# Michael Aschbacher's Books

We list below ten books by Michael Aschbacher, some single author, some multiple authors. For each we give some information such as extracts from reviews, extracts from Prefaces and/or publisher's information.

**Click on a link below to go to that book**- The finite simple groups and their classification (1980)

- Finite Group Theory (1986)

- Sporadic Groups (1994)

- 3-transposition groups (1997)

- Finite Group Theory. Second Edition (2000)

- The classification of quasithin groups. I. Structure of strongly quasithin K-groups (2004) with Stephen D Smith

- The classification of quasithin groups. II. Main theorems: the classification of simple QTKE-groups (2004) with Stephen D Smith

- The classification of finite simple groups. Groups of characteristic 2 type (2011) with Richard Lyons, Stephen D Smith and Ronald Solomon

- Fusion systems in algebra and topology (2011) with Radha Kessar and Bob Oliver

- Quaternion fusion packets (2021)

**1. The finite simple groups and their classification (1980), by Michael Aschbacher.**

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

*The Mathematical Gazette*

**64**(430) (1980), 302.

This is a delightful little book. For one thing, it contains no proofs whatever! It describes the bare bones of a programme for the classification of the finite simple groups. This is a problem which at present engages the attention of some of the best minds in mathematics. Those who really understand what is happening form a tightly knit body of perhaps one hundred people. It is a measure of the quality of Professor Aschbacher's book that he succeeds in conveying something of the flavour of the subject, and of the energy and enthusiasm of the protagonists. to the reader.

What is a finite simple group? It is a finite group G in which the only normal subgroups are G itself and the unit subgroup. Such groups are (via extension theory) the building blocks for all finite groups. Many years ago developments in the theory of Lie algebras led to the idea that finite simple groups might conceivably be classified. At that time seven infinite sequences of such groups (other than cyclic groups) were known, as well as a few "sporadic" groups which did not fit into any infinite sequence.

For many years classification remained a remote hope. But a cataract of new results began about 1963, in particular John Thompson and Walter Feit's Odd Order Paper, so called because it contains the theorem that every finite simple group must have elements of order 2. We now know of 15 infinite sequences. All except the alternating groups are "of Lie type", which means that we have a unified theory of them. About 26 sporadic groups are also known.

What is the problem? It is to prove that these are all the finite simple groups, excepting perhaps a few more sporadic groups. The proof may well run to several thousand pages of detailed mathematics. Large parts of it are already in existence, and few doubt that it will be completed in the next year or two.

Who should read this book? It is a "must" for all group theorists. Indeed all research mathematicians should read it. At the same time they might look at the papers by Thompson et al. in the

*Bulletin of the London Mathematical Society*, part 3 of volume 11 (1979) where there is material about the astonishing connection between the sporadic group called the Monster and the elliptic modular function. But one must emphasise that Aschbacher's book deserves a much wider readership. It can profitably be read by anyone with an interest in mathematics beyond the most elementary. Its content is as accessible as it is interesting and important. If an introduction is needed, there are excellent expository articles by Stewart and Gardiner in the

*New Scientist*, volume 82 No. 1149 (April 1979).

**1.2. Review by: Arthur Reifart**.

*Mathematical Reviews*MR0555880

**(81e:20021)**.

The purpose of this survey article is to inform interested mathematicians how and how far the determination of all finite simple groups has gone. But group theorists should also use the article for the following two reasons: (i) to see which parts of the classification have been done and which parts are still open; (ii) to see what group theorists must do once the finite simple groups are classified. Perhaps the following provocative statement describes the situation quite well and, at least in the reviewer's opinion, it can also be read between the lines of this survey article. Very soon the classification of all finite simple groups will have been achieved but nobody will be too surprised if one or more groups are missing in this classification.

**1.3. Review by: Daniel Gorenstein.**

*American Scientist*

**68**(6) (1980), 708.

In these 1978 James K Whittemore Lectures, Michael Aschbacher, of the California Institute of Technology, one of the world leaders in finite group theory, presents some of the highlights of the classification of finite simple groups. The book, intended for the general professional mathematician, provides a highly readable description of the basic subdivision of the proof into groups of component and noncomponent type, as well as several of the major techniques underlying the analysis, and also includes a brief discussion of the known finite simple groups. It is an excellent introduction to one of the most active areas of mathematical research.

