Symmetry and the Monster (5): Thirty years, ten thousand pages, and a proof no one can read to the end

Reductio ad absurdum (a method of proving theorems), the weapon Euclid loved so much, is one of the finest weapons a mathematician has. It is far more refined than any brilliant move in a game of chess: a chess player at most sacrifices a pawn, or even a piece, while what the mathematician stakes is

the whole game

. — G. H.

Hardy

(G. H. Hardy),

A Mathematician’s Apology

(quoted in translation)

[Author’s note — Yeqiu]

Honestly, at this point in the series my own mathematical background is no longer enough. I am not a professional researcher in this field; I have been making these episodes one by one purely out of interest, and as an experiment in how AI and people can work together at present. In these very difficult times it has also helped me focus on what I am interested in and taken me away, for a while, from the many troubles of real life, so what I can do is try as hard as I can to understand the mathematical parts and share them with whoever is interested. This blog is less something written for others than my public tree hollow; I hope that focusing on what I want to do will help me get through this hard time.

The companion video is online (YouTube)

Prologue · A proof no one can read to the end

There is a theorem like this.

Its proof is about ten to fifteen thousand pages long, scattered across about 500 research papers, from the hands of about 100 authors, written over a span of about 30 years. And the most unsettling sentence about it is this: today, no single person in the world understands the whole proof completely.

It is the longest proof in history written by people working together, by hand, page by page, and to this day no other proof written by people exceeds it. It is not “once the longest, now surpassed”; it still is the longest now. What has to be excluded are computer-assisted proofs: the four color theorem (which we covered in a separate episode outside this series), the machine proof of about 200 TB for a combinatorial coloring problem in 2016 (the Boolean Pythagorean triples), the formal verification of the Kepler conjecture. Those are larger, but they were all generated by machines, not written by people page by page. The distinction takes only one sentence: written by people, or generated by a machine. We are talking about the former.

The suspense left at the end of the last episode has thus, unexpectedly, been pushed in front of a sharper question.

In EP4 we watched Lie’s mathematics of continuous symmetry being carried back to the finite world and lined up in neat families. But the “atoms of symmetry”, the finite simple groups (the simple groups covered in EP2 and EP3: groups that cannot be taken apart any further, which apart from the trivial ways have no normal subgroup that can shrink them as a whole; the “atoms” in the world of symmetry that cannot be split any further), how many kinds of them are there, in the end? In this episode, about a hundred mathematicians answered that over thirty years and gave a complete list. But the price was this: the answer is so long that no single person can verify it from beginning to end.

The four color theorem was the first to put the same question on the table. In 1976 that computer checked more than a thousand cases one by one and got them right, but the volume was so large that no one could recheck it all by hand, page by page; its difficulty was “computed correctly, yet no one can survey it all”. In this episode the same question comes back in another, perhaps more fundamental, way: this time every page can be understood by a person, yet no one can read it all.

A proof so long that no single person can verify it from beginning to end: is it still a proof?

In this episode we follow this line through three people and three chapters. The three chapters seem to tell separate stories, but they are really three steps of the same thing: first build the atoms, then learn to recognize them, and finally prove that not one is missing. First Chevalley finds a method that builds all the regular atoms of symmetry in one go (§1); then Tits puts on a different pair of eyes, sees these groups as “buildings”, one after another, and can therefore recognize them (§2); finally, about a hundred mathematicians spend thirty years proving that this list has no atom that slipped through the net, and we see the most unexpected ending of the whole affair (§3).

The most dramatic thing about this story is not “it could not be proved”, but this: faced with this pile of proof that no one can check alone, the response humankind made was, of all things,

to invent a whole new piece of engineering and spend decades rewriting this proof, just so that it could be checked.

A proof so long that it pushed humankind to build a new quality-control assembly line for “proof” itself. This reversal is revealed at the end of §3.

