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YouTube LECTURE COURSES/TEXTS : “Humanity’s Greatest Intellectual Achievement : Classification Theorem of the Finite Simple Groups (CTFSG)”

YouTube LECTURE COURSES/TEXTS : “Humanity’s Greatest Intellectual Achievement : Classification Theorem of the Finite Simple Groups (CTFSG)”

For a concrete course (actually, a series of courses) on the CTFSG, go to the bottom of this page.

Anyone who knows about the Classification Theorem of the Finite Simple Groups (CTFSG) knows that it is enormous. Gorenstein, the genius who first conceived a concrete proposal on how to classify the finite simple groups, described the effort he organized, which really only finished in 2004, as “The 30 Years War” (making an analogy with the 30 Years War in Germany in the 1600s between the Catholics and the Protestants).

It was a humungous effort, involving hundreds of research pure mathematicians, 10,000-15,000 pages of research journals, hundreds of articles, and took about 30 years, mostly during the 50s, 60s, and 70s. There was a gap (the classification of the quasi-thin simple groups) that was not tidied up until 2004 (which took 1300 pages!)

In the early 80s, most simple group researchers thought that the back of the problem of classification had been broken, so immediately a “revision proof” was begun to unify, simplify and condense it to about 3000-5000 pages, i.e. into roughly a dozen books that could be placed on a single book shelf. At the time of writing (2016) 6 such volumes have been published, of roughly 400 pages each.

Each book would correspond to a lecture course, so with about a dozen or more courses, the CTFSG could be covered. That is my ambition, and it is ambitious, but doable, I think.

I hope to live another 20-30 years. I’ll be 70 in 2017, so I’m thinking long term. My father who is still alive (in 2016) is 96 and still in good health, and I think I have his longevity genes because I always look a decade younger than I am, so I may have enough time to complete this very ambitious task?!


Claiming that the CTFSG is humanity’s greatest intellectual achievement is a VERY strong statement, so needs some justification. I’ve heard people say that in their view the greatest, most significant idea that humanity has had was Darwin’s “natural selection.” Indeed, that idea has profoundly shaped the way we see ourselves and the world, but it was not humanity’s greatest INTELLECTUAL achievement, requiring enormous intellects (i.e. male genii) to make humungous intellectual efforts to achieve their goal. The major players of the CTFSG were super genii, guys like John Thompson (who is considered a mathematical god amongst his peers) and Michael Aschbacher, who almost single handedly, turned Gorenstein’s vision into a concrete reality. These two math gods are unknown to the general public, because their works are so far above the IQ levels of the vast majority of humanity, that most of humanity will never even know about them.

The CTFSG is a work of genius, created by the planet’s smartest men. Thompson and Aschbacher must be in the top handful of the planet’s smartest humans. When one studies their works, one is awe struck, that the human mind is capable of creating such wonders, such cathedrals of mathematical logic. One is in AWE.

I cant think of anything that can match the CTFSG in terms of intellectual achievement, and I know a lot of math and physics. It’s what I do all day in my ARCing (after retirement careering). The pure mathematicians and the theoretical physicists are the smartest profs on campus (as shown by psychometricians in the 1950s.) Pure mathematicians generally consider that the CTFSG is the toughest task that math has ever attempted. It attracted the most brilliant of the brilliant, and its stars, Gorenstein, Thompson, Aschbacher, are considered to be math gods.

In light of the above, I don’t think its too much of an exaggeration to say that the CTFSG is humanity’s greatest intellectual achievement.


Why my fascination with the CTFSG? Because, the more I study pure math and math physics, the more convinced I become that the laws of physics have been designed, i.e. architected by some hyper intelligence, who was a mathematician, who used the simple groups to design the laws of physics. Anyone who studies modern particle physics knows that the laws of interaction between particles obey symmetry principles, i.e. they use some of the simple groups (the so-called “gauge groups”)  in the very interaction laws themselves (see my courses on particle physics).

For example, why on earth, should the so called elementary particles be classifiable according to various representations of Lie algebras (e.g. the Lie algebra of the SU(3) Lie group, or why does the largest of the finite simple groups, the “Monster” with nearly 10 to power 54 elements, play a crucial role in the mathematical model of a 26 dimensional string theory, which got its discoverer, Borcherds, a Fields Medal in the 90s?