**1.4. Review by: Neil K Dickson.**

*Proceedings of the American Mathematical Society*

**26**(1) (1983), 116-117.

The complexity of the problem of classifying the finite simple groups has fascinated many mathematicians and the quantity of the work carried out has necessitated the production of a number of surveys. These range from Daniel Gorenstein's book, Finite Simple Groups (1968), which can be regarded as the finite simple group theorist's bible, to the most recent comprehensive survey "The classification of finite simple groups", also by Daniel Gorenstein, part I of which appeared in

*Bull. American Math. Soc. (New Series)*

**1**(1979), 43-199 and the remainder of which has still to appear. However, most of the surveys have been for the use of experts and there has been a distinct lack of exposition suitable for the wide range of mathematicians who are interested in the subject.

I was therefore pleased to read that Michael Aschbacher's book, which is based on four special lectures delivered at Yale, is "intended for a general mathematical audience". Unfortunately this intention has not been carried out. The author has not thought carefully enough about the existing knowledge of his reader. The result is a rather uneven set of assumptions. Thus the reader finds that he is told more than once the definitions of the terms involution and elementary abelian p-group but is expected to cope without help with a variety of terminology and notation concerning normalizers, commutators and conjugates and to absorb quickly some rather complex ideas. So I disagree with the claim on the back cover that the contents are "accessible to most mathematicians".

**2. Finite Group Theory (1986), by Michael Aschbacher.**

**2.1. From the Publisher.**

During the last 30 years the theory of finite groups has developed dramatically. Our understanding of finite simple groups has been enhanced by their classification. Many questions about arbitrary groups can be reduced to similar questions about simple groups and applications of the theory are beginning to appear in other branches of mathematics. The foundations of the theory of finite groups are developed in this book. Unifying themes include the Classification Theory and the classical linear groups. Lie theory appears in chapters on Coxeter groups, root systems, buildings and Tits systems. There is a new proof of the Solvable Signalizer Functor theorem and a brief outline of the proof of the Classification Theorem itself. For students familiar with basic abstract algebra this book will serve as a text for a course in finite group theory. Finite Group Theory provides the basic background necessary to understand the research literature and apply the theory. It will become the standard basic reference.

**2.2. Review by: Ronald Solomon.**

*American Scientist*

**76**(2) (1988), 213.

The theme of the book is the study of finite groups via their representations: permutation, linear, and adjoint. Most of the material is standard, but the choice of topics is personal, and the proofs are sharp and crisp. Genuinely new is the author's proof of the solvable signalizer functor theorem in chapter 15. Also the treatment in chapter 9 of change of field and of the field of definition of a representation has interesting new features and is good preparation for a study of the author's work on the maximal subgroups of the classical linear groups. Chapter 7 is an excellent presentation of these classical linear groups and their geometries over arbitrary fields, including orthogonal groups in characteristic 2 without extra fuss or fanfare. Combining this with a treatment of Coxeter groups in chapter 10 and BN-pairs in chapter 14, he provides quite an extensive overview of these important groups. There is an excellent brief survey of the classification proof in section 48. The size of the book is misleading. It is crammed with enough material for a solid year's graduate course. Indeed the density makes it a bit forbidding as a text. However, I suggest viewing it as an opportunity for the lecturer to clothe the lean torso of the text in his favourite garb of examples and digressions.

**2.3. Review by; Stephen D Smith.**

*Mathematical Reviews*MR0895134

**(89b:20001)**.

This book is likely to become a standard text for graduate students beginning work in finite group theory. It will also be of interest to a wider audience interested in the foundations of the classification of finite simple groups, a great mathematical achievement completed several years earlier with many essential contributions by the author.

...

As a text, it would probably be most useful at the second-year graduate level, for students contemplating thesis work in finite groups. (Rumours of the death of the field have been greatly exaggerated. The years after the classification have seen vigorous development in areas such as linear representation theory, maximal subgroups, and discrete geometries.) The author indicates as prerequisite just a first algebra course at the level of I N Herstein's

*Topics in algebra*, but probably only a few universities would present this material in an undergraduate course. The appropriate and inevitable comparison is with D Gorenstein's

*Finite groups*[1968], which introduced research-level group theory to several generations of students (including the reviewer). In many respects the book under review can be regarded as a natural successor to Gorenstein's, updated according to the experiences of the intervening years during which simple-group classification progressed from a seemingly distant goal to a reality. ... overall the treatment seems designed rather similarly to equip the student with the basic tools needed for contemporary research, leading up to consideration of a few representative results, with an overview of the simple groups and the classification project.