Contents

• §1 · After the war: one method that builds all the “atoms” in one go
• §2 · The man from Uccle: seeing a group as a building
• §3 · The great theorem: thirty years, ten thousand pages, a list no one can check alone
• Epilogue · Twenty-six zero remainders
• Appendix · Standing on EP4, learning this episode step by step
• Sources
• Image sources

§1 · After the war: one method that builds all the “atoms” in one go

At the end of EP4 we were standing in front of something half-finished. Dickson (Leonard Dickson, 1874–1954) “carried Lie groups over to finite fields” and did build a batch of finite atoms of symmetry. For example, carrying type $A_1$ over to a finite field with only $p$ numbers, $\mathbb{F}_p$, gives $\mathrm{PSL}(2,p)$, and inside it hides the $A_5$ from EP3. But Dickson verified them family by family, almost one by one, by hand: the classical families (corresponding to $A,B,C,D$ in the EP4 periodic table) he could handle, but the 5 exceptions that abruptly cap off in the corner of EP4, $G_2,F_4,E_6,E_7,E_8$ (EP4 only named them and left them to later episodes), were extremely awkward to verify, and there was no unified way of building them. In other words: EP4 had already listed the “periodic table” of continuous symmetry in full, but the method for carrying the whole table back to the finite world had not yet been fully found.

Figure 1: Leonard Dickson (1874–1954). He was the first to “carry Lie groups over to finite fields” and built a batch of finite atoms of symmetry, but he could only verify them family by family, almost one by one, by hand. Source: MacTutor portrait archive.

A word, in passing, on why they are “odd”: the four great families $A$–$D$ all come, in the end, from three “normed division algebras” in which the associative law holds: the real numbers $\mathbb{R}$, the complex numbers $\mathbb{C}$, the quaternions $\mathbb{H}$. And there are only four such algebras in all (Hurwitz’s theorem); the last one is the octonions $\mathbb{O}$ (octonions: the “last” number system after the quaternions, whose multiplication has lost even the associative law). These 5 exceptions all grow, without exception, on this fourth and most anomalous algebra, the octonions; “why exactly 5” has its root here. The cleanest example is $G_2$: it is exactly the symmetry group of the octonions $\mathrm{Aut}(\mathbb{O})$. The octonions have seven “imaginary units”, and their pairwise products obey a fixed set of rules (which can be laid out as an oriented Fano plane); $G_2$ is all the symmetries that can continuously “stir” these seven imaginary directions while leaving the whole set of multiplication rules completely unchanged, $14$ degrees of freedom in all (for the concrete construction see Appendix J). The remaining $F_4,E_6,E_7,E_8$ follow the Freudenthal–Tits magic square and grow along the geometry of the octonions all the way to the $248$-dimensional $E_8$, where they abruptly cap off; beyond that the constraints of the Dynkin diagram no longer allow it. $E_8$ later showed up in string theory (the $E_8\times E_8$ of the heterotic string) and in optimal sphere packing in $8$ dimensions (Viazovska 2016). Their complete individual constructions need Jordan algebras over the octonions and exactly how each cell of the magic square is filled in; this episode explains the first level thoroughly in Appendix J (the octonions, “why exactly these few exceptions”, and that $G_2$ is exactly the symmetry group of the octonions); the magic square for $F_4$–$E_8$ goes deeper, the appendix only names it, and for details see Baez’s The Octonions at the end of the article. (The later episodes are about another patch of truly strange things, the sporadic groups, not these few exceptional Lie types.)

Figure 2: The Cayley–Dickson ladder: each step up doubles the dimension and gives up one property: order → commutativity → associativity → division. The octonions $\mathbb{O}$ are the “last” normed division algebra, and those 5 exceptional Lie types all grow on it.

The wars left two scars on this line. First the First World War: it almost wiped out an entire generation of French mathematicians. At the most renowned of schools, the École Normale Supérieure in Paris, about half of the students on the wartime roll died at the front. In the words Henri Cartan later used (here in translation), “we were the first generation after the war; in front of us was a blank, and everything had to be done from the beginning.” It was in this blank that a group of young people came together in 1934 and decided to write books under a shared pen name, rewriting the whole of modern mathematics on the most rigorous and abstract foundations. The pen name was Bourbaki (Nicolas Bourbaki): not one person, but the pseudonym of a collective. The origin of the name is itself a joke: the École Normale Supérieure once had a prank lecture in which a disguised student solemnly “proved” a string of theorems that were all wrong, and the most impressive of them was given the name “Bourbaki’s theorem” (Bourbaki was originally the name of a French general in the Franco-Prussian War); a few young people found this amusing and borrowed the name, and the first name “Nicolas” was chosen by the wife of André Weil. Later the Second World War interrupted everything once again, and one of the youngest founding members of Bourbaki is precisely the protagonist of this section: Chevalley (Claude Chevalley, 1909–1984).

