1.1. From the Preface.

Near-rings can be said to have reached a milestone when the first, and so far the only, book on the subject was reprinted. That book, Near-rings by G Pilz, which first appeared in 1977, and whose second edition appeared in 1983, is a very thorough and almost complete survey on the subject. This book is intended as a much easier paced introduction to the subject suitable for first-year postgraduates or even final year undergraduates. It also contains some material in more depth on the group theoretic aspects. ...

This book is organised in two parts. The first consisting of chapters 1 to 8 is a general introduction to the subject, suitable for an introductory postgraduate course. It does not attempt to be complete or as general as possible, but it aims at the main structural results now available. The second part consists of chapters 9 to 14 and is more strongly linked to group theory. It presupposes a more thorough background in group theory than the earlier part and it develops more deeply the strong connection between groups and distributively generated (d.g.) near-rings. This area is my own main interest and much of the material in it is not available in the books by G Pilz. It should be easily accessible to any one with a good group-theoretical back ground who has read the first part. Chapter 11 in particular is best understood by someone with detailed knowledge of the relevant groups, but it can be used by any worker in near-ring theory.

1.2. Review by: Homer F Bechtell.
zbMATH 0658.16029.

This treatise is extremely well-written, very readable, and highly recommended for that first taste of this subject. The author favours the left distributive law and at times prefers his own terminology rather than that normally used, for example an $R$-submodule here is an $R$-subgroup in other treatments. This is clearly pointed out with a Warning. In the closing chapters, the reader is brought to the fore of current investigations and open questions within the author's area of interest. There is ready use of references and an ample supply of examples. Several results appear in English for the first time.

The book is refreshing and highly recommended for that first taste of this subject. The author is to be commended on a contribution that will further investigative interest in this area.

1.3. Review by: Günter F Pilz.
Bulletin of the American Mathematical Society 17 (1) (1987), 157-160.

Meldrum's book is an excellent piece of work. Part 1 gives an easy introduction into the subject, most useful both for beginners as well as other mathematicians who just want to see the main ideas. Part 2 goes considerably deeper and reflects the author's mastery of group theory; it also contains research results of the author which appear in a book for the first time.

The book is written very carefully and thoughtfully. This applies to the mathematical precision as well as to the style and also to the typing; I know that the author put the manuscript on a word processor himself; this guaranteed that everything is so arranged that it provides maximal clarity. The reader might miss more detailed descriptions of applications of near-rings, like the ones to block designs, but this can be done in further reading. Altogether, this book should not stand on a shelf: it should be read and used.

2.1. From the Publisher.

Wreath products have arisen in many situations in both group and semigroup theory, often providing examples of unexpected behaviour, but also in quite fundamental settings. They occur in many applications in science, particularly in physics and chemistry.

2.2. Preface.

Most algebraists meet wreath products purely as a tool. They provide a means of constructing a group or semigroup with certain properties, generally in a slightly unexpected setting. They also appear in certain natural settings, such as the Sylow subgroups of appropriate symmetric groups. Also most group theorists will have heard of the Krasner-Kaloujnine embedding theorem and most semigroup theorists will have heard of the Krohn-Rhodes theorem. So the topic is well-known, but mostly as a somewhat unusual construction method.

On the other hand, the very fact that wreath products appear in such fundamental theorems, and also often in natural settings, would indicate that the concept is an important one. However, apart from one book published in what was then Eastern Germany (L A Kaluzhnin, P M Beleckij and V Z Feinberg), and which was rather restricted in scope, no sustained presentation of wreath products has been made. In many text books, there is a section or chapter on wreath products. There have been some survey papers, notably by C Wells. But in almost all cases where wreath products are used, in book or paper, the author has to develop the theory himself, using material from a variety of sources. With this book we seek to make a start at redressing this situation.

When I first started work on this book, over ten years ago, I expected that most of the major results on wreath products of groups and semigroups could be presented in one reasonably sized volume. A literature search using one of the computerised databases soon put paid to that idea. The number of papers which were relevant to wreath products, either in proving results about them or in applying them in a remarkably wide variety of situations, was very large. It became obvious that a good proportion would need to be left out, and that the whole process would be much harder and take much longer than expected. A great deal of selection would have to be done.

