1.1. Review by: Edward F Assmus Jr.
Mathematical Reviews MR0506700 (58 #22254).

This delightful monograph of less than two hundred pages with a German text and an English "Vorwort" is the result of fifteen lectures on groups of prime degree that the author delivered in Giessen in the winter of 1976. Much of the author's easy speaking style comes through the printed page and his historical asides give even the most well-known results a fresh life. The first two lectures introduce the problem and reduce it to a study of those nonsolvable groups of order divisible by p and contained in Sym(p). The third lecture is devoted to Burnside's proof that these groups must act doubly-transitively and in the fourth he tabulates all the known examples and sketches a proof (modulo the determination of all finite simple groups) that his list is complete. Lectures 5 through 8 pertain to the representation theory, especially certain ideas of J A Green concerning the relationships among the $G$-modules of functions from $X$ to $F_{p}$ where $X$ is either the $p$-set on which $G$ acts, its collection of 2-subsets, or the set of ordered pairs (but not including the diagonal). They serve, among other things, to illustrate the importance of the parameter $t = N_{G}(P)$, where $P$ is a Sylow $p$-subgroup (the cyclic group of order $p$, of course) of $G$. In the next three lectures the author experiments with various values of $t$. In Lectures 11 and 12 he experiments with $p = 2q+1$, $q$ prime, and in Lecture 13 with small values of $p$. In the final two lectures the author discusses groups of degree $2p$. There is an appendix which contains a review of the necessary ordinary and modular representation theory.

2.1. From the preface.

These notes have been written for undergraduates in Oxford. In part they carry out the promise made many years ago to expand and polish the brief Notes on Algebra that two of us wrote for publication by the Mathematical Institute in 1970 . But over the intervening years attitudes and syllabuses have changed and as a result this new set of notes bears very little relation to the old. Our intention has been to provide a text suitable for the group theory and geometry parts of the revised Section 1 of the Final Honour School, and we have therefore taken the syllabus for the 1981 examination as our guide: in the 1979-80 edition of the Examination Decrees and Regulations the relevant passage reads as follows: Action of a group on a set, orbits, transitivity, coset spaces. The symmetric groups $S_{n}$ and alternating groups $A_{n}$, with particular reference to $n ≤ 5$. Normalizers, conjugates, elementary applications of Sylow's theorems. Formulae for counting orbits.

The affine group of transformations of $\mathbb{R}^{n}$. Affine equivalence of figures in $\mathbb{R}^{2}$, Euclidean isometries of $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$; orthogonal transformations. Finite groups of isometries; the Platonic solids and their symmetry groups. The complex plane and Riemann sphere; Möbius transformations. Inversion.

We hope that we have interpreted this syllabus correctly: we believe, covered by Chapters 3, 4, 5, 6, 9, 12, 14, 15, 17 and parts of Chapters 2, 8, 11, 16. The remainder of the mathematics that we present is included for various reasons. Some of it is introductory: for example, Chapter 1 is a survey of the group theory that we expect undergraduates to have met in their first year , and Chapters 2 , 10 , 11 also contain preparatory material . Some is intended to complete our treatment of one topic or another and to establish links with options available in Section 2 of the Oxford syllabus . We believe that Chapter 7 , part of Chapter 8 , Chapter 13 , Chapter 16 , Chapter 18 and Chapter 19 consist mainly of material that is not in the syllabus for Section 1 .

The learning of theories and theorems is only a small part of the process of learning mathematics, the larger part of which involves acquiring the art of solving problems, hence of thinking creatively in mathematics. For this reason we have included a number of exercises. Some are intended to be easy, most require some thought, and a few are intended to be hard (nevertheless, we have tried to avoid open problems). One or two of the exercises develop parts of the theory for which we (all three of us) felt there was no space in the text, and on occasion we have taken the liberty of citing these where we needed the results in our exposition. The exercises therefore form an important part of the book, arguably the most important, and we hope that the reader will enjoy and benefit from them.