It looks as though the deity (creator, architect) of our universe (i.e. the laws of physics)  used finite simple groups as the intellectual tool to design it. So, in a sense, the FSGs preexisted the deity, so in some sense, are more fundamental than the deity. If the deity wanted to use symmetry in its designs for the universe, then it was obliged to use these symmetry groups, because THERE ARE NO OTHERS, as stipulated by the CTFSG.

This idea fascinates me, so provided one major motivation to study them and then to make lecture video courses on them, hoping to generate the same fascination in other students of the CTFSG. By understanding the finite simple groups, and their classification, one is probably understanding the intellectual tools that the deity used to construct its universe. One is, in a manner of speaking, “speaking the language of the deity”   or as Einstein put it “I want to know god’s thoughts”, so if I’m right, the deity thought in terms of the finite simple groups and was obliged to choose from a limited range of them, because the CTFSG proves that there are a limited number of them, with the implication that “Math is a higher god than the deity, the creator of the universe!”

(The only case I’ve come across where the situation is reversed was in Carl Sagan’s science fiction novel “Contact” in which he talks about a hyper advanced species who built worm hole transport systems across the galaxies and designed the laws of space so that the value of pi if expanded to enough places, displayed a circle, making it the “signature of the artist” (i.e. the signature of the designer of the universe.) Needless to say, I admire Sagan’s extraordinary vision.


There is another main motive I have. I call it “Building CATHEDRALS of mathematical logic” The CTFSG is the most beautiful and most powerful piece of mathematics I have ever seen. I call it a cathedral of mathematical construction. It is dense, tight, enormous, extremely rich in structure (e.g. look at the Monster with its 10 to power 54 elements) and has the capacity, for anyone with a good math brain, to become totally engrossed by.

I notice, that when I get heavily involved with it, I begin to feel a happiness that ordinary living, with all its daily frustrations, does not evoke. This total engrossing provides enormous satisfaction as one masters the CTFSG step by step, absorbing and memorizing its huge pile of definitions, and then playing with them, to build massive mathematical structures, i.e. “cathedrals” of logic, mathematical logic.


A finite SIMPLE group is one which has no proper normal subgroups (i.e. it cannot be deconstructed into a smaller normal subgroup and its corresponding smaller factor/quotient group). To classify the finite simple groups means to make a list of them all, and then to PROVE that that list is complete, i.e. that any finite simple group one comes across has to be isomorphic to one of the finite simple groups in the list.

These normal and factor groups that are smaller than the original group can perhaps themselves be deconstructed into yet smaller normal subgroups and factor subgroups, etc, until one eventually reaches undeconstructible subgroups, i.e. the finite  simple groups. They are simple in the sense of being undeconstructible, not simple in the sense of being easy to understand. In the case of the Monster simple group, it has links with number theory and physics that remain mysterious to this day.

Simple groups are thus analogous to the primes for integers, or the atoms for molecules, i.e. they are the building blocks for all finite groups. There are famous theorems in group theory showing this, i.e. that a given finite group can be deconstructed, broken down, into components that are simple groups, in a way similar to the famous “fundamental theorem of arithmetic” which says that any integer can be uniquely decomposed into a product of prime numbers raised to integer powers.

Since understanding the laws of physics implies understanding symmetries, to be able to express the mathematics of these symmetries, one needs group theory, and if the groups are finite (with a finite number of elements or members) then understanding the finite groups ultimately boils down to understanding their “simple group” building blocks.

With roughly one book (of the dozen or so) of the simplified, condensed, compact revision CTFSG, per course, I anticipate I will need about a dozen or so courses, probably more, but that is doable over a time frame of several decades. So I’m thinking long term. For someone of nearly 70, another decade or so (provided I stay healthy, my top priority at my stage in life) is not a big deal. Another decade would be just 1/8 of my life!



What now follows is a list of the text books and journal articles that I plan to use in teaching these YouTube lecture courses. I will hunt down links to the full contents of each if possible (using Google Scholar, Google, Research Gate, individual authors’ websites, etc) and provide the link to each publication, so that students can start studying the CTFSG immediately, and not have to wait for years for me to make the video courses.