...

In conclusion we mention one other factor which may attract readers. While in some areas (such as the chapter on $p$-groups) the treatment is more or less standard, in others (for example, the material on Witt's lemma and the classical groups) the development seems to be much more distinctively the author's own. Those who have read his papers will be familiar with his ability to rediscover entire bodies of theory without recourse to the standard references. The elementary material of this book is naturally more accessible, and many mathematicians will want to dip into it just to get a glimpse of the author's deep insights and original approaches, which were crucial in making simple-group classification a reality in our time.

**3. Sporadic Groups (1994), by Michael Aschbacher.**

**3.1. From the Preface.**

The classification of the finite simple groups says that each finite simple group is isomorphic to exactly one of the following: a group of prime order; an alternating group An of degree n; a group of Lie type; one of twenty-six sporadic groups.

As a first step in the classification, each of the simple groups must be shown to exist and to be unique subject to suitable hypotheses, and the most basic properties of the group must be established. The existence of the alternating group An comes for free, while the representation of An on its n-set makes possible a first uniqueness proof and easy proofs of most properties of the group. The situation with the groups of Lie type is more difficult, but while groups of Lie rank 1 and 2 cause some problems, Lie theory provides proofs of the existence, uniqueness, and basic structure of the groups of Lie type in terms of their Lie algebras and buildings.

However, the situation with the sporadic groups is less satisfactory. Much of the existing treatment of the sporadic groups remains unpublished, and the mathematics which does appear in print lacks uniformity, is spread over many papers, and often depends upon machine calculation.

Sporadic groups represents the first step in a program to provide a uniform, self-contained treatment of the foundational material on the sporadic groups. More precisely, our eventual aim is to provide complete proofs of the existence and uniqueness of the twenty-six sporadic groups subject to appropriate hypotheses, and to derive the most basic structure of the sporadics, such as the group order and the normalizers of subgroups of prime order.

While much of this program is necessarily technical and specialised, other parts are accessible to mathematicians with only a basic knowledge of finite group theory. Moreover, some of the sporadic groups are the automorphism groups of combinatorial objects of independent interest, so it is desirable to make this part of the program available to as large an audience as possible. For example, the Mathieu groups are the automorphism groups of Steiner systems and Golay codes while the largest Conway group is the automorphism group of the Leech lattice.

Sporadic groups begins the treatment of the foundations of the sporadic groups by concentrating on the most accessible chapters of the subject. It is our hope that large parts of the book can be read by the nonspecialist and provide a good picture of the structure of the sporadics and the methods for studying these groups. At the same time the book provides the basis for a complete treatment of the sporadics.

The book is divided into three parts: Part I, introductory material (Chapters 1-5); Part II, existence theorems (Chapters 6-11); and Part III uniqueness theorems (Chapters 12-17).

The goal of the existence treatment is to construct the largest sporadic group (the Monster) as the group of automorphisms of the Griess algebra. Twenty of the twenty-six sporadic groups are sections of the Monster. We establish the existence of these groups via these embeddings. To construct the Griess algebra one must first construct the Leech lattice and the Conway groups, and to construct the Leech lattice one must first construct the Mathieu groups, their Steiner systems, and the binary Golay code.

There are many constructions of the Mathieu groups. Our treatment proceeds by constructing the Steiner systems for the Mathieu groups as a tower of extensions of the projective plane of order 4. This method has the advantage of supplying the extremely detailed information about the Mathieu groups, their Steiner systems, and the Golay code module and Todd module necessary both for the construction of the Leech lattice and the Griess algebra, and for the proof of the uniqueness of various sporadics.

**4. 3-transposition groups (1997), by Michael Aschbacher.**

**4.1. Review by: Jonathan I Hall.**

*Mathematical Reviews*MR1423599

**(98h:20024)**.

A group is called a 3-transposition group if it is generated by a conjugacy class of 3-transpositions, that is, a class of elements of order 2 with pairwise products of order at most 3. The study of such groups was initiated by Bernd Fischer in the late 1960s. Around 1970 he proved the most striking theorem in the subject, classifying all finite 3-transposition groups with trivial centre and simple derived subgroup. The basic examples of 3-tranposition classes are the transpositions of symmetric groups (the motivating example), the transvections of orthogonal and symplectic groups over $GF(2)$ and unitary groups over $GF(4)$, and the reflections of orthogonal groups over $GF(3)$. In his classification, Fischer discovered only three additional groups, which he denoted $M(22), M(23)$, and $M(24)$ (because of connections with the corresponding Mathieu groups). The first two are simple while the third has simple derived subgroup of index 2 (here denoted $F24$). These are three of the twenty-six sporadic finite simple groups.