2.2. Contents.

Vol. I: Chapter 1. A survey of some group theory; Chapter 2. A menagerie of groups; Chapter 3. Actions of groups; Chapter 4. A garden of G-spaces; Chapter 5. Transitivity and orbits; Chapter 6. The classification of transitive G-spaces; Chapter 7. G-morphisms; Chapter 8. Group actions in group theory; Chapter 9. Actions count; Chapter 10. Geometry: an introduction; Chapter 11. The axiomatisation of geometry.

Vol. II: Chapter 12. Affine geometry; Chapter 13. Projective geometry; Chapter 14. Euclidean geometry; Chapter 15. Finite groups of isometries; Chapter 16. Rotations in Euclidean space; Chapter 17. Inversive geometry; Chapter 18. Topological considerations; Chapter 19. The group theory of the Hungarian magic cube; Index.

3.1. From the Preface.

Few of us can have a hand in the writing of our own memorial. We (Peter M Neumann and Gabrielle A Stoy) dedicate this book to the memory of our much loved and greatly respected late colleague and collaborator Edward Thompson, so that he may become one of those select few. He deserves and has earned it. He was an enthusiast who taught us much delightful mathematics. But he was much more than a mathematician, he was a wise man, and we both learned from him much more than mathematics.

This book is about the measurement of symmetry. That is what groups are for. Symmetry is visible in all parts of mathematics - and in many other arts too - and geometrical symmetry is the most visible of all symmetries. That is why groups and geometry are such close neighbours.

In its first edition (1980, reprinted with corrections 1982) the book was a two-volume set of cyclostyled notes produced and distributed by the Oxford Mathematical Institute for second- and third-year undergraduates and first year postgraduate students. The main changes to this Oxford University Press edition are a complete re-write of Chapter 16 (completed by Edward Thompson before his demise), and, on advice from a benevolently critical referee, the addition of specimen solutions to some more-or-less randomly selected exercises in the first half of the book. Although we were guided in our choice of material by the Oxford syllabus of that earlier time, we allowed ourselves to present much other material. It was included for various reasons. Some was introductory or preparatory, some was intended to round out topics that had been, for excellent pedagogical reasons, excluded or curtailed in the syllabus, some was intended to establish links with other subjects, and, although we did not acknowledge in the preface of that earlier work what many readers will have recognised, some was included for the best of all possible reasons, namely, that we liked it. The result may be an idiosyncratic work. We make no apology for that: there is no point in publishing a book that is just like many others.

One or its idiosyncrasies is that it divides clearly into two halves: what you have purchased amice lector is really two books for the price of one. Nevertheless. those two books enjoy a symbiotic relationship which more than justifies the union. For, although Edward Thompson's geometrical part, Chapters 10-18, is written in a different style from Chapters 1-9 and 19, it is based on our treatment of group actions, which in turn was adapted to geometrical requirements. He and we worked closely together. There are small discrepancies of notation in the two halves, but we estimate that these should not be confusing: therefore we have chosen to maintain our own usages and not to interfere in any way with Edward's preferences and style. We hope that his enthusiasm for geometry and its history, about which he was writing a treatise when he died aged 72 in May 1991, will shine through.

The learning of theories and theorems is only a small part of the process of learning mathematics, the larger part of which involves acquiring the art of solving problems, hence of thinking creatively in mathematics. For this reason we have included a number of exercises. Some are intended to be easy, most require some thought, and a few are intended to be hard (nevertheless. we have tried to avoid open problems). One or two of the exercises develop parts of the theory for which we (all three of us) felt there was no space in the text, and on occasion we have taken the liberty of citing these where we needed the results in our exposition. The exercises therefore form an important part of the book, arguably the most important, and we hope that the reader will enjoy and benefit from them. The specimen answers in Chapters 1-9 are intended to be of the nature of an extension of the text and are placed between the main body of each chapter and the exercises. The wise and conscientious reader will, however, try the exercises unaided before attempting to evaluate what we have offered.

