Relationships of the almost layer finite groups with similar group classes

This paper is devoted to the study of the relationships between classes of groups with finiteness conditions. Finiteness conditions superimpose limitations on a number of elements of a certain order in a group, on the order of its elements and on the cardinality of conjugate classes. The paper provides examples of groups that share class of almost layer-finite groups and those close to that class of groups: layer-finite groups, periodic groups, Chernikov’s groups, locally normal groups and the groups with the finite classes of conjugate elements. We establish the properties of the relationships of the considered classes of groups. In particular, we prove the coincidence of the classes of almost layer-finite groups and Chernikov’s groups in the class of primary groups. Our results will be used in the study of infinite groups with finiteness conditions.