Fischer's classification is important not only for the discovery of these interesting new groups, but also for the basic philosophy of his proof in which he used elementary group-theoretic properties of the 3-transposition class to reconstruct the natural geometry on which each of the example groups acts. This was the beginning of what Gorenstein referred to as "internal geometric analysis" and had a profound effect on the classification of finite simple groups as it unwound over the next ten years. The author, Michael Aschbacher, was one of the major players in the classification. Recently he has embarked upon a program to present the sporadic groups and to prove their uniqueness in an appropriately consistent and useful manner. This book is his second devoted largely to this project, the first being Sporadic groups [1994].

...

Aschbacher's book is not uniformly easy to read, but he has worked at making it understandable and complete. He has kept the prerequisites to a minimum, particularly in the elementary first part. When he does need external material, he either quotes from one of his two previous books or provides the material himself. This is always done in an efficient and enlightening manner. The whole book is recommended for all people interested in finite simple groups, and the first section is recommended for anyone who wishes for some insight into this fascinating and important field.

**4.2. Review by: Jonathan I Hall.**

*Bulletin of the American Mathematical Society*

**53**(2) (1998), 161-169.

We begin with a quote from Daniel Gorenstein, the prime force behind the classification of finite simple groups:

To the non-expert, the name Bernd Fischer is known solely for its connection with a number of sporadic simple groups, but to the practitioner, he is recognised as the founder of internal geometric analysis ... a fundamental general technique for studying simple groups, which can reasonably be regarded as second in importance for the classification only to local group-theoretic analysis.

In contrast to local analysis, Fischer's work appears as a personal creation, and aspects of its subsequent development have an almost magical quality. ... in Fischer's case, originality begins with the very question he raised:

Which finite groups can be generated by a conjugacy class of involutions, the product of any two of which has order 1, 2, or 3?

Here an involution is an element of order 2, and a class as described is called a conjugacy class of 3-transpositions. The group it generates is then a 3-transposition group. Aschbacher's excellent book presents a proof of Fischer's elegant and remarkable theorem of classification and goes on to describe many properties of the groups that arise, particularly the three sporadic Fischer groups. The book is bro- ken into three parts. The first presents Fischer's classification, the second deals with existence and uniqueness issues for the sporadic Fischer groups, and the third details their local structure.

...

In summary, Aschbacher's excellent book is required for all with specific interest in finite simple groups, but those with a less direct connection will also find much of value. In particular, they will find the only available collected treatment of Fischer's classification of 3-transposition groups, one of the most important and historic results from the theory of finite simple groups, presented lucidly by one of the most original minds in that area.

**5. Finite Group Theory. Second Edition (2000), by Michael Aschbacher.**

**5.1. From the Preface.**

Finite group theory develops the foundations of the theory of finite groups. It can serve as a text for a course on finite groups for students already exposed to a first course in algebra. For the reader with some mathematical sophistication but limited knowledge of finite group theory, the book supplies the basic background necessary to begin to read journal articles in the field. It also provides the specialist in finite group theory with a reference in the foundations of the subject.

The second edition of Finite group theory has been considerably improved, with a completely rewritten Chapter 15 considering the 2-signalizer functor theorem and the addition of an appendix containing solutions to exercises.

**5.2. Review by: Brian Denton.**

*The Mathematical Gazette*

**85**(504) (2001), 546-547.

This book was first published in 1986. It is number 10 in the prestigious series of pure mathematics books 'Cambridge studies in advanced mathematics'.

This book is not for the faint-hearted. It starts by assuming that the reader has some knowledge of a substantial nature. For example it says that Herstein's book

*Topics in algebra*would be a typical text for an introductory course and goes on to say that Lang's book

*Algebra*has deeper topics that are needed. With these assumptions readers know they are in for a hard time!

The second edition has substantial changes in it making it more useful for present-day group theorists and researchers. The book is intended as a basic text as well as a work of reference. If it is used as a text the lecturer would have to supply many of the proofs and other missing pieces. It would not be at the undergraduate level in most universities. The sixth page, for example, has an introduction to Categories, the previous five pages having worked through elementary group theory which includes homomorphisms, isomorphisms, normal subgroups, direct products of subgroups, normalisers, centralisers and $p$-groups, to name but a few.