3.2. Review by: Robert J Bumcrot.
Mathematical Reviews MR1283590 (95f:20001).

As the first two authors remark, this work is almost two books in one. They are responsible for the first nine and last (nineteenth) chapters, which present portions of group theory that are of great use in geometry, and the (late) third author for the rest, which presents portions of geometry the understanding of which is greatly enhanced by group theory. Thus the main focus of the group theory is on $G$-spaces and permutations, while much of the geometry follows the Erlanger program.

The material is suitable for undergraduates at about the level of those who have recently done well in a strong linear algebra course. Its limited coverage renders it unsuitable as a text for a first course in abstract algebra, and its level of abstraction makes it an unlikely candidate for most advanced geometry courses as they are now organised. But the book would be an excellent choice for a special elective.

The writing is very clear, often witty, and sometimes unusually graceful, as for example in the first pages of Chapter 11 on the axiomatisation of geometry. Chapters one through eighteen close with exercises, and the last chapter, on the Rubik's cube puzzle, has exercises throughout. Some exercises explore new topics, such as the wreath product. In the first nine chapters, detailed solutions and discussions of selected exercises are presented.

4.1. From the Publisher.

The book, based on a course of lectures by the authors at the Indian Institute of Technology, Guwahati, covers aspects of infinite permutation groups theory and some related model-theoretic constructions. There is basic background in both group theory and the necessary model theory, and the following topics are covered: transitivity and primitivity; symmetric groups and general linear groups; wreath products; automorphism groups of various treelike objects; model-theoretic constructions for building structures with rich automorphism groups, the structure and classification of infinite primitive Jordan groups (surveyed); applications and open problems. With many examples and exercises, the book is intended primarily for a beginning graduate student in group theory.

4.2. From the Preface.

The oldest part of group theory is that which deals with finite groups of permutations. One of the newest is the theory of infinite permutation groups. Much progress has been made during the last two decades, but much remains to be discovered. It is therefore an excellent research area: on the one hand there is plenty to be done; on the other hand techniques are becoming available. Some research takes as its goal the generalisation, extension or adaptation of classical results about finite permutation groups to the infinite case. But mostly the problems of interest in the finite and infinite contexts are quite different. In spite of its newness, the latter has already developed a momentum and an ethos of its own. It has also developed strong links with logic, especially with model theory.

To bring this lively and exciting subject to the attention of mathematicians in Assam-both students and established scholars - a course of sixteen lectures was presented in August and September 1996 at the Indian Institute of Technology, Guwahati, In the available time it was not possible to explore more than a fraction of the area. The lectures were therefore conceived with restricted aims: firstly, to expound a useful amount of general theory; secondly, to introduce and survey just one of the rich areas of recent research in which considerable progress has been made, namely, the theory of Jordan groups. That limited aim is reflected in this book. Although it is a considerable expansion of the lecture notes (and we have included many exercises, which range in difficulty from routine juggling of definitions to substantial recent research results), it is not intended to introduce the reader to more than a small-though important and interesting-part of the subject.

A course at this level can only succeed if it is a collaboration between lecturers and audience. That was so in this case and although the four speakers are those listed as authors of this volume, credit would be spread much more widely if title-page conventions permitted. In particular, we record our very warm thanks to the audience for being so enthusiastic and alert, and especially to the three note-takers Dr B K Sharma, Ms Shreemayee Bora and Ms Shabeena Ahmed.

4.3. Review by: Cheryl Praeger.
Mathematical Reviews MR1632579 (99e:20003).

This book was developed from a series of lectures on permutation groups given by the four authors in Guwahati, Assam, in 1996. Its main focus is the recent classification of infinite primitive Jordan groups. An outline of the proof and a description of the main classes of examples are given. The book is an excellent introduction to infinite permutation groups, and, in particular, is accessible to graduate students.