Popular Text

A popular book, written at undergrad level, that got me interested in the whole CTFSG enterprise years ago, was –

Ronan, “Symmetry and the Monster, One of the Greatest Quests of Mathematics” (2006) (undergrad level)  (unfree)  (free)  (free)  (free)  (free)



(full text) links from

Here is a list of PRELIMINARY text books that I will give courses on. They provide the tools and knowledge as prerequisites for the tougher PhD2 level material that comes later.

Robinson, “A Course in the Theory of Groups” (1982) (M2 level)  (unfree)  (full text tick selected book then download it);

Kurzweil, Stellmacher, “The Theory of Finite Groups : An Introduction” (2004) (M2 level)  (unfree)  (full text);

The following preliminary text books are frequently cited in the (six) “revision proof” volumes Vol.1 – Vol.6 below, that I will give courses on (most of them) if I can live long enough. There are 17 of them.

Gorenstein, “Finite Groups” (1968) (PhD1 level)  (unfree)  (free)  (free)  (full text tick selected book then download it);

Aschbacher, “Finite Group Theory” (1986) (PhD1 level)  (unfree)  (full text tick selected book then download it);

Huppert, “Endliche Gruppen I” (1967) (Ph1 level)  (unfree)

Huppert-Blackburn, “Finite Groups III” (1985) (PhD1 level)  (unfree)

Suzuki, “Group Theory I” (1982) (M2 level)  (unfree)

Suzuki, “Group Theory II” (1986) (PhD1 level)  (unfree)

Isaacs, “Character Theory of Finite Groups” (1976) (M2-PhD1 levels)  (unfree)  (free)  (full text tick selected book then download it);

Feit, “The Representation Theory of Finite Groups” (1982) (PhD1 level)  (unfree)

Gorenstein, Lyons, “The Local Structure of Finite Groups of Characteristic 2 Type” (1983) (PhD2 level)   (unfree), (free), (free); (Note, this text does NOT include the classification of the quasi-thin finite simple groups, a task not completed until 2004.)

Bender , Glauberman, “Local Analysis for the Odd Order Theorem” (1994) (PhD1 level)  (unfree)

Aschbacher, “3-Transposition Groups” (1997)  (PhD1 level)  (unfree)

Aschbacher, “Sporadic Groups” (1994) (PhD1 level)  (unfree)  (free)  (free)  (free)  (free) (free)  (free)  (free)  (free);

Carter, “Simple Groups of Lie Type” (1972) (PhD1 level)  (unfree)  (free), (full text tick selected book then download it);

Conway et al, “Atlas of Finite Groups” (1985) (PhD2 level)  (unfree)

Dieudonne, “La Geometrie des Groupes Classiques” (1955) (PhD1 level)  (unfree)

Steinberg, “Lectures on Chevalley Groups” (1968) (PhD1 level)  (unfree)

Steinberg, “Endomorphisms of Linear Algebraic Groups” (1968) (PhD1 level)  (unfree)


Here is a list of the main texts that provide an OVERVIEW (but not the detailed proofs) of the CTFSG.

Gorenstein, “Finite Simple Groups, An Introduction to their Classification” (1982) (PhD1 level)  (unfree) (free), (free), (free), (free), (free), (free), (free), (free), (free), (free), (free), (free), (free), (free), (free), (free), (free), (free), (free), (free), (free), (free), (free);

Gorenstein, “The Classification of the Finite Simple Groups : Vol. 1, Groups of Non-Characteristic 2 Type” (1983) (PhD1 level)  (unfree) (free) (free);

Aschbacher et al, “The Classification of the Finite Simple Groups : Vol. 2, Groups of Characteristic 2 Type” (2011) (PhD1 level)  (unfree), (free), (free), (free), (free), (free), (free), (free);


Here is a list of main texts that provide parts of the detailed proofs on the CTFSG

Aschbacher, Smith, “The Classification of the Quasithin Groups, I. Structure of Strongly Quasithin K-Groups” (2004)  (PhD2 level)  (unfree), (free), (free), (full text);

Aschbacher, Smith, “The Classification of the Quasithin Groups, II. Main Theorems : The Classification of Simple QTKE-Groups” (2004)  (PhD2 level)  (unfree), (free), (free), (full text);