Chapter headings will give a flavour of what is to come: Permutation representations; Representations of groups on groups; Linear representations; Permutation groups; Extensions of groups and modules; Spaces with forms; $p$-groups; Change of field of a linear transformation; Presentation of groups; The generalised Fitting subgroup; Linear representations of finite groups; Transfer and fusion; The geometry of groups of Lie type; Signaliser functors; Finite simple groups.

Here we have the essence of the book, the classification of finite simple groups. Some will recognise the name of Michael Aschbacher (the Shaler Arthur Hanisch Professor of Mathematics at the California Institute of Technology) as a great contributor to this classification. Cambridge University Press have published two other books by him:

*Sporadic Groups*(1994) and 3

*-Transposition Groups*(1997). On my shelf I have another of his books [

*The finite simple groups and their classification*], a slim volume written when the final stages of the classification theorem were looming large. The final chapter in the book under review has an outline of the classification theorem, thus bringing it right up-to-date.

This book is a must for devotees of group theory and in particular the classification of finite simple groups.

**6. The classification of quasithin groups. I. Structure of strongly quasithin K-groups (2004), by Michael Aschbacher and Stephen D Smith.**

**6.1. From the Publisher.**

Around 1980, G Mason announced the classification of a certain subclass of an important class of finite simple groups known as "quasithin groups". The classification of the finite simple groups depends upon a proof that there are no unexpected groups in this subclass. Unfortunately Mason neither completed nor published his work. In the Main Theorem of this two-part book (Volumes 111 and 112 in the AMS series, Mathematical Surveys and Monographs) the authors provide a proof of a stronger theorem classifying a larger class of groups, which is independent of Mason's arguments. In particular, this allows the authors to close this last remaining gap in the proof of the classification of all finite simple groups.

An important corollary of the Main Theorem provides a bridge to the program of Gorenstein, Lyons, and Solomon (Volume 40 in the AMS series, Mathematical Surveys and Monographs) which seeks to give a new, simplified proof of the classification of the finite simple groups.

Part I contains results which are used in the proof of the Main Theorem. Some of the results are known and fairly general, but their proofs are scattered throughout the literature; others are more specialised and are proved here for the first time.

The book is suitable for graduate students and researchers interested in the theory of finite groups.

**6.2. From the Preface.**

The classification of the quasithin simple groups of even characteristic can be thought of as roughly one fourth of the classification of the finite simple groups. The two volumes in this series provide the first proof that each group in this class is a known simple group. This result closes a gap in the classification of the finite simple groups which has existed for over twenty years.

In addition the series is part of an ongoing effort to reorganise and simplify the original proof of the classification of the finite simple groups, and to write the proof down carefully in a relatively short number of pages (e.g., less than ten thousand). The effort includes the "GLS" series of Gorenstein, Lyons, and Solomon, which at the moment consists of the five volumes, but it also includes smaller projects.

A detailed discussion of these matters appears in the introductions to each of the two volumes in our series. Roughly speaking, the first volume consists of fairly general results on finite groups (with emphasis on quasithin groups) which serve as the foundation for the classification of the quasithin groups. The second volume consists of a proof that the groups listed in our Main Theorem are the simple quasithin groups of even characteristic, all of whose proper simple sections are known simple groups.

We would be remiss if we did not acknowledge the assistance of a number of people:

During the many years we have worked on this project, each of us visited and benefited from the hospitality of many universities and faculties, whose assistance we gratefully acknowledge.

...

Most importantly, we would like to thank John Thompson for reading large portions of the two volumes and suggesting numerous improvements and simplifications. The authors, and indeed the finite group theory community, owe him a great debt of gratitude for his selfless work benefiting us all.

**6.3. Review by Ronald Solomon.**

*Mathematical Reviews*MR2097623

**(2005m:20038a)**.

In 1983, Danny Gorenstein announced the completion of the classification of the finite simple groups. All of the major constituent theorems were published by 1983 with one exception. This exception was at last removed and the classification has now been completed with the publication of the two monographs under review. These volumes, classifying the quasithin finite simple groups of even characteristic, are thus a major milestone in the history of finite group theory.

...

The book is written with remarkable care and clarity, especially in view of its extraordinary length and depth. It is an amazing tour de force. Volume I has been read carefully by John Thompson.