The first chapters contain basic theory of permutation groups, followed by descriptions of several families of (mainly infinite) permutation groups, including symmetric groups, linear groups, wreath products of permutation groups, and the group of order automorphisms of the rational numbers. Jordan groups are introduced, and the examples of finite and infinite primitive Jordan groups are defined and discussed. Then results of Cameron, Adeleke, Neumann and Macpherson, which led to a classification of the infinite primitive Jordan groups, are presented. Some basic concepts of model theory are given and its connections with Jordan groups are illustrated using a construction of Hrushovski. The final chapter contains a number of open problems.

5.1. From the Publisher.

How many groups of order n are there? This is a natural question for anyone studying group theory, and this Tract provides an exhaustive and up-to-date account of research into this question spanning almost fifty years. The authors presuppose an undergraduate knowledge of group theory, up to and including Sylow's Theorems, a little knowledge of how a group may be presented by generators and relations, a very little representation theory from the perspective of module theory, and a very little cohomology theory - but most of the basics are expounded here and the book is more or less self-contained. Although it is principally devoted to a connected exposition of an agreeable theory, the book does also contain some material that has not hitherto been published. It is designed to be used as a graduate text but also as a handbook for established research workers in group theory.

5.2. From the Preface.

This book has grown out of a series of lectures given in the Advanced Algebra Class at Oxford in Michaelmas Term 1991 and Hilary Term 1992, that is to say from October 1991 to March 1992. The focus was - and is - the big question

how many groups of order n are there?

Two of the lectures were given by Professor Graham Higman, FRS, two by Simon R Blackburn and the rest by Peter M Neumann. Notes were written up week by week by Simon Blackburn and Geetha Venkataraman and those notes formed the original basis of this work. They have, however, been re-worked and updated to include recent developments.

The lectures were designed for graduate students in algebra and the book has been drafted with a similar readership in mind. It presupposes undergraduate knowledge of group theory - up to and including Sylow's theorems, a little knowledge of how a group may be presented by generators and relations, a very little representation theory from the perspective of module theory and a very little cohomology theory - but most of the basics are expounded here and the book should therefore be found to be more or less self-contained. Although it remains a work principally devoted to connected exposition of an agreeable theory, it does also contain some material that has not hitherto been published, particularly in Part IV.

We owe thanks to a number of friends and colleagues: to Graham Higman for his contribution to the lectures; to members of the original audience for their interest and their comments; to Laci Pyber for comments on an early draft; to Mike Newman for permission to include unpublished work of himself and Craig Seeley; to Eira Scourfield for guidance on the literature of analytic number theory; to Juliette White for comments on the earlier chapters of the book and for help with proofreading our first draft. We would also like to acknowledge the support of the Mathematical Sciences Foundation, St Stephen's College, Delhi, The Indian Institute of Science, Bangalore and our respective home institutions. Geetha Venkataraman would also like to acknowledge the encouragement and support extended by Uttara, Mahesh and Shantha Rangarajan and her parents WgCdr P S Venkataraman and Visalakshi Venkataraman. Professor Dinesh Singh has been a mentor providing much needed support, encouragement and intellectual fellowship. We record our gratitude to an anonymous friendly referee for constructive suggestions and for drawing our attention to some recent references that we had missed. We are also very grateful to the editorial staff of Cambridge University Press for their great courtesy, enthusiasm and helpfulness.

Turning lecture notes into a book involves much hard work. Inevitably that work has fallen unequally on the three authors. The senior author, too happy to have relied on the excellent principle juniores ad labores (which he admits to having embraced less enthusiastically when he was younger), is glad to have the opportunity to acknowledge that all the hard work has been done by his two colleagues, whom he thanks very warmly.

