Shunkov groups

The paper is devoted to the study of a class of conjugately biprimitively finite groups named as groups of Shunkov. Finiteness condition in such groups is superimposed on the subgroup generated by two conjugate elements of the group and group sections on finite subgroups. The paper presents results concerning groups of Shunkov. The relations between the class of groups of Shunkov with classes of groups of Chernikov, groups of Aleshin, almost layer-finite groups and periodic groups are shown. We have proven two results establishing the properties of groups of Shunkov. V. P. Shunkov in his first theorem dedicated to the class of groups of Shunkov established their connection with Chernikov groups in the class of primary groups. Further the groups of Shunkov together with the minimal condition for Abelian subgroups, with a primary minimality condition and with different conditions for systems of subgroups are studied. V. P. Shunkov establishes the existence of infinite Abelian subgroups in an arbitrary infinite Shunkov group. A. I. Sozutov described the structure of complement of group of Shunkov which is a Frobenius group or constituting a Frobenius pair with a proper subgroup. The structure of periodic groups of Shunkov with Chernikov Sylow 2-subgroups was studied. Several authors have established relationships of Shunkov groups with similar classes of groups. The existence of Shunkov groups without periodic part was proved. A. V. Rozhkov using techniques for working with automorphisms of trees divided an infinite set of classes of subgroups, generalizing the concept of Shunkov group by transferring finiteness conditions from the subgroup generated by two elements conjugate to the subgroup generated by any of its n conjugate elements. The results on Shunkov groups with the condition of saturation have been intensively studied in recent years were not included in this work because they can be found in the review of A. A. Kuznetsov and K. A. Filippov in the Siberian Electronic Mathematical News. Our results will be used in the study of infinite groups with finiteness conditions.