Here is a list of the first 6 volumes that have appeared so far on the “revision proof” of the CTFSG, by Gorenstein (who died in 1992), Lyons, and Solomon (GLS)

Vol.1  Gorenstein, Lyons, Solomon, “The Classification of the Finite Simple Groups, Part I, Ch.1 : Overview; Ch.2 : Outline of Proof” (1994)  (PhD2 level)  (unfree)  (full text tick selected book then download it);

Vol.2  Gorenstein, Lyons, Solomon, “The Classification of the Finite Simple Groups, Part I, Ch.G, General Group Theory” (1996)  (PhD2 level)  (unfree)  (full text tick selected book then download it);

Vol.3  Gorenstein, Lyons, Solomon, “The Classification of the Finite Simple Groups, Part I, Ch.A, Almost Simple Groups” (1998)  (PhD2 level)  (unfree)

Vol.4  Gorenstein, Lyons, Solomon, “The Classification of the Finite Simple Groups, Part II, Chs.1-4, Uniqueness Theorems” (1999)  (PhD2 level)  (unfree)

Vol.5  Gorenstein, Lyons, Solomon, “The Classification of the Finite Simple Groups, Part III, Chs.1-6, The Generic case” (2002)  (PhD2 level)  (unfree)

Vol.6  Gorenstein, Lyons, Solomon, “The Classification of the Finite Simple Groups, Part IV, The Special Odd Case” (2005)  (PhD2 level)  (unfree)

Here is a list of supporting texts, that I may use.

Aschbacher, “3-Transposition Groups” (1997)  (PhD1 level)  (unfree)

Aschbacher, “Sporadic Groups” (1994) (PhD1 level)  (unfree)  (free)  (free)  (free)  (free) (free)  (free)  (free)  (free);

Conway, “Sphere Packings, Lattices and Groups” (1999) (PhD1 level)  (unfree)

Michler, “Theory of Finite Simple Groups” (2006) (PhD1 level)  (unfree)

Michler, “Theory of Finite Simple Groups II : Commentary on the Classification Problems” (2010) (PhD1 level)  (unfree)

Griess, “Twelve Sporadic Groups” (1998) (PhD1 level)  (unfree)

Ivanov, “Geometry of Sporadic Groups I, Petersen and Tilde Geometries” (1999) (PhD1 level)  (unfree)

Timmesfeld, “Abstract Root Subgroups and Simple Groups of Lie Type” (2001) (PhD1 level)  (unfree)

Wilson, “The Finite Simple Groups” (2009) (PhD1 level)  (unfree)



Over the months, I will build up an e-library of full-content electronic CTFSG journal articles, to help students of the CTFSG, to have easy, clickable access to them. There will be several hundred of them, enough to be challenging, but not so many, that one feels drowned by their number.

CTFSG Journal Articles e-Library  (link)     (currently doing)


(full text) links from

The CTFSG is the deepest, richest, most powerful and most beautiful branch of math that I have ever come across. I also believe, given the enormous effort involved and the number of the smartest humans on the planet who created it, that it is also the greatest intellectual achievement in human history.

Studying the CTFSG totally engulfs me. I feel I’m participating in the construction of a “CATHEDRAL of mathematical logic.” I would love to pass on this feeling to the planet, by giving a series of courses on the CTFSG, to students around the world, who love math, and love getting their mathematical teeth into some real meat. So if you’re a real “math brain” with an IQ in the top percentile (the top 1%) and love the rigor and power of the logic of group theory, then perhaps committing yourself to the study of the CTFSG is the thing for you. It will take you years (40+ courses), so is not a decision to be taken lightly, but on the other hand, you will obtain the satisfaction of having mastered “humanity’s greatest intellectual achievement.”

What now follows is a concrete suggestion for a series of courses on the CTFSG, involving the texts and research papers needed. You can start studying these texts/papers NOW. You don’t have to wait many years for me to reach the filming of PhD1 and PhD2 level video courses. (I’m planning to give about 120 YouTube video courses in Pure Math, and Math Physics, staring at undergrad level and working my way up, M1, M2, PhD1, PhD2 levels. For a concrete list of these courses and their (full text) links to their text books click (here)).