Figure 3: A Bourbaki gathering in 1938 (stylized reconstruction). “Bourbaki” was not one person but the collective pseudonym of a group of young mathematicians; Chevalley was the youngest of the nine founders. (Stylized reconstruction, not an original group photo.)

Chevalley was a thoroughgoing cosmopolitan: born in Johannesburg, South Africa, raised in France, teaching at Columbia University in New York from 1949 to 1957, then back to Paris. When Bourbaki first met at the end of 1934 he was there, the youngest of the nine founders. Bourbaki’s conviction that “everything should be built on unified, abstract, rigorous structures” was exactly the ground tone of the work he did later.

Figure 4: Claude Chevalley (1909–1984). Source: Konrad Jacobs / MFO, CC BY-SA 2.0 de

In 1955, Chevalley published a paper with a title so plain as to be unremarkable: “On certain simple groups” (“Sur certains groupes simples”, published in the Japanese Tôhoku Math. J., second series, vol. 7, 1955, pp. 14–66). What this paper did can be summed up in one sentence: he built the machine.

The work Dickson computed by hand family by family, Chevalley gave a unified recipe for: take any one type from the Killing–Cartan periodic table of EP4 (the 4 great families plus the 5 exceptions, 9 in all), and with the same set of mechanical steps the corresponding group can be built over any field: the reals, the complexes, and even any finite field $\mathbb{F}_q$ (the kind of number system covered in EP4 §4 that has only $q$ elements yet still allows addition, subtraction, multiplication and division; $q$ is necessarily a power of some prime). One recipe: feed in a Lie type plus a field, and out comes a group. Through this method, every continuous symmetry in the EP4 periodic table got its finite version, including those 5 exceptions. Chevalley was the first to build systematically the finite simple groups of $G_2,F_4,E_6,E_7,E_8$ over finite fields; no one had constructed these groups before him.

Chevalley’s unified construction: carrying groups of Lie type from “the reals” to “any finite field”

The “skeleton” of a Lie group (for example all real matrices with determinant $1$ of size $2\times2$, $\mathrm{SL}_2(\mathbb{R})$, which is the kind of matrix group we used in EP4 §4 to build $A_5$ and $\mathrm{PSL}(2,7)$), the set of numbers describing “how two of the most basic infinitesimal rotations compose into a third”, is called the

structure constants

, and they can in fact all be written as

integers

. Integers make sense in any number field. So Chevalley said: stop using the reals $\mathbb{R}$, switch to a finite field with only $q$ elements, $\mathbb{F}_q$ ($q$ a power of some prime), and the same skeleton still builds a group. A continuous, infinitely large symmetry group is in this way “pixelated” into a

finite

group, and often a simple group.

Small example: replace $\mathbb{R}$ with $\mathbb{F}_q$, and $\mathrm{SL}_2$ becomes the finite group $\mathrm{SL}(2,q)$; then divide out the center $\{\pm I\}$ to get the

projective special linear group

$\mathrm{PSL}(2,q)$, whose

order

(that is, the number of elements in the group) has a clean formula:

$$|\mathrm{PSL}(2,q)| \;=\; \frac{q\,(q^2-1)}{\gcd(2,\,q-1)}.$$

Substitute $q=7$: $\frac{7\cdot 48}{2}=168$, and this is the

second smallest non-abelian simple group

(the smallest has order $60$). Substitute $q=5$ as well: you get $60$, and that is exactly the $A_5$ that appeared in EP4 ($\mathrm{PSL}(2,5)\cong A_5$). The same simple group of order $60$ is both “the even permutations of five things” and “the projective matrix group over $\mathbb{F}_5$”: two completely different roads leading to the same end. (For where the order formula comes from, see Appendix A at the end of the article; it is one of the few small pieces of this episode that can be proved step by step to the end.)

Figure 5: “Count” $\mathrm{GL}$ out column by column, then “peel” twice (dividing out the $\det=1$ layer, dividing out the center $\pm I$ layer) down to $\mathrm{PSL}$; substitute $q=7$ and you get $168$.