5.3. Review by: Benjamin Klopsch.
Mathematical Reviews MR2382539 (2009c:20041).

This monograph has its origin in a series of lectures on the enumeration of finite groups given by Graham Higman, Simon R Blackburn and Peter M Neumann at the University of Oxford in 1991-92. Considerable effort has been made to supplement, update and extend the original notes by Blackburn and Geetha Venkataraman. The result is a comprehensive and reasonably self-contained introduction to the subject of group enumeration which focuses on the basic question: "How many groups are there of any given order $n$?", and its variations. The book covers classical results of Hölder from the late 19th century as well as key theorems of Higman, Sims and Pyber from the 20th century. It provides an indication of recent developments up to 2006 and ends with a substantial list of open problems for future research. Where needed, brief explanations of the group theoretic notions and techniques required are incorporated into the text. The careful exposition gives a clear sense of direction and renders the book accessible to graduate students.
...
The book is a valuable contribution to this particular area of group theory. In addition to covering key results and techniques, it points towards a broad range of recent results and open problems. A minor shortcoming of the book is the following. Despite having been drafted with non-expert readers in mind, it contains no exercises and surprisingly few examples.

6.1. From the Publisher.

Before he died at the age of twenty, shot in a mysterious early-morning duel at the end of May 1832, Évariste Galois created mathematics that changed the direction of algebra. This book contains English translations of almost all the Galois material. The translations are presented alongside a new transcription of the original French and are enhanced by three levels of commentary. An introduction explains the context of Galois' work, the various publications in which it appears, and the vagaries of his manuscripts. Then there is a chapter in which the five mathematical articles published in his lifetime are reprinted. After that come the testamentary letter and the first memoir (in which Galois expounded on the ideas that led to Galois Theory), which are the most famous of the manuscripts. These are followed by the second memoir and other lesser known manuscripts. This book makes available to a wide mathematical and historical readership some of the most exciting mathematics of the first half of the nineteenth century, presented in its original form. The primary aim is to establish a text of what Galois wrote. The details of what he did, the proper evidence of his genius, deserve to be well understood and appreciated by mathematicians as well as historians of mathematics.
A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

6.2. From the Preface.

Before he died aged twenty, shot in a mysterious early-morning duel at the end of May 1832, Évariste Galois created mathematics which changed the direction of algebra. His revolutionary ideas date from around May 1829 to June 1830, the twelve to thirteen months surrounding his eighteenth birthday. An article published in June 1830 created the theory of Galois imaginaries, a fore-runner of what are now known as finite fields; his so-called Premier Mémoire created group theory and Galois Theory - the modern version of the theory of equations. The Lettre testamentaire, the letter that he wrote to his friend Auguste Chevalier on 29 May 1832, the eve of the duel, is an extraordinary summary of what he had achieved and what he might have achieved had he lived to develop and expound more of his mathematical ideas.

Although there have been several French editions of his writings, there has never until now been a systematic English translation. Translations of historical material are of little use without the originals alongside, however. What is offered here therefore is a bilingual edition. The French transcription is a new one. Following precedents set by Tannery in 1906/07 and by Bourgne and Azra in 1962 it is as close to the original manuscripts as I have been able to make it. Main text, afterthoughts, deletions, insertions, over-writings - all are recorded as faithfully as I could manage within the inevitable constraints imposed by the differences between manuscript and print.

In addition I offer three levels of commentary. First there is general contextual information; secondly there are notes on the physical state of the manuscripts and on the disposition of their content; third, there are comparisons of the various previous editions, including variant readings, in minutely pedantic and minutely printed marginal notes. Little of the commentary here is mathematical. It is focussed on the symbols on the page, on the syntax, on establishing an accurate text. Commentaries on the semantics, the meaning of what Galois wrote, would be a quite different exercise. That comes next, but must be the subject of other studies. I have neither the space nor the time. Space is a concern because the book is already substantially longer than I had anticipated in light of the shortness of Galois' productive life. Time is short because a proper modern study of his writings would take years, whereas it is planned that this book should appear on 25 October 2011 as homage to Galois on the 200th anniversary of his birth.

