Some lattice-theoretical problems in the theory of groups

The study of group properties and group structure can proceed from various initial principles. Usually, properties of a group are obtained from the properties of its elements. However, in a number of investigations essentially the converse problems are solved: to study the influence of the properties of the set of all subgroups of a group on the properties of its elements. This explains to what extent purely set-theoretical operations given on the set of all subgroups of a group $G$ (this set forms a lattice $S(G)$) determine the properties of the group operation itself.
The investigation of connections between the structure of a group and the structure of its lattice of subgroups has become one of the main general approaches in the study of groups. There are many interesting results in this field, but also many important unsolved problems. These problems seem to be difficult, and apparently new ideas and mathematical techniques are needed for their solution.
The present survey (without claiming to exhaust the whole material) is devoted to the so-called lattice-theoretical questions of group theory and contains recent results. The main attention is given to infinite groups. The structures of various classes of finite groups, their lattice isomorphisms and homomorphisms, and many related questions are treated in Suzuki's comprehensive monograph [1]. General surveys of the results (up to 1961) for infinite soluble, nilpotent and radical groups are contained in the papers of Kontorovich, Pekelis, and Starostin [2] and Plotkin [3]. Finally, the basic notions (for group theory see [4], [5], for lattice theory [6]) and a brief survey of results in this field are contained in the new edition of Kurosh's monograph [5] Under the circumstances there is naturally some overlap between this work and the above articles. This can be justified by the intention to make the report as complete as possible.