**6.4. Review by Ronald Solomon.**

*Bulletin of the American Mathematical Society*

**43**(1) (2006), 115-121.

In 1983, Danny Gorenstein announced the completion of the Classification of the Finite Simple Groups. This announcement was somewhat premature. The Classification of the Finite Simple Groups was at last completed with the publication in 2004 of the two monographs under review here. These volumes, classifying the quasithin finite simple groups of even characteristic, are a major milestone in the history of finite group theory. It is appropriate that the great classification endeavour, whose beginning may reasonably be dated to the publication of the monumental Odd Order Paper [FT] of Feit and Thompson in 1963, ends with the publication of a work whose size dwarfs even that massive work.

...

Both of these volumes are written beautifully and with great care. Part I is a particular gem, being an encyclopaedic exposition of the principal methods for treating groups of even characteristic: failure of factorisation analysis, pushing up, weak closure methods and amalgam methods. Stellmacher's fundamental qrc-Lemma is presented. This focusses attention on certain important classes of modules for finite simple groups, and these modules are discussed in considerable detail. Anyone wishing to gain mastery of these important tools would do well to read Part I, which could serve as a text for an advanced graduate course on local group theory.

**7. The classification of quasithin groups. II. Main theorems: the classification of simple QTKE-groups (2004), by Michael Aschbacher and Stephen D Smith.**

**7.1. From the Publisher.**

Around 1980, G Mason announced the classification of a certain subclass of an important class of finite simple groups known as "quasithin groups". The classification of the finite simple groups depends upon a proof that there are no unexpected groups in this subclass. Unfortunately Mason neither completed nor published his work. In the Main Theorem of this two-part book (Volumes 111 and 112 in the AMS series, Mathematical Surveys and Monographs) the authors provide a proof of a stronger theorem classifying a larger class of groups, which is independent of Mason's arguments. In particular, this allows the authors to close this last remaining gap in the proof of the classification of all finite simple groups.

An important corollary of the Main Theorem provides a bridge to the program of Gorenstein, Lyons, and Solomon (Volume 40 in the AMS series, Mathematical Surveys and Monographs) which seeks to give a new, simplified proof of the classification of the finite simple groups.

Part II of the work (Volume 112) contains the proof of the Main Theorem, and the proof of the corollary classifying quasithin groups of even type.

The book is suitable for graduate students and researchers interested in the theory of finite groups.

**7.2. Review by Ronald Solomon.**

*Mathematical Reviews*MR2097624

**(2005m:20038b)**.

In 1983, Danny Gorenstein announced the completion of the classification of the finite simple groups. All of the major constituent theorems were published by 1983 with one exception. This exception was at last removed and the classification has now been completed with the publication of the two monographs under review. These volumes, classifying the quasithin finite simple groups of even characteristic, are thus a major milestone in the history of finite group theory.

...

The book is written with remarkable care and clarity, especially in view of its extraordinary length and depth. It is an amazing tour de force. Volume II has been checked by a team of referees.

**8. The classification of finite simple groups. Groups of characteristic 2 type (2011), by Michael Aschbacher, Richard Lyons, Stephen D Smith and Ronald Solomon.**

**8.1. From the Publisher.**

The book provides an outline and modern overview of the classification of the finite simple groups. It primarily covers the "even case", where the main groups arising are Lie-type (matrix) groups over a field of characteristic 2. The book thus completes a project begun by Daniel Gorenstein's 1983 book, which outlined the classification of groups of "noncharacteristic 2 type".

However, this book provides much more. Chapter 0 is a modern overview of the logical structure of the entire classification. Chapter 1 is a concise but complete outline of the "odd case" with updated references, while Chapter 2 sets the stage for the remainder of the book with a similar outline of the "even case". The remaining six chapters describe in detail the fundamental results whose union completes the proof of the classification theorem. Several important subsidiary results are also discussed. In addition, there is a comprehensive listing of the large number of papers referenced from the literature. Appendices provide a brief but valuable modern introduction to many key ideas and techniques of the proof. Some improved arguments are developed, along with indications of new approaches to the entire classification - such as the second and third generation projects - although there is no attempt to cover them comprehensively.

The work should appeal to a broad range of mathematicians - from those who just want an overview of the main ideas of the classification, to those who want a reader's guide to help navigate some of the major papers, and to those who may wish to improve the existing proofs.