What Chevalley’s method builds are called Chevalley groups: they are the 9 families among the finite simple groups of Lie type that “have not been modified” (that is, the 4 great families + 5 exceptions just mentioned). But the story is not over: this method can also be “improved a little”.

The three people who added gears to the machine: twisted groups

Chevalley’s recipe is very regular, but someone noticed that the “diagrams” of those Lie types from EP4 (Dynkin diagrams, which you can think of as a stick-figure fingerprint of each Lie type) themselves carry symmetry. To take a concrete case: the diagram of type $A_2$ is just two dots with a line between them, and type $A_n$ is $n$ dots joined in a chain; this chain is symmetric left to right. Type $D_4$ is one central dot with three short branches sticking out, carrying a three-pronged threefold symmetry. Since the diagram itself can be folded over, if Chevalley’s construction is twisted along this folding, will new groups grow out of it?

Figure 6: Every Lie type has a Dynkin diagram (a stick-figure fingerprint), and the diagram itself carries symmetry: fold it along this symmetry, “twist” the construction along with it, and new families grow out: $A_n\to{}^2A_n$, $D_4\to{}^3D_4$.

They will. Steinberg (Robert Steinberg, 1922–2014), born in what is now Moldova, raised in Canada, who finished his doctorate in Toronto under Brauer (Richard Brauer, the one who will come back in §3) and taught at UCLA for his whole career, did exactly this in his 1959 paper “Variations on a theme of Chevalley”: using the symmetry of the Lie-type diagram itself to “twist” Chevalley’s construction, he grew 4 more families of finite simple groups (the notation carries a number at the upper left: $^2A_n,^2D_n,^2E_6,^3D_4$). These are called the Steinberg twisted groups; among them $^2A_n$ is the branch of the unitary groups (the unitary groups are not exactly a familiar face to the general public: roughly speaking, they are a class of matrix groups obtained by replacing the transpose in “orthogonal matrices preserve length” with the “conjugate transpose”; for details see Appendix D). The paper’s title says “variations”, but that was modest: he fitted Chevalley’s method with a new gear.

Something more unexpected came from a completely different direction. Suzuki (Michio Suzuki, 1926–1998), Japanese, who taught for a long time at the University of Illinois in the United States, was studying a class of highly symmetric permutation groups in 1960 when he ran into a family of new simple groups that no one had expected, written $^2B_2(q)$ (the Suzuki groups). They differ from everything else in two ways: they exist only over fields of characteristic $2$; and their order is not divisible by $3$: they are the only family of non-abelian finite simple groups with no factor $3$ in the order (the smallest one has $29120$ elements). At first it looked like a lone exception; only later was it seen clearly that it too is something “twisted” out of Chevalley’s method, only with a more hidden twist.

As soon as Suzuki’s discovery came out, Rimhak Ree(Rimhak Ree, 1922–2005) quickly followed up. Ree was born in Hamhung, now in North Korea, later taught at the University of British Columbia in Canada, and was a Korean-Canadian mathematician. Inspired by Suzuki, in 1960–61 he found the last two families under a similar mechanism, $^2G_2(q)$ and $^2F_4(q)$, called in Chinese 里群 (“Lǐ groups”, the Ree groups).

Figure 7: The three people who “added gears” to Chevalley’s machine (from left): Robert Steinberg (1922–2014, the 4 families of twisted groups $^2A_n$ and others), Michio Suzuki (1926–1998, the Suzuki groups $^2B_2$), Rimhak Ree (1922–2005, the Ree groups $^2G_2$, $^2F_4$). Sources: Steinberg — David Weisbart / NAS Biographical Memoirs & UCLA Math; Suzuki — MacTutor / AMS Notices; Ree — MacTutor.

⚠️ A small reminder about a Chinese rendering (readers, please take note): in Chinese, these groups built by Ree are written ”

里群

” (“Lǐ groups”), and

not “李群”

(“Lǐ groups” written with a different character): “李群” is the protagonist of EP4, the Lie group (continuous symmetry group), and the two have nothing to do with each other. The character “里” is used here purely to avoid a literal collision with Lie. For the person we use “李林学” (Rimhak Ree), and for the groups always “里群” (Ree groups).