If you’re what I call an “alfa” (alpha) i.e. with an IQ in the top percentile, and love math, then you don’t really need the YouTube video courses I eventually will make on the CTFSG. Its possible I might die before I reach the PhD2 level courses. All you need to be provided with are the (full text) text books and the research papers, then you can teach yourself. If you’re an alfa, you have the brains enough to be able to do that. So here is a list of the texts and their (full text) links so that you can start studying NOW.

Pure math, and especially the CTFSG, is highly sequential in terms of its knowledge prerequisites. You need to study undergrad group theory to be able to understand M1 group theory, which is a prerequisite for M2 level group theory, which is needed to understand the CTFSG at PhD1 and PhD2 levels. So I give now a list of text books you will need to study, and you will need to study them in the order in which I give them. An earlier text is a prerequisite for a later one.

These courses and their texts are part of the 120 “deGarisMPC” YouTube video courses, so are presented here in the same format. They are limited to the group theory courses and related topics (e.g. abstract algebra, field theory, Galois theory, Lie theory, representation theory, differential geometry, etc)

Course 1 : “Mathematical Groups,” 1, Jun-Sen, Barnard, Neill, 220pp  (link);  (videolink);

Course 4 : “Linear Algebra,” 4, Jun-Sen, Lipschutz, Lipson, 420pp  (link), (full text);

Course 5 : “Undergraduate Algebra,” 5, Sen-M1, Lang, 360pp  (link), (full text tick selected book then download);

Course 9 : “A Course in Group Theory,” 9, Sen-M1, Humphreys, 270pp  (link), (free), (free), (free), (free), (close to full text);

Course 11 : “Basic Abstract Algebra,” 11, M1, Ash, 400pp  (link), (full text);

Course 15 : “Representations and Characters of Groups,” 15, M1,  James, Liebeck, 410pp  (link), (free), (free), (full text);

Course 22  : “Differential Geometry of Curves and Surfaces,” 22, Sen-M1, do Carmo, 500pp  (link), (free), (free), (free), (free), (full text);

Course 30 : “Introduction to Field Theory,” 30, Sen-M1, Adamson, 170pp  (link), (full text);

Course 31 : “Groups and Representations,” 31, M1, Alperin, Bell, 180pp  (link), (full text);

Course 32 : “Galois Theory,”  32, Sen-M1, Rotman, 150pp  (link), (full text);

Course 33 : “Fields and Rings, ” 33, M1, Kaplansky, 200pp  (link), (full text);

Course 34 : “Matrix Groups : An Introduction to Lie Group Theory,” 34, M1, Baker, 320pp  (link), (free), (free), (free), (free), (approx. full text);

Course 38 : “Semi-Simple Lie Algebras and their Representations,” 38, M1, Cahn, 150pp  (link), (free), (free), (full text);

Course 49 : “The Lie Algebras su(N) : An Introduction,” 49, M1, Pfeifer, 110pp  (link), (full text);

Course 51 : “Lie Groups, Lie Algebras, and Representations : An Elementary Introduction ” 51, M1, Hall, 350pp  (link), (free), (free), (free), (free), (full text);

Course 52 : “Introduction to Lie Groups and Lie Algebras,” 52, M1, Sagle, Walde,  350pp  (link), (free), (full text);

Course 53 : “An Introduction to Differentiable Manifolds and Riemannian Geometry,” 53, M1, Boothby, 400pp  (link), (free), (full text);

Course 58 : “A Course in the Theory of Groups,” 58, M2, Robinson, 460pp  (link), (full text tick selected book then download it);

Course 61 : “Modern Differential Geometry for Physicists,” 61, M2, Isham, 280pp  (link), (full text);

Course 63 : “The Theory of Finite Groups : An Introduction,” 63, M2,  Kurzweil, Stellmacher, 370pp  (link), (full text);

Course 66 : “Algebra : A Graduate Course,” 66, M2, Isaacs, 500pp  (link), (free), (full text tick selected book then download it);

Course 67 : “Character Theory of Finite Groups,” 67, M2, Isaacs, 290pp  (link), (free), (free), (full text tick selected book then download it);

Course 79 : “From Holomorphic Functions to Complex Manifolds,” 79, M2, Fritzsche, Grauert, 370pp  (link), (full text);