**8.2. From the Preface.**

The present book, "The Classification of Finite Simple Groups: Groups of Characteristic 2 Type", completes a project of giving an outline of the proof of the Classification of the Finite Simple Groups (CFSG). The project was begun by Daniel Gorenstein in 1983 with his book [Daniel Gorenstein, The classification of finite simple groups (1983)] - which he subtitled "Volume 1: Groups of Noncharacteristic 2 Type". Thus we regard our present discussion of groups of characteristic 2 type as "Volume 2" of that project.

The Classification of the Finite Simple Groups (CFSG) is one of the premier achievements of twentieth century mathematics. The result has a history which, in some sense, goes back to the beginnings of proto-group theory in the late eighteenth century. Many classic problems with a long history are important more for the mathematics they inspire and generate, than because of interesting consequences. This is not true of the Classification, which is an extremely useful result, making possible many modern successes of finite group theory, which have in turn been applied to solve numerous problems in many areas of mathematics.

A theorem of this beauty and consequence deserves and demands a proof accessible to any mathematician with enough background in finite group theory to read the proof. Unfortunately the proof of the Classification is very long and complicated, consisting of thousands of pages, written by hundreds of mathematicians in hundreds of articles published over a period of decades. The only way to make such a proof truly accessible is, with hindsight, to reorganise and rework the mathematics, collect it all in one place, and make the treatment self-contained, except for some carefully written and selected basic references. Such an effort is in progress in the work of Gorenstein, Lyons, and Solomon (GLS) in their series beginning with [Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups (1994)], which seeks to produce a second-generation proof of the Classification.

However in the meantime, there should at least be a detailed outline of the existing proof, that gives a global picture of the mathematics involved, and explicitly lists the papers which make up the proof. Even after a second-generation proof is in place, such an outline would have great historical value, and would also provide those group theorists who seek to further simplify the proof with the opportunity to understand the approach and ideas that appear in the proof. That is the goal of this volume: to provide an overview and reader's guide to the huge literature which makes up the original proof of the Classification.

Soon after the apparent completion of the Classification in the early 1980s, Daniel Gorenstein began a project aimed at giving an outline of the original proof. He provided background in a substantial Introduction [Daniel Gorenstein,

*Finite simple groups*(1982)], in particular discussing the partition of simple groups into groups of odd characteristic and groups of characteristic 2 type. Then in Volume 1 [Daniel Gorenstein,

*The classification of finite simple groups*(1983)] he described the treatment of the groups of odd characteristic in detail. However he did not complete the rest of his project, in part because the proof for groups of characteristic 2 type remained incomplete, specifically that part of the proof treating the quasithin groups undertaken by Mason. This gap was recently filled by the Aschbacher-Smith classification of the quasithin groups. Hence it is now possible to finish Gorenstein's project by outlining the proof for the groups of characteristic 2 type. We accomplish that goal here, adopting his title, and regarding the work as "Volume 2" in the series.

While we recommend that the interested reader consult Gorenstein's books, we also intend that our treatment should be sufficiently self-contained that those works will not be a prerequisite. Therefore in Chapter 1, we supply an overview of the treatment of the groups of odd characteristic, which is much briefer than Gorenstein's detailed treatment.

In fact, throughout our exposition, we will be less detailed than Gorenstein, since we believe that a briefer outline of the main steps will be more accessible and useful to most readers. On the other hand, we are careful to honour the important fundamental goal of explicitly listing those works in the literature which make up the proof that all simple groups of characteristic 2 type are known.

**8.3. Review by: Tanaka, Yasuhiko.**

*Mathematical Reviews*MR2778190

**(2012d:20023)**.

The book consists of eight chapters plus two appendices covering the background. People having an interest in finite groups probably know that simple groups are divided into two types by inner structure, and that the existing classification proceeds separately for each type of group. Nevertheless, the first introductory chapter, and the two subsequent chapters as well, is filled with the idea that a unified approach could be applicable and should be developed for both types of groups. The existing proof is in fact a large collection of contributions by a lot of mathematicians. Apart from the historical development and the logical ordering, they adopt a path of reorganisation and simplification in order to clarify the essence of the proof. Though they use the word "oversimplification" like an excuse, it is quite desirable for most readers so that they can grasp the entire idea of the monstrous proof. The three chapters are not just for the groups of characteristic 2 type. Rather, the purpose of the authors is to convey their opinion, with a bit of emotion, that the existing proof is more structured and sophisticated than generally accepted. Although some readers might want to hurry to the subsequent chapters, I would recommend that they devote more time to these chapters to taste their flavour well.