First, a word on what “twisting” actually is

(intuition, no proof): a finite field comes with a natural “flipping” operation of its own, raising every number to the $p$-th power ($x\mapsto x^p$), which in mathematics is called the

Frobenius map

. It is like “complex conjugation” in a finite field: doing it repeatedly turns things back to the original, and it is a symmetry of this number system itself. What “twisting” means is taking the left-right symmetry of the Lie-type diagram and, through this Frobenius flip,

weaving it into the definition of the group

. The most familiar product is the unitary group: twist the untwisted $A_2(q)=\mathrm{PSL}(3,q)$ once, and it becomes $^2A_2(q)=\mathrm{PSU}(3,q)$; what is added is precisely a “conjugate transpose” constraint written with Frobenius (for a concrete $3\times3$ matrix, see Appendix D).

At this point, the parts of that assembly line mass-producing “atoms” of symmetry were complete. Add up Chevalley (9 families) + Steinberg (4 families) + Suzuki (1 family) + Ree (2 families), and you get exactly 16 families of finite simple groups, collectively called the finite simple groups of Lie type. Add the alternating groups $A_n$ from EP3 (also an infinite family) and the plainest of all, the cyclic groups of prime order, and the part of the finite simple groups that “is regular and lines up in infinite families” comes to exactly 18 infinite families, 16 of them of Lie type.

The “periodic table of continuous symmetry” of EP4 was, at this point, carried back to the finite world in full: four regular families + five exceptions, each with its finite version, lined up neatly in 16 families.

One method seems to have built all the regular atoms of symmetry. But if you keep asking, are these 16 families (together with the cyclic groups and the alternating groups) really all of them? Could there be some atom that does not come out of this method at all?, then to answer this question you have to change the angle and look at these groups afresh. The first person to change the angle saw them as geometry. Let us first see who he was.

§2 · The man from Uccle: seeing a group as a building

Chevalley, Steinberg and the others used matrices, generators and equations over fields to compute these groups. But a more fundamental question was hanging in the air: these groups are symmetries of “what”, exactly? In EP1 the groups of the regular polyhedra are the symmetries of some Platonic solid; in EP4 $SO(3)$ is the symmetry of a sphere. Behind every such group there seems to stand a geometric object that it “rotates”. So these 16 families of groups of Lie type: what are they rotations of?

The man who answered this question was Tits (Jacques Tits, 1930–2021). He was born in 1930 in Uccle, a small town on the southern edge of Brussels, Belgium; the title Ronan gave this chapter is “the man from Uccle”. Tits was a child prodigy and went to university very young; he became a French citizen in 1974 and for a long time held the chair of algebra at the Collège de France.

Figure 8: Jacques Tits (1930–2021), “the man from Uccle”. Source: Harald Hanche-Olsen, CC BY 3.0

The answer Tits gave was a completely new geometric concept: the building (in the original French, immeuble). Roughly speaking, he proved that every group of Lie type is the symmetry group of some “building”. This building is not made of bricks and stones; it is a high-dimensional geometric skeleton, pieced together from many flat layers called “apartments”, each layer in turn tiled into cell after cell of “chambers”. The group acts on this building and moves the chambers around, and the geometric shape of the building encodes the abstract group completely.

Figure 9: A Tits building: apartments (Coxeter reflection tilings) glued together along chambers; the group moves the chambers and encodes the geometry as a group; when rank ≥ 3 the geometry determines the group uniquely; the smallest one = the Fano plane = the building of $\mathrm{PSL}(3,2)$.

What a building is (intuition + a small example)

A building is a geometry pieced together from many pieces of “flat ground”; each piece of flat ground is called an

apartment, and all of them have the same regular tiling pattern (the symmetry pattern of a reflection group, a

Coxeter group

). What a group of Lie type does is move back and forth between these apartments and fit them together. The beauty of it is that when the group is large enough,

the geometry in turn constrains the group

: as long as you draw this building diagram correctly, the group is uniquely determined. Tits could therefore “discover” and “identify” groups of Lie type by pure geometry, without relying purely on algebraic derivation.

