Lukashova T. Groups with the restrictions on generalized norms of given systems of subgroups.

The thesis is devoted to research of properties of various classes of groups depending on properties of their generalized norms for the given systems  of subgroups. In other words, we study the groups in which restrictions are imposed on the intersection of normalizers of all subgroups of some system of subgroups . This intersection is called the -norm of group if the system  consists of all subgroups of the group with a certain theoretical-group property. In many cases, the influence of the -norms satisfied restrictions on the properties of the group is so strong that it allows us to describe the structure of the group constructively. Therefore, since the 30th of the XX century, interest in studying the -norms of the group for different systems of subgroups  and the impact of these norms on the properties of the group is in focus. The thesis studies the relations between the properties of groups and their -norms, provided that the system  consists of all Abelian non-cyclic, all decomposable, and all cyclic subgroups of a non-prime order. As a restriction, which satisfy the specified -norms, their non-Dedekindness is chosen. In the first part of the work the properties of the norm of Abelian non-cyclic subgroups are investigated and the influence of this norm on the properties of the group under the condition of non-Dedekindness of in the class of locally finite groups is studied. The structural description of infinite locally finite p-groups (p is prime), finite 2-groups and non-prime locally finite groups in which the norm is non-Dedekind, is obtained. The class of periodic locally nilpotent groups with non-Dedekind norm of Abelian non-cyclic subgroups is characterized. It is proved that all such groups are direct products of norm of Abelian non-cyclic subgroups and a finite p'-subgroup, all Abelian subgroups of which are cyclic. The properties of infinite locally finite non-locally nilpotent groups in which the norm of Abelian non-cyclic subgroups is non-Dedekind locally nilpotent group are investigated and a structural description of such groups is obtained. The second part of the work is devoted to the study of the properties and influence of the norm of decomposable subgroups of a group, on various characteristics of the group. The norm of decomposable subgroups of group G is the intersection of normalizers of all decomposable subgroups this group (if the system of such subgroups is empty, we have G= ). The non-Dedekindness of this norm is also chosen as a defining restriction. The descriptions of locally finite p-groups, periodic locally nilpotent and infinite locally finite non-locally nilpotent groups with the non-Dedekind locally nilpotent norm are obtained. It is proved that the class of non-periodic locally nilpotent groups in which the norm is a non-Dedekind subgroup coincides with the class of non-periodic locally nilpotent groups whose all decomposable subgroups are normal and G= . The properties of non-periodic locally solvable and non-locally nilpotent groups in which the norm of decomposable subgroups is a locally nilpotent non-Dedekind group are investigated. It is proved that under this condition the norm cannot be a periodic group. The structural description of non-periodic locally solvable groups with the above restrictions is found. The norm turned out to be quite closely related to the norm of Abelian non-cyclic subgroups of the group. It is proved that in the class of locally finite groups these norms either coincide or one of them is a subgroup of the other one. Similar relations between the specified norms take place in a class of non-periodic locally solvable groups, provided that at least one of these norms is non-Dedekind. The properties of another generalized norm – the norm of cyclic subgroups of non-prime orders of a group G, which is the intersection of normalizers of all subgroups of a group which have a compound or infinite order, is investigated only for non-periodic groups. It is established that in torsion-free groups such a norm is Abelian and its relations with the center of a group are analyzed. It is proved that the norm is Abelian if it does not contain elements of non-prime orders of a group. The structural description of non-periodic groups with non-Abelian norm is found. It is proved that in such groups each cyclic subgroup of non-prime order is invariant.