Subgroup lattices of groups.

*(English)*Zbl 0843.20003
De Gruyter Expositions in Mathematics. 14. Berlin: Walter de Gruyter. xv, 572 p. (1994).

The subject of this book was taken up before by M. Suzuki in 1956 in an “Ergebnisbericht” [Structure of a group and the structure of its lattice of subgroups (1956; Zbl 0070.25406)]. To given a rough indication on the contents of the book under review, we list first the chapter headings: 1. Fundamental concepts, 2. Modular lattices and abelian groups, 3. Complements and special elements in the subgroup lattice of a group, 4. Projectivities and arithmetic structure of finite groups, 5. Projectivities and normal structure of finite groups, 6. Projectivities and normal structure of infinite groups, 7. Classes of groups and their properties, 8. Dualities of subgroup lattices, 9. Further lattices. Chapter 9 deals with lattices of normal subgroups, subnormal subgroups, centralizers, and of cosets. The bibliography consists of 364 items.

Looking at the size of the book the reader will find it reasonable to restrict to subgroups lattices of discrete groups, and a quick glimpse at the bibliography shows how much this area has developed since 1956 and how much of this has been included in this book. The concept of the book is that of a basic reference, a text for graduate studies and a source for research ideas. Information is added in the exercises at the end of each chapter section. In the opinion of the reviewer the book is very well written – to wait for a new book in this area almost 40 years has proved to be worthwhile.

Looking at the size of the book the reader will find it reasonable to restrict to subgroups lattices of discrete groups, and a quick glimpse at the bibliography shows how much this area has developed since 1956 and how much of this has been included in this book. The concept of the book is that of a basic reference, a text for graduate studies and a source for research ideas. Information is added in the exercises at the end of each chapter section. In the opinion of the reviewer the book is very well written – to wait for a new book in this area almost 40 years has proved to be worthwhile.

Reviewer: H.Heineken (Würzburg)

##### MSC:

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |

20E15 | Chains and lattices of subgroups, subnormal subgroups |

20D30 | Series and lattices of subgroups |

06B15 | Representation theory of lattices |