Small example: take the smallest one, the projective plane over $\mathbb{F}_2$, which is the smallest kind of “plane geometry” in mathematics, called the

Fano plane: the whole plane has only $7$ points and $7$ lines, $3$ points on each line and $3$ lines through each point, small enough to draw in full on one sheet of paper. Draw the “points” and the “lines” each as one kind of vertex, and join them by an edge whenever a point lies on a line; the graph you get (the

Heawood graph

) is the building of $\mathrm{PSL}(3,2)$ (order $168$; it is in fact the same group as the $\mathrm{PSL}(2,7)$ of §1,

proof in Appendix F

): $14$ vertices, $21$ edges (the details of the seven lines are all listed in Appendix E). Each of its “apartments” is a regular hexagon in the graph.

A building is where a group shows itself in its own geometry.

Figure 10: The smallest kind of plane geometry, the Fano plane ($7$ points, $7$ lines, $3$ points on each line, $3$ lines through each point); its incidence graph (the Heawood graph, $14$ vertices, $21$ edges) is the building of $\mathrm{PSL}(3,2)$.

The real weight of this step by Tits is that it gave the whole family of groups of Lie type a unified geometric point of view: from then on these groups had a double identity, both algebraic objects each defined by matrices and symmetries of the same kind of geometry (the building). This is another road running parallel to the line of algebraic construction in §1: one road builds groups with a recipe, the other looks at groups through geometry. (This road of “working back from the group to the geometry” in fact continues the Erlangen program of Klein in the 19th century. Klein said “a geometry is its symmetry group”: what a geometry studies is what stays invariant under the action of some group of transformations; Tits went the other way: given a group of Lie type, build a geometry on which it acts as the symmetry. Taken together, the two of them are the two sides of the dictionary “group ⟷ geometry”.)

This geometric road later carried great weight. In 2008, Tits, together with the protagonist of the next section, John Thompson (John Thompson), received the Abel Prize. The prize is named after Niels Abel of EP3, who died young (the Abel of the Abel–Ruffini theorem), and is regarded as one of the highest honors in mathematics. The citation honored them for profound achievements in algebra, and in particular for shaping modern group theory. One who re-examined groups through geometry, one who listed the whole list one item at a time: the two sides of modern group theory were named together by this year’s Abel Prize.

Tits gave us the eyes to see these groups clearly. But the question “is the list really complete” cannot be seen by looking alone; someone had to actually go and prove it. And that used up about a hundred mathematicians for thirty whole years.

§3 · The great theorem: thirty years, ten thousand pages, a list no one can check alone

The first step: the odd order theorem

To prove that “the list of atoms of symmetry is complete”, the first step is to narrow the search range. In 1955, exactly the same year in which Chevalley published that paper, Brauer (Richard Brauer, 1901–1977) and K. A. Fowler proved an inconspicuous but important theorem (“On groups of even order”, Annals of Math. 62, 1955): fix the centralizer of an “involution” (that is, an element of order $2$), and there are at most finitely many finite simple groups that match it (this “finiteness” can be proved just by counting involutions; the complete proof is in Appendix I at the end of the article). In plain words: inside every finite simple group there is a small piece of structure, “which elements commute with a certain element of order $2$”; once this small structure is fixed, only finitely many finite simple groups fit it. This theorem turned “looking for a needle in a haystack among infinitely many groups” into “narrowing the candidates down to finitely many by one small feature”, and gave the whole classification its first foothold. To take an example you can work by hand: in $A_5$ (order $60$), take the element of order $2$ $(1\,2)(3\,4)$; only $4$ elements commute with it, and they form exactly a small group of order $4$ (§3 will compute it for you below). Brauer’s theorem says, conversely: as soon as you stipulate that “the structure commuting with some element of order $2$ is exactly such a small group of order $4$”, the finite simple groups in the whole world satisfying this add up to only finitely many, and $A_5$ is one of them. So in this one year, 1955, two lines started at the same time: Chevalley’s line, in charge of building the regular atoms; Brauer’s line, in charge of framing where to go looking for the irregular atoms.

The real turning point came in 1963, from the hands of Feit (Walter Feit, 1930–2004) and Thompson (John Thompson, 1932– ). Feit’s life story is itself a footnote to the war: born in Vienna, at the end of August 1939 he was put by his parents on a “children’s evacuation train” (Kindertransport) and left, on the very two days before the Second World War broke out; according to Ronan, his parents had planned to follow two weeks later but never managed to, and he never saw them again in his life. That train was the last one to leave Vienna. Years later, Ronan’s book still preserves the letter that the young Feit wrote to his aunt from America. It was this Feit who, together with Thompson, proved a theorem later recognized as the most influential theorem in the history of group theory:

The odd order theorem (Feit–Thompson, 1963).

