the local nilpotence of the group or one of its derived groups (as in the case of radical CZ-groups of finite rank), and also by the imposition of various finiteness conditions using arguments on the upper Hirsch-Plotkin series which utilize some of the results obtained earlier for locally nilpotent CZ-groups.