**1. The theory of groups (1959), by Marshall Hall, Jr.**

**1.1. Review by: Wilhelm Magnus.**

*Bull. Amer. Math. Soc.* **66** (3) (1960), 144-146.

When, in 1911, W Burnside published the second edition of his 'Theory of groups of finite order', his work contained all the essential group theoretical knowledge of his time with the exception of the theory of continuous groups (as they were called fifty years ago). In spite of the large number of results found and of methods developed since Burnside's time, Hall's 'Theory of groups' can claim to be a "Burnside brought up-to-date." Clearly, this cannot mean any more an account of all things known which now would require a book of thousands of pages. But Hall introduces the reader to the most important concepts and methods available in group theory (outside of the theory of Lie groups and topological groups), and he leads him in many cases to the frontier of our knowledge. ... the style of the book is concise and demanding like that of a research paper, - a well written and lucid research paper, but with few "asides" and with little leisurely exposition. (Incidentally, the same may be said about large portions of Burnside's book.) A student who studies Hall's book should find himself well equipped with both the mathematical background and maturity required for the reading of current literature in group theory.

**1.2. Review by: William J Turanski.**

*SIAM Review ***2** (2) (1960), 161-163.

The theory of groups provides one of the earliest examples in mathematics of the power of abstraction. By the middle of the nineteenth century it had become apparent to mathematicians that many properties of certain collections of specific entities (numbers, transformations, symmetries of an object, etc.) did not depend on the actual nature of the entities themselves, but upon the existence within the collections of rules for combining the entities. Here, the specific nature of the rule of combination did not matter as long as it satisfied certain simple properties (closure, associativity, possession of inverse), i.e., the collection with its rule of combination formed a group. Thus abstract group theory was born and with it came coherent unification of many loosely organized domains within mathematics. ... it became important to study groups as such. Now whenever an abstract entity, such as a group, is introduced into mathematics, natural development of the theory proceeds along two main lines. First is the theory of representations - i.e., what are general methods for producing concrete models of the abstract entity? Second is structure theory; i.e., in what ways can an entity be compounded out of smaller entities, what are the elementary building blocks, in what ways can one entity be embedded in another? It seems unfortunate that the prevailing style of writing mathematical textbooks prohibits the presentation of a theory along these lines, together with enough historical background to place the theory in its proper relationship to the whole of mathematics. The rule, rather, is to present a sequence of theorems in "logical" order, each theorem derivable from previously given theorems. This usually leaves the reader in an unresolved state of suspense as he works from proof to proof and from chapter to chapter. ... Within the constraints imposed by the prevailing style, Professor Hall has produced an excellent text.

**1.3. Review by: Eugene Guth.**

*American Scientist* **49** (1) (1961), 111A-112A.

This is a book on the modern theory of groups. It emphasizes finite groups, without neglecting infinite groups. Group theory has gone a long way since 1911, when Burnside summarized the knowledge of his time in his 'Theory of Groups of Finite Order'. Nevertheless, Hall succeeds to develop the most important concepts and methods and bring complete proofs of almost all theorems stated on about 450 pages! ... Mastering the contents of Hall's book will lead a student to the frontiers of group theory. He will be well equipped to read any recent literature and start original research himself in this field. ... This remarkable book undoubtedly will become a standard text on group theory.

**1.4. Review by: Richard Hubert Bruck.**

*Amer. Math. Monthly* **67** (2) (1960), 194-195.

This is a book which I wish I could put in the hands of every graduate student who has shown an interest in the elements of group theory. The first ten chapters would give him an excellent foundation in group theory, and there would still remain ten chapters for his delight. I will speak of the latter, some of which are unique. Many mathematicians are vaguely aware that there is a famous problem for groups, called the Burnside problem. But what proportion, even if we restrict attention to algebraists alone, can state the problem, let alone distinguish it from the several forms of the restricted Burnside problem? This book supplies most of the tools currently used for the Burnside problem, proves many of the known theorems in the subject (with a few omissions where the only known proofs would severely try the patience of the reader) and brings the history of the problem as close to the present moment as is humanly possible. No other book goes half so far in this direction. ... To this reviewer, the book seems like a lively and attractive grandson of Burnside's 'Theory of Groups of Finite Order', a grandson, let it be added, with a marked personality of his own.

**2. The theory of groups, 2nd edition (1976), by Marshall Hall, Jr.**

**2.1. Review by: Ian D Macdonald.**

*The Mathematical Gazette* **61** (418) (1977), 308.