Every finite group of

odd order

is

solvable

.

What the odd order theorem says, and why it is the starting point of this classification (intuition)

“Solvable” groups are the most

docile

kind of group: they can be built up layer by layer from abelian groups, and they contain no unbreakable hard building block of the “non-abelian simple group” kind. Put the other way round, this theorem amounts to declaring:

you can never put together a non-abelian simple group from a pile of parts of odd order

: every non-abelian simple group necessarily has an order that is

even, and therefore (by

Cauchy’s theorem, proved in EP3: if a prime $p$ divides the order of a group, the group must have an element of order $p$; here take $p=2$) it certainly contains an element of order $2$, that is, an “involution”. With this, the range to be searched is thoroughly pinned down: every non-abelian simple group carries involutions, so one can start from such an involution and look at “the elements that commute with it”; the group these elements form is called its

involution centralizer

$C_G(t)$. Tying back to EP3/EP4: $60=2^2\cdot 3\cdot 5$ and $168=2^3\cdot 3\cdot 7$ both contain the factor $2$, exactly in line with this theorem.

What made the whole mathematical world gasp at the time was the length of its proof: this paper ran to a full 255 pages (“Solvability of groups of odd order”, Pacific J. Math. vol. 13, 1963, pp. 775–1029, filling almost an entire issue of the journal). One theorem, 255 pages. If this single small step, “odd order → solvable”, alone needs 255 pages, then how vast, even endless, a project would it be to classify all finite simple groups? The odd order theorem was both something like a formal starting point and something like an ominous omen. The method it used, local analysis, became everyone’s template from then on.

Figure 11: The two authors of the odd order theorem: Walter Feit (1930–2004, left) and John Thompson (1932–, right). Sources: Feit — Konrad Jacobs / MFO, CC BY-SA 2.0 de; Thompson — © C. J. Mozzochi, Princeton N.J.

The intuition of local analysis

Instead of tackling the whole huge group $G$ head-on, pay attention only to its

local fragments

: the involution centralizers, the small subgroups belonging to each prime. Each fragment is not large and can be seen clearly; the whole craft of the classification project is

putting the whole of $G$ back together from these local fragments

. An analogy: instead of looking at the whole building directly, check the floor plan of each household and its relations with its neighbors, and piece together the blueprint of the whole building. Why always pay attention to the prime $2$? Because the odd order theorem guarantees that every non-abelian simple group contains involutions (elements of order $2$), so $2$ is the entrance that can always be opened.

Figure 12: Vienna at the end of August 1939: a “children’s evacuation train” pulls out; on this very day the Second World War broke out. Nine-year-old Walter Feit was one of the children on the train, and he never saw his parents again. (Stylized reconstruction.)

Small example: $A_5$ again (order $60$, even, in line with the theorem). Take one of its involutions $t=(1\,2)(3\,4)$; the elements that commute with $t$ are exactly

$$C_{A_5}(t)=\{\,e,\;(1\,2)(3\,4),\;(1\,3)(2\,4),\;(1\,4)(2\,3)\,\}\;\cong\;V_4,$$

the Klein four-group,

of order $4$

($A_5$ has $15$ involutions of this kind in all, and $60/15=4$). Such a small group of order $4$ is one piece of “local information” in the hands of the classification project. It is hard to imagine, but the whole classification was completed by piecing together thousands upon thousands of small fragments like this.

The chief designer Gorenstein: a project he called the “Thirty Years’ War”

After the odd order theorem, the work spread out on every front. Throughout the 1960s and 70s, first a few, then dozens, and in the end about a hundred group theorists pushed the classification forward piece by piece. Collaboration on this scale needs a commander-in-chief, and that person was Gorenstein (Daniel Gorenstein, 1923–1992).