The book is conceived as a contribution to the history of mathematics. I hope, however, that it may bring the mathematical writings of this extraordinary genius to a wider mathematical public than has hitherto been able to appreciate them. At the very least it may serve to dispel some of the common myths that surround Galois and his understanding of mathematics. It is simply not true, for example, that he proved and used the simplicity of alternating groups. He did not need to: he was much cleverer than that; his treatment of solubility of equations is at once simpler and more elegant than what has now become textbook tradition. The details of what he did, the proper evidence of his genius, deserve to be as well understood and appreciated amongst mathematicians as amongst historians of mathematics. If this edition extends his readership beyond the bounds presently imposed by linguistic constraints it will have succeeded.

Review by: Charles W Curtis.
Notices of the American Mathematical Society 59 (11) (2012), 1565-1568.

The book that is the subject of this review, by Peter M Neumann, contains an English translation of all of Galois's writings published by Liouville and Tannery and more, with the original on the facing page for purposes of comparison. The author has done a painstaking analysis of the manuscripts, so that the French edition is not simply a reissue of older ones, and has included images of some of the Galois manuscripts from the archives de l'Institut de France. The reproductions provide a fascinating counterpoint to the texts and the translations. The author has also included extensive commentary "focussed on the symbols on the page, on the syntax, on establishing an accurate text." He acknowledges that commentaries "on the semantics, the meaning of what Galois wrote, would be a quite different exercise" and "must be the subject of further studies." Hints of what further mathematical studies might be are included at several places in the notes on individual items. Comments on some of the existing English translations appear, with comparison with the author's version.

The translations begin with the five mathematical articles published in Galois's lifetime, beginning with an article on continued fractions published when he was seventeen in Annales de Mathématiques pures et appliquées, edited and published by J D Gergonne. In an introductory note, the author states that the article received a friendly review and was cited as recently as 1962 by Davenport, who commented, however, that the material was implicit in earlier work of Lagrange. The fourth, "Sur la théorie des nombres", published when Galois was eighteen, was of great interest and importance, as it contained the original version of a substantial part of the theory of finite fields.

A translation of the testamentary letter of 29 May 1832 addressed to Chevalier appears next. The letter contains a survey of Galois's ideas on the subject of what is now called Galois theory, along with a remarkable new result (with a proof) concerning the simple groups $PSL(2, p)$ and some material on integrals of algebraic functions and the theory of elliptic functions.
...
The book, in the author's words, "is conceived as a contribution to the history of mathematics." It also gives a mathematically knowledgeable reader a chance to follow a series of groundbreaking mathematical events, from Galois's invention of what he needed from group theory in order to solve the problem stated in the title of the first memoir to his first steps towards establishing group theory as an independent subject, beginning with the second memoir and continuing with his determination, in the letter to Chevalier, of the exceptional permutation representations of the linear fractional groups. It's quite a story.

6.3. Review by: Graham Hoare.
The Mathematical Gazette 97 (538), 187-188.

Romanticised and erroneous versions of some aspects of Galois' short life and work have appeared from time to time. Although it is a peripheral consideration in the author's project to rebut or demythologise such accounts, Peter Neumann, nevertheless, does debunk some misconceptions that have persisted. Galois lived in turbulent times but despite becoming enmeshed in political activities he changed the course of algebra by transforming the classical theory of equations into what is now called Galois Theory, together with its abstract algebra including the theory of groups and fields. Failed political revolutionary he may have been but in mathematics he was a géomètre révolutionnaire.

Until now there has been no systematic English translation of Galois' writings. Here, they are presented alongside a new transcription of the original French, enhanced by three levels of commentary; Neumann discusses the possible influences on Galois, and the physical state of the manuscripts. He also compares previous editions of his writings, from Liouville's of 1846 to the monumental work published by Bourgne and Azra in 1962. These preliminaries, which comprise the bulk of the Introduction (Chapter I), are concluded with a glossary of technical terms and words used by Galois, some of which may have changed since he penned them.