It is obviously important to include the two famous theorems mentioned above. Also a good look at the basic structure of wreath products of groups and semigroups is necessary. Although a majority of papers using wreath products deal with the standard wreath product, both for groups and semigroups, the natural setting for wreath products is undoubtedly in permutation groups, or transformation semigroups respectively, and I have tried to put as many results as possible into this setting. Beyond that I have tried to include a range of topics of interest both in their own right and because of the techniques needed in their proofs. But of necessity the selection was governed to some extent by personal taste and personal knowledge. To try to compensate for this I have included in the bibliography a large number of papers which are not referred to directly in the text. Even so these do not include all the papers mentioning wreath products of which I am aware. Papers which I have knowingly omitted are either about very specific applications, or about very remote topics, or making very small contributions to wreath products. Many papers have been omitted due to my being unaware of them. Also there are a number of papers on wreath products of other structures of which only a few have been included. So although the bibliography is large, it is not meant to be exhaustive.

When this book was still only a project, I discussed it with Professor B H Neumann, and he was most encouraging, for which I am very grateful. I would also like to thank Dr R B J T Allenby and Dr P M Neumann, who helped me in the early stages, and Dr J B Fountain who helped me with the semigroup part of the work. The editorial staff at Pitman's as it was in 1984, and at Longman's subsequently have been very patient and helpful. There have been so many delays and readjustments that I am surprised that they did not call the whole thing off a long time ago. I am most grateful to them for all they have done over the years, especially to the staff who have finally seen this project to fruition.

Above all though my wife Patricia deserves a great deal of the credit. Her support and encouragement over the whole period made it possible to write this book in spite of an ever increasing load of teaching and administration at my university. None of this would have been possible without her, and she bas my heartfelt thanks.

2.3. Contents.

Preface

Notation

Part I: Wreath product of groups

1. Construction and basic properties;

2. Centralizers;

3. Conjugacy and direct decomposition;

4. Nilpotent wreath products;

5. Automorphisms of wreath products;

6. Classes of groups;

7. Generalized wreath products;

8. Applications of generalized wreath products;

9. Some generalizations and applications.

Part II: Wreath products of semigroups

10. The wreath products of semigroups;

11. Regular semigroups;

12. The Krohn-Rhodes theorem;

13. Some applications;

14. Generalizations.

2.4. Review by: P Lakatos.
Mathematical Reviews MR1379113 (97j:20030).

This book is a particularly well-prepared account of a very useful construction in group theory; in other words, this is a book specifically devoted to wreath products of groups and semigroups. The wreath product is widely used for the construction of counterexamples and for the proofs of existence theorems and occurs in other non-mathematical areas. In spite of this there has been no dedicated survey of the ideas and methods involved until this book.

This book is addressed to all who may find wreath products of use and who have a working knowledge of group theory and semigroup theory. An important point to notice is the existence of an extended bibliography. Up to now the material of the book has appeared mostly in journal articles, and its appearance here is most welcome - only the book by L A Kaluzhnin, P M Beletskii and V Z Feinberg was devoted to this construction restricted to the wreath product of permutation groups and transformation semigroups.

The material has been chosen to provide both an account of important work and a taste of the various techniques that arise in the theory. Several generalisations, improved proofs and extensions are also presented. The book contains many applications of a theoretical nature, and is written clearly and accurately.

2.5. Review by: U Knauer.
zbMATH 0833.20001.

The book contains 9 chapters in Part I (Wreath products of groups, about 180 pages) and 5 chapters in Part II (Wreath products of semigroups, about 120 pages). Although wreath product constructions have been used in group theory for many years as well as in semigroup theory, so far there has been no monograph devoted to this subject. The basic approach in both parts are permutation groups and transformation semigroups as the constituents of the considered wreath products and, according to the preface, the wreath product Embedding Theorem of Krasner-Kaloujnine and the Theorem of Krohn-Rhodes are the musts of the book.

Some familiarity with group theory will be important since quite a few concepts are used without explication or definition, especially for some parts of notation this may be inconvenient for some readers. References, far from being complete, are sometimes just mentioned but do not appear anywhere in the text, which the author himself points out.

Chapter 1: Basic definitions: Chapter 1 contains most of the basic definitions for the group theoretic part of the book. It starts with the definitions of the complete and the restricted permutational wreath product and the standard wreath product. The mentioned result by Krasner and Kaloujnine from 1950/51 follows already in section 1.3. (a group can be embedded into a manifold wreath product constituted by factors of a normal series).