This is a second edition, accomplished by a photographic process, of Marshall Hall's celebrated text on group theory. It corresponds with the original 1959 edition word for word and symbol for symbol, minor corrections apart. There has been no rewriting, not even of preface or references. Hall's book is a significant and well written text, a fact which was more obviously true in 1959 when the number of books on group theory was much smaller than it is today. Users in schools should be warned: they will find it strong meat. The book is written at the abstract level; even so it goes at a fast pace for a first course, and it makes little concession to concrete matters. There are only twelve diagrams, all of them hardly bigger than a postage stamp, for Hall follows the usual convention that diagrams for abstract theory are strictly a do-it-yourself activity. A feature of the original text was the almost infinite number of misprints, spelling mistakes, etc. To be fair, few of these were seriously misleading. Their provenance was perhaps bad handwriting and hasty proof-reading. They should be charitably regarded - a challenge to the reader, perhaps, or an irritant for reviewers. Though most of them have now gone, comparison with the 1959 edition reveals a sizable residue which it is far beyond the scope of this review to list.

**3. The Groups of Order 2 ^{n} (n ≤ 6) (1964), by Marshall Hall, Jr. and James K Senior.**

**3.1. From the Preface:**

No single presentation of a group or list of groups can be expected to yield all the information which a reader might desire. Here, each group is presented in three different ways: (1) by generators and defining relations; (2) by generating permutations; and (3) by its lattice of normal subgroups, together with the identification of every such subgroup and its factor group. In this lattice the characteristic subgroups are distinguished. For each group, additional information is given. Here are included the order of the group of automorphisms and the number of elements of each possible order 2, 4, 8, 16, 32, and 64 ..... All the groups are divided into twenty-seven families, following Philip Hall's theory of isotopy. Chapters 3 and 4 give the theoretical background for the construction of the tables. But these chapters are not necessary for the use of those tables; for that purpose Chapter 2 is adequate. Chapter 5 draws attention to a number of the more interesting individual groups.

**3.2. Review by: D.S.**

*Mathematics of Computation* **19** (90) (1965), 335-337.

The preparation of these tables was begun by the "senior" author way back in 1935. For a while Philip Hall was directly involved, and though he later withdrew as a co-author, the classification used is still based largely upon his ideas. ... The tables are nicely printed. The lattice diagrams, however, were not drawn by a professional draftsman, and exhibit much shaky lettering and uneven inking. This economy on the part of the publisher is somewhat regrettable, especially since the groups will be with us forever. Nonetheless, the diagrams are legible, and their interest and value are not negated by their lack of artistic perfection. Apparently the tables were constructed entirely by hand. It would be an interesting challenge to an experienced programmer with the requisite algebraic knowledge and interest to attempt to reproduce and extend these tables with a computer.

**3.3. Review by: Olga Taussky Todd.**

*American Scientist* **53** (2) (1965), 230A.

In a difficult subject like finite group theory it is of great importance to have examples to test theories. However, examples are not readily available. E.g., the problem of finding the exact number of (non isomorphic) groups of a given order is forbiddingly difficult. Everything is easy up to order 8 where there are three abelian and two non abelian groups, the quaternion group and the dihedral group. The next complicated order is 12, then comes 16. There are 14 groups of order 16 and 51 of order 32. This book gives a table of the 267 groups of order 64. Even the number was not known previously. Each group is presented (i) by generators and defining relations; (ii) by generating permutations; (iii) by a diagram of normal sub groups, the normal subgroup and factor group being identified in each case. Further, each group is given three identification marks, the order, its "family" and its "place" in the family. Some classification principles devised by P Hall and published elsewhere are used. ... The work was originally started separately by Senior (a chemist) and P Hall. Their later collaboration was interrupted by the war. When P Hall found himself unable to return to the plan, M Hall, Jr. used his wide experience in group theory to fill his place and to explain the theoretical background of the construction. All the work was done by hand; since electronic computers are now being employed for algebraic and combinatorial problems it is possible that this enormous task could have been eased. Patience and time were needed for this work, but the reader will soon notice that far more than that was involved.

**3.4. Review by: Michael Rosen.**

*Mathematical Reviews*, MR0168631 **(29 #5889)**.

