Structure of a Group and the Structure of its Lattice of Subgroups

The central theme of this monograph is the relation between the structure of a group and the structure of its lattice of subgroups. Since the first papers on this topic have appeared, notably those of BAER and ORE, a large body of literature has grown up around this theory, and it is our aim to give a picture of the present state of this theory. To obtain a systematic treatment of the subject quite a few unpublished results of the author had to be included. On the other hand, it is natural that we could not reproduce every detail and had to treat some parts some wh at sketchily. We have tried to make this report as self-contained as possible. Accordingly we have given some proofs in considerable detail, though of course it is in the nature of such areport that many proofs have to be omitted or can only be given in outline. Similarly references to the concepts and theorems used are almost exclusively references to standard works like BIRKHOFF [lJ and ZASSENHAUS [lJ. The author would like to express his sincere gratitude to Professors REINHOLD BAER and DONALD G. HIGMAN for their kindness in giving hirn many valuable suggestions. His thanks are also due to Dr. NOBORU ITo who, during stimulating conversations, contributed many useful ideas. Urbana, May, 1956. M. Suzuki. Contents.

Accordingly we have given some proofs in considerable detail, though of course it is in the nature of such areport that many proofs have to be omitted or can only be given in outline.

I. Groups with a special kind of subgroup lattice.- 1. The distributive law in subgroup lattices.- 2. Modular identity in subgroup lattices.- 3. The Jordan-Dedekind chain condition and lower semi-modularity.- 4. Finite groups with a modular lattice of subgroups.- 5. Structure of infinite M-groups.- 6. Structure of UM-groups.- 7. Complemented groups.- II. Isomorphisms of subgroup lattices.- 1. Projectivities.- 2. Projectivities of abelian groups.- 3. Projectivities of locally free groups.- 4. Projectivities of finite groups.- 5. Projectivities of modular groups.- 6. Index-preserving Proj ectivities.- 7. The images of normal subgroups under projectivities of finite groups.- 8. The number of finite groups with given lattice of subgroups.- 9. The group of auto-proj ectivities.- 10. Projectivities of simple groups.- 11. Characteristic chains of subgroup lattices.- 12. Representation of lattices as subgroup lattices.- 13. The situation-preserving mappings.- III. Homomorphisms of subgroup lattices.- 1. The kernels of a homomorphism of a subgroup lattice.- 2. Complete L-homomorphisms onto cyclic groups.- 3. General properties of complete L-homomorphisms.- 4. L-homomorphisms induced by group-homomorphisms.- 5. Incomplete L-homomorphisms.- 6. L-homomorphisms of finite groups.- 7. The meet-homomorphisms.- 8. Structure of finite groups which admit a proper L-homomorphism.- 9. L-homomorphisms onto a nilpotent group.- IV. Dualisms of subgroup lattices.- 1. Dualisms (of abelian groups).- 2. Nilpotent groups with duals.- 3. Finite solvable groups with duals.