In the next five chapters almost all of Galois' papers are presented, 'warts and all', in parallel form with French on the left page and an English translation facing. Editing is sensitive but instructive; exegesis is minimal. In Chapter II we have reprints of the five mathematical articles published in Galois' lifetime. The first appeared when he was seventeen, the last two years later. The most significant of these, Sur La Théorie Des Nombres, contains the theory of Galois fields. Next we have the Testamentary Letter written to his friend Auguste Chevalier on the eve of the fatal duel. It is full of brilliant insights. We have, for example, the reduction of the modular equation and the groups now called $PSL(2, p)$, for $p = 5, 7, 11$, which Galois explicitly recognized as being 'simple'. The letter contains summaries of his discoveries and refers to three memoirs, the now famous Premiere Mémoire, the incomplete Second Mémoire and the Troisième Mémoire, the last of which was lost. The Theory of Numbers paper, the two memoirs and the letter to Chevalier, which comprise Galois' main work, were eventually published by Liouville in 1846.

In the remarkable Premiere Mémoire Galois introduced the Galois group of an equation and showed how solubility of a polynomial equation in terms of radicals could be characterised by a property of the group. He deals explicitly with the quartic equation, known to be solvable by radicals, and the irreducible quintic, which he shows to be solvable by radicals just in case its group is either of order 20 or a subgroup of that group.

The second memoir (Chapter V) focuses on aspects of what we now regard as the theory of groups. Its main theorem is that the degree of every primitive soluble permutation group is the power of a prime number. In Chapter VI Neumann selects nineteen minor mathematical manuscripts, although as he suggests in the Introduction, some would be more appropriately described as 'philosophical-polemical'.

The text is concluded by a short epilogue devoted to myths and mysteries. The best known myth, that Galois created his theory of groups on the evening before the infamous duel, is easily dispatched by the author, who traces it back to the chapter on Galois in E T Bell's (1937) Men of Mathematics. Another, firmly dispelled by Neumann, is that Galois based his proof of the insolubility of the general quintic by radicals on the simplicity of the alternating group $A_{s}$. Indeed there is no evidence that Galois proved that the alternating groups $A_{n}$ are simple for $n ≥ 5$. As Neumann writes, "his treatment of solubility of equations is at once simpler and more elegant than what has now become textbook tradition". As for mysteries we can but speculate as to why Galois failed to write up his memoirs in a better form in his last two years. Puzzling, too, is the lack of reaction to Galois' output in the decade following his death and, even more so, the abruptness of the ending of this period of silence.

This book represents a contribution to the history of mathematics and a resource for mathematicians interested in what Galois actually wrote. It is beautifully produced and is enhanced by some illuminating facsimiles of manuscript pages.

6.4. Review by: E J Barbeau.
Mathematical Review MR2882171 (2012j:01032).

Upon his death, the manuscripts of Évariste Galois (1811-1832) passed into the hands of Auguste Chevalier. Included among them was a Testamentary Letter to Chevalier written the night before Galois died that reviewed his mathematical results and instructed Chevalier to have it printed in the Revue Encyclopédique (where it appeared in September, 1832). Chevalier made copies of many of them and, in 1843, passed them on to Joseph Liouville, editor of the Journal de Mathématiques Pures et Appliquées. Liouville published them in 1846. Later, Galois' collected works were produced by Jules Tannery in 1906-1907 and Robert Bourgne and Jean-Pierre Azra in 1962. In addition, there were five articles by Galois that were published in 1829 and 1830 while he was yet alive.