Chapter 2: Centralizers: This chapter contains results by L G Kovacs from 1985 to 1988. Elements with trivial centralizer and self- centralizing elements in wreath products are considered.

Chapter 3: Conjugacy and direct decomposition: Here a generalization of results obtained by P M Neumann in 1964 is presented. In 3.2 and 3.3 necessary and under same conditions also sufficient conditions are given for a wreath product to be a direct product.

Chapter 4: Nilpotent wreath products: Following Baumslag (1959) results of several authors are presented and characterisations of nilpotent restricted wreath products are given in 4.1. The nilpotency classes of two standard wreath products are described in section 4.2. In 4.3 the Sylow $p$-subgroup of the symmetric group on $p^{r}$ elements is presented as a certain wreath product.

Chapter 5: Automorphisms of wreath products: The chapter starts with the paper by P M Neumann from 1964 concerning the basic group in a standard wreath product. Now this is used to describe isomorphisms between standard wreath products and automorphism groups of standard wreath products. The latter question is briefly mentioned also for the permutational wreath product, following papers of Hassanabadin (1978), Lentoudis and Tits (1985 and 1987).

Chapter 6: Classes of groups: Restricted wreath products which are supersoluble groups (i.e. groups with normal series whose factors are abelian) are characterized with some finiteness condition, several sufficient and several necessary conditions are given for other cases. Similar questions for locally nilpotent wreath products and wreath products which are so-called Engel groups are considered. The material comes from Durblin (1966), Sonneborn (1966), A I Scot (1978) and an (unpublished) thesis by May (1967). Finally necessary and sufficient conditions for a wreath product belonging to a finite soluble group (following Gaschütz (1962)) are presented, thereby using results by D B Parker (1970).

Chapter 7: Generalized wreath products: From Chapter 7 to 9 several generalizations of wreath products are discussed. First a construction by Behrendt (1990) is presented for a complete and a restricted generalised wreath product using more than two groups, some associativity results are proved in both cases, and several other constructions are detected as special cases.

Chapter 8: Applications of generalised wreath products: Two embedding theorems into generalised complete and restricted wreath products are considered and normal subgroups of generalised restricted wreath products are discussed.

Chapter 9: Generalization and application: Here so-called verbal and twisted wreath products and the crown product are presented (following B H.Neumann (1956 and 1963), Moran (1956 and 1958), and Shmelkin (1964)). The chapter and this part of the book finishes with a link between wreath products and group rings.

Chapter 10: Wreath product of semigroup: This chapter is the first in the semigroup part of the book. Thus the transformation wreath product, the standard wreath product, and the with respect to an idempotent of the passive semigroup restricted transformation wreath products are defined. Discussions of associativity and Greens relations follow together with some more detailed analysis of the restricted wreath product.

Chapter 11: Regular semigroups: After discussing the difficulties with regularity of the wreath product, sufficient conditions for the wreath product to be regular and inverse are given. Following C H Houghton (1976) an embedding result of an inverse semigroup in some wreath product is shown and after the definition of an inverse wreath product another embedding result of a 0-bisimple inverse semigroup is proved. In accordance to R J Warne (1983, 1986) similar results for regular semigroups are presented.

Chapter 12: The Krohn-Rhodes-Theorem: This rather self-contained chapter presents the proof of the Krohn-Rhodes-Theorem (following C Wells (1976)).

Chapter 13: Some applications: Following B H Neumann (1960) it is proved that every countable semigroup with 1 and 0 can be embedded in a two-generator semigroup, using the wreath product construction. In a certain analogy with the first part of the book centre and commutativity of wreath products are discussed by slightly generalising older papers by Hunter (1966) and Moors (1969). The last section of this chapter presents quite a few new results on simplicities of the wreath product.

Chapter 14: Generalizations: To some extent this final chapter parallels the approach for groups in Chapter 7 with generalised complete and restricted wreath products. Brief discussions of filter products and their links with wreath products (following Bryant and Groves (1978) and Meldrum (1992)) and of partial wreath products (following Petrich (1973)) contain a variant of the well-known description of endomorphism monoids of free semigroup actions with zero.