The ambitious task the authors set themselves in this book is the determination of all groups of order 2^{n,}where n ≤ 6. They not only succeed, but also supply us with a great deal of information about the groups on their list. ... The classification begins with the notion of "family'' which was introduced by Philip Hall (1940). ... The first tables deal with family invariants, i.e., those invariants of a group which depend only on the family to which the group belongs. The length of the lower central series is an example. Following these are tables which list the groups of order 2^{n,}n ≤ 6, in each of the 27 families. The groups are given by generators and relations. For each group we are supplied, among other things, with the number of elements of each order, and the order of the automorphism group. Finally, there are 130 pages of lattice diagrams giving abundant information about the lattice of normal subgroups of all the various groups under consideration. The format of this book is rather unusual. The dimensions are 14 inches high, 17 inches wide, and 1/2 inch thick. It is interesting to conjecture about the eventual location of the book in a typical library. Moreover, the first ten pages have three pages per page. More precisely, the first ten sides have three columns, each column being given a page number. Thus the eleventh side turns out to be page 31. Undoubtedly this book will serve as a useful testing ground for group-theoretical conjectures. It is perhaps unfortunate that the testing ground has such extremely awkward dimensions.

**4. Combinatorial theory (1967), by Marshall Hall Jr.**

**4.1. From the Preface:**

[Hall describes Combinatorics.] Like many branches of mathematics, its boundaries are not clearly defined, but the central problem may be considered that of arranging objects according to specified rules and finding out in how many ways this may be done. If the specified rules are very simple, then the chief emphasis is on the enumeration of the number of ways in which the arrangement may be made. If the rules are subtle or complicated, the chief problem is whether or not such arrangements exist, and to find methods for constructing the arrangements. An intermediate area is the relationship between related choices, and a typical theorem will assert that the maximum for one kind of choice is equal to the minimum for another kind.

**4.2. Review by: William G Brown.**

*Amer. Math. Monthly* **76** (7) (1969), 851-852.

Although this book is entitled 'Combinatorial Theory', most of it is confined to the theory of designs. As such it is a comprehensive polished survey. There are extensive discussions of difference sets, finite geometries, orthogonal Latin squares (an example of order ten is illustrated on the dust jacket), Hadamard matrices, and completion and embedding theorems. The treatment is lucid and rigorous, suitable for seniors or graduate students who have had courses in abstract and linear algebra. This book could well be included in the training of every linear algebraist, to offset the present danger of becoming lost in homological generalities.

**4.3. Review by: Herbert John Ryser.**

*Mathematical Reviews*, MR0224481 **(37 #80)**.

The text is divided into three major subdivisions. Chapters 1 through 4 deal with problems of enumeration; Chapters 5 through 9 deal with the intermediate area of theorems on choice; Chapters 10 through 16 deal with the existence and construction of block designs. But the last of these subdivisions is by far the lengthiest and accounts for approximately 200 of the 300 pages of text. This allows the author to investigate block designs in considerable detail and reach the frontiers of our present day knowledge. Indeed, throughout this portion of the text the reader is made aware of the many basic problems involving block designs and the various techniques available for dealing with them. The first two subdivisions of the text are of necessity less thorough in their coverage, and no attempt is made to match the scope of the detailed treatises available on enumeration theory, network flow theory, graph theory, and related topics.

**5. Combinatorial theory, 2nd edition (1998), by Marshall Hall Jr.**

**5.1. Review by: Steve Abbott.**

*The Mathematical Gazette* **83** (497) (1999), 355.

Combinatorial theory encompasses a wide variety of topics, from simple counting of permutations and use of the pigeonhole principle to partitions, map colourings, latin squares, rook polynomials, design of experiments and Ramsey theory. In chapters 1-4 Hall covers the tools of enumeration, such as the inclusion-exclusion principle, generating functions and Möbius inversion and applies some of these in investigating partition functions, which count the number of ways of writing a number as a sum with specified properties. The middle part of the book deals with theorems of P Hall, Konig, Ramsey and Dilworth on distinct representations and assignment problems and includes a short section on graphs. The major part of the book concentrates on experimental designs, with chapters on block designs, difference sets, finite geometries, latin squares, Hadamard matrices, theorems on completion and embedding, and coding theory. There is nothing in this book on the enumeration of edge and vertex colourings.

**5.2. Review by: Giuseppe Pellegrino.**

*Mathematical Reviews*,** **MR0840216 **(87j:05001).**

This book (together with '*Combinatorial mathematics'* by H J Ryser [1963], to whose memory the book is dedicated) is a reference point for those interested in combinatorics, both for the selection of the topics - fruit of a deep knowledge of the field - and for the clearness of exposition that makes the reading agreeable. ... This book preserves the structure of the first edition; substantial variations do not appear in either the propositions or their proofs; the modifications are a consequence of the need to connect the subjects of a first edition with the added ones. In conclusion, as to the value of the work, we concur with the author's comment: "This book is not an encyclopaedia. But a conscious effort has been made to bring it up to date in the areas it covers."