This book, written for an English reader, provides a version of Galois' work in French that faithfully reflects the appearance of the manuscript along with the emendations made by the author accompanied by an English translation, hitherto unavailable in most cases. The volume includes everything in the Liouville and Tannery collections, but leaves out "many scraps and partial calculations" that need no translation as well as items of schoolwork omitted through space limitations. The only complete edition of Galois' works is that of Bourgne and Azra.

The French and English texts of the items appear on facing pages. Each is preceded by a description of the manuscript, its context and publication history. Copious notes are provided, either in the margin of the French version or at the end. The English translation is designed to give the flavour of the original, including for example some of the crossed out material, but has been edited to conform to modern conventions. The treatment is largely editorial and the author does not provide a mathematical exegesis of Galois' work. The are photocopies of selected pages of the manuscripts.

The introductory chapter provides a thumbnail sketch of Galois' biography, a discussion of what he might have read and how his ideas were received, and an overview of the manuscripts and their publication history. The author discusses at length other editions and his own approach to the works, including how he arrived at appropriate English renditions of French terminology and conventions.

The second chapter reprints the five articles published before 1832, while the third contains the Testamentary Letter to Chevalier.

In Chapter IV, we have the text of the "First Memoir". This is a draft of Galois' third submission to the Académie des Sciences; it was received by the Académie on January 17, 1831 and returned to Galois in July 1831. We see in this chapter the rejection report of S-F Lacroix and S D Poisson and Galois' letter in March inquiring about the fate of his submission.

The next chapter is devoted to the "Second Memoir" left by Galois at the time of his death. In Chapter VI, we find most of Galois' remaining manuscripts in the order into which they were sorted into dossiers by Liouville's daughter, Madame de Blignières, around the end of the 19th century. Some are mathematical: work related to the two memoirs, drafts of articles, and sketches of his ideas and plans. Others are more philosophical. The author discusses in detail their dating and intent, explaining how he agrees with or differs from the conclusions of earlier editors.

The brief epilogue treats various myths and mysteries about Galois' work, particularly why a decade elapsed between his death and the published appearance of his manuscripts. This source book, with its careful analysis and essentially complete English translation of Galois' manuscripts, is a useful resource for both the historian and the general mathematical reader who wish to gain a first-hand appreciation of Galois' achievement.

6.5. Review by: Edmund F Robertson
Bulletin of the London Mathematical Society 44 (2012), 1303-1307.

This review is available at THIS LINK.

7.1. From the Publisher.

Although Évariste Galois was only 20 years old when he died, shot in a mysterious early-morning duel in 1832, his ideas, when they were published 14 years later, changed the course of algebra. He invented what is now called Galois Theory, the modern form of what was classically the Theory of Equations. For that purpose, and in particular to formulate a precise condition for solubility of equations by radicals, he also invented groups and began investigating their theory. His main writings were published in French in 1846 and there have been a number of French editions culminating in the great work published by Bourgne and Azra in 1962 containing transcriptions of every page and fragment of the manuscripts that survive. Very few items have been available in English up to now.

The present work contains English translations of almost all the Galois material. They are presented alongside a new transcription of the original French, and are enhanced by three levels of commentary. An introduction explains the context of Galois' work, the various publications in which it appears, and the vagaries of his manuscripts. Then there is a chapter in which the five mathematical articles published in his lifetime are reprinted. After that come the Testamentary Letter and the First Memoir (in which Galois expounded the ideas now called Galois Theory), which are the most famous of the manuscripts. There follow the less well known manuscripts, namely the Second Memoir and the many fragments. A short epilogue devoted to myths and mysteries concludes the text.

The book is written as a contribution to the history of mathematics but with mathematicians as well as historians in mind. It makes available to a wide mathematical and historical readership some of the most exciting mathematics of the first half of the 19th century, presented in its original form. The primary aim is to establish a text of what Galois wrote. Exegesis would fill another book or books, and little of that is to be found here.

This work will be a resource for research in the history of mathematics, especially algebra, as well as a sourcebook for those many mathematicians who enliven their student lectures with reliable historical background.