**9. Fusion systems in algebra and topology (2011), by Michael Aschbacher, Radha Kessar and Bob Oliver.**

**9.1. From the Publisher.**

A fusion system over a $p$-group $S$ is a category whose objects form the set of all subgroups of $S$, whose morphisms are certain injective group homomorphisms, and which satisfies axioms first formulated by Puig that are modelled on conjugacy relations in finite groups. The definition was originally motivated by representation theory, but fusion systems also have applications to local group theory and to homotopy theory. The connection with homotopy theory arises through classifying spaces which can be associated to fusion systems and which have many of the nice properties of p-completed classifying spaces of finite groups. Beginning with a detailed exposition of the foundational material, the authors then proceed to discuss the role of fusion systems in local finite group theory, homotopy theory and modular representation theory. This book serves as a basic reference and as an introduction to the field, particularly for students and other young mathematicians.

**9.2. Review by: Charles W Eaton**.

*Mathematical Reviews*MR2848834

**(2012m:20015)**.

The theory of abstract fusion systems, whilst having old and well-established roots in group theory and topology, is a young subject progressing rapidly and yet to fully settle on all of its basic definitions. Early work on fusion in groups included results by Burnside and Frobenius concerning instances of control of fusion in Sylow $p$-subgroups of finite groups and, much later, Alperin's fusion theorem provided the key idea. Its main themes developed from modular representation theory, the local analysis in the classification of finite simple groups and work by Cartan and Eilenberg and others relating fusion in Sylow p-subgroups to group cohomology, arriving with Puig's definition of a Frobenius category (now more commonly referred to as a saturated fusion system) on a finite $p$-group $S$ without relation to a group containing $S$. This very welcome book originates from a workshop on fusion systems held at the University of Birmingham in 2007 and is one of the first two texts on the subject, appearing at approximately the same time as [D A Craven, The theory of fusion systems, 2011]. The three authors each gave a series of lectures on a different aspect of the theory, and this is reflected in the structure of the book, which consists of four main parts: the first a general introduction and the other three surveys are handled by each author individually. The book under review complements Craven's detailed textbook very nicely, being a wide-ranging and reasonably comprehensive survey of a large part of the subject as it stood in about 2011 (inevitably there is somewhat less detail on developments since 2008). Each author gives a generous and detailed list of open problems or a discussion of possible directions of research at the end of his or her respective sections. The book serves as an excellent introduction to the subject and a guide up to the cutting edge of current research.

The authors are sympathetic to the problem that definitions are not yet settled, and take care to reconcile alternative definitions. Some particularly troublesome definitions are those of normal fusion subsystems and solvable fusion systems. In the case of normal subsystems, the authors define normal, weakly normal and invariant subsystems, these accounting (up to equivalence) for the definitions found in the literature. Definitions are given of solvable and Puig solvable saturated fusion systems, the second condition stronger than the first.

**10. Quaternion fusion packets (2021), by Michael Aschbacher.**

**10.1. From the Publisher.**

Let $p$ be a prime and $S$ a finite $p$-group. A $p$-fusion system on $S$ is a category whose objects are the subgroups of $S$ and whose morphisms are certain injective group homomorphisms. Fusion systems are of interest in modular representation theory, algebraic topology, and local finite group theory.

The book provides a characterisation of the 2-fusion systems of the groups of Lie type and odd characteristic, a result analogous to the Classical Involution Theorem for groups. The theorem is the most difficult step in a two-part program. The first part of the program aims to determine a large subclass of the class of simple 2-fusion systems, while part two seeks to use the result on fusion systems to simplify the proof of the theorem classifying the finite simple groups.

**10.2. Review by: Ronald Solomon.**

*Mathematical Reviews*MR4240597.

As Aschbacher indicates in his Introduction to the volume under review, his work was inspired by a 1974 series of AMS lectures by John G. Thompson proposing an approach to proving Thompson's B-Conjecture.

Aschbacher has done a fine job of providing a careful overview and development of background results with supporting examples in Chapters 1 through 5 of this volume, before plunging into the proofs of the main supporting Theorems 2-8 in Chapters 6-15, and concluding with the proofs of Theorem 1 and the Main Theorem in Chapter 16. The pace and intensity of work on the classification of the finite simple groups during the 1970s were extraordinary. In consequence, exposition often suffered. Aschbacher has ably remedied this in the current volume, for which he deserves much thanks.

Last Updated March 2024