Course 80 : “An Invitation to Morse Theory,” 80, M2, Nicolaescu, 230pp  (link), (full text);

Course 81 : “Noncommutative Rings,” 81, M2, Herstein, 190pp  (link), (free), (full text tick selected book then download it);

Course 82 : “Finite Group Theory,” 82, PhD1, Aschbacher, 270pp  (link), (full text tick selected book then download it);

Course 84 : “Simple Groups of Lie Type,” 84, M2, Carter, 310pp  (link), (free), (full text tick selected book then download it);

Course 88 : “Finite Groups,” 88, PhD1, Gorenstein, 500pp  (link), (free), (free), (full text tick selected book then download it);

Course 89 : “Sporadic Groups,” 89, PhD1, Aschbacher, 300pp  (link), (free), (free), (free), (free), (free), (free), (free), (free);

Course 91 : “An Introduction to Nonassociative Algebras,” 91, PhD1, Schafer, 150pp  (link), (free), (free), (free), (free), (partial text);

Course 99 : “Finite Simple Groups : An Introduction to their Classification,” 99, PhD1, Gorenstein, 310pp  (link), (free), (free), (free), (free), (free), (free), (free), (free), (free), (free), (free), (free), (free), (free), (free), (free), (free), (free), (free), (free), (free), (free), (free);

Course 100 : “The Classification of Finite Simple Groups, Vol. 1 : Groups of Noncharacteristic 2 Type,” 100, PhD1, Gorenstein, 470pp  (link), (free), (free);

Course 101 : “The Classification of Finite Simple Groups : Vol. 2, Groups of Characteristic 2 Type,” 101, PhD2, Aschbacher et al, 310pp  (link), (free), (free), (free), (free), (free), (free), (free);

Course 102 : “The Classification of Quasithin Groups” 102, PhD2, Aschbacher, Smith, 1220pp  (link1), (link2), (free), (free), (full text);

Course 104 : “The Local Structure of Finite Groups of Characteristic 2 Type,” 104, PhD2, Gorenstein, Lyons, 720pp  (link), (free), (free);

Course 105 : “The Classification of the Finite Simple Groups : Vol. 1, Overview, Outline of Proof,” 105, PhD2, Gorenstein, Lyons, Solomon, 140pp  (link), (full text tick selected book then download it);

Course 106 : “The Classification of the Finite Simple Groups : Vol. 2, General Group Theory,” 106, PhD2, Gorenstein, Lyons, Solomon, 200pp (link), (full text tick selected book then download it);

Course 107 : “The Classification of the Finite Simple Groups : Vol. 3, Almost Simple K-Groups,” 107, PhD2, Gorenstein, Lyons, Solomon, 400pp  (link);

Course 109 : “The Classification of the Finite Simple Groups : Vol. 4, Uniqueness Theorems,” 109, PhD2,  Gorenstein, Lyons, Solomon, 330pp (link);

Course 111 : “The Classification of the Finite Simple Groups : Vol. 5, The Generic Case,” 111, PhD2, Gorenstein, Lyons, Solomon, 460pp  (link);

Course 112 : “The Classification of the Finite Simple Groups : Vol. 6, The Special Odd Case,” 112, PhD2,  Gorenstein, Lyons, Solomon, 520pp  (link), (free);

Course 119 : “Vertex Operator Algebras and the Monster,” 119, PhD2, Frenkel et al, 480pp (link), (free), (free), (free), (free), (free), (free), (free), (free);

Course 120 : “Moonshine beyond the Monster : The Bridge Connecting Algebra, Modular Forms and Physics,” 120, PhD2, Gannon, 430pp (link), (free), (free), (free), (free);

That’s 43 courses, about a third of the total number of courses (120) in the whole “deGarisMPC” series of courses.


I am currently creating a list of research papers referred to in the above texts. I already have several hundred on them. I will put links to them shortly on this page. Personally, I have a Sony “Digital Paper” A4 size, ultra light, e-reader, with a 32G memory chip. I put the above books and the research papers in it. These research papers I obtained largely from Google Scholar, and some from full content links from Google. Obtaining these research papers made me curse pluddite (paper luddite) institutions such as Springer, Vstor, etc for not providing these papers for free, the way the publisher Elsevier does.

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