Figure 13: Daniel Gorenstein (1923–1992), the “godfather” and chief designer of the classification project. Source: Konrad Jacobs / MFO, CC BY-SA 2.0 de

Others each proved one hardest lemma after another, while Gorenstein did something else they could not do: survey the whole picture and plan the campaign. His colleagues privately called him the “godfather”: he could coordinate more than a hundred researchers at once, so that they “each did their own work, yet it could all be put together in one place”. In 1972, in Chicago, London and Israel’s Weizmann Institute in turn, he publicly laid out a program for completing the whole classification, a road map of about sixteen steps (he formally published this program in 1974; the number “exactly sixteen steps” is a list drawn up by Gorenstein himself when he looked back later, not an objective count by a third party). With this road map, research that had been scattered became a coordinated advance with an overall plan to follow.

Gorenstein himself gave this project a name: the “Thirty Years’ War”. And most incredibly, as the nominal “marshal” of this “war”, he was at the same time running the entire mathematics department of Rutgers University. The price was a staggering volume: by the time the project was nearing its end, the proof had grown to this scale: about ten to fifteen thousand pages, scattered across about 500 papers, from the hands of about 100 authors, written over about 30 years.

The final structure of the classification: an overview

What these thirty years bought is a list that can be said in one sentence: every finite simple group belongs to exactly one of the four classes below, with nothing missing and no fifth class:

The first three classes are a regular method: name a prime, name an $n$, name a field and a family, and it hands you a simple group, as many as you want. The real strangeness is in the fourth class: those 26 sporadic groups do not fit into this method; they are the isolated “leftover remainders” that are left over after all the regular families have been counted. The largest of them is the Monster group (commonly called “the Monster”), of order about $8\times10^{53}$ (a $54$-digit number when written out): it belongs to no pattern at all, and yet it definitely exists.

A small note (“how many classes, how many groups” can actually be pressed further): whether the Lie type counts as ”$16$ families”, or, together with the cyclic and alternating groups, as ”$18$ infinite families”, depends on how you count the odds and ends; even the number "$26$" has people pressing the point:

the Tits group $^2F_4(2)'$

(the commutator subgroup of the Ree group $^2F_4(2)$, of index $2$) still has no agreed classification to this day; some treat it as the $27$th sporadic group, some count it among the odds and ends of the Lie type. Even “how many rows the classification list really has” has to be pressed this hard, and this pressing itself is the theme of this episode.

Honesty: is it still a proof?

Around 1981 (as Gorenstein announced; popular sources also give 1983, while Aschbacher’s own wording is “around 1980”; even this “year of completion” does not line up), the whole group theory community declared: the finite simple groups have been classified. The list is the four classes above. This project that had lasted thirty years seemed finally to be wrapped up.

But this “completion” was wrapped from the start in a layer of four-color-theorem-style unease, and more unsettling than the four color theorem.

The trouble with the four color theorem was: the machine checked more than a thousand cases one by one and got them right, but the volume was so large that no one could recheck it all by hand, page by page; the difficulty lay in the volume, and people could actually understand what the machine was doing at each step. The trouble with the classification of finite simple groups fell on the other side: this time the work to be read was all written by hand by people; every page can be understood by a person, yet no one can read it all. About ten thousand pages, scattered across 500 papers, each piece written by one of 100 people: no one in this world has read the whole proof from beginning to end, let alone verified it independently. More seriously, Aschbacher (Michael Aschbacher, 1944– ) later pointed out honestly that many “well-known” results used in the original proof were in fact scattered all over the literature, or worse, had never appeared in the literature at all.

Then the most embarrassing thing happened: that “completion” had been announced too early.

More than ten years after completion was announced, people found gaps in the original proof one after another. Most gaps were quickly filled, but one was especially hard: the large class of so-called “quasithin groups” had in fact never been properly dealt with (that part of the manuscript was never really finished). This gap was filled by Aschbacher and Stephen Smith, the two of them working away for quite a few years, and it took about 1221 more pages (two volumes, 2004; Vol I about $477$ pages + Vol II about $744$ pages). That is to say: “it is done” was said in 1981, and the last gap was really filled only in 2004, more than twenty years later.

This puts the central question of this episode right in front of us: a proof so long that no one can check it alone, where even “which year it was completed” cannot be stated for sure, and even “whether it was really completed” took another twenty years before anyone dared confirm it.

Is it still a proof, then?

The reversal: to check a proof, humankind invented a new kind of engineering

If the story stopped here, it would be a depressing ending: “the greatest theorem turns out to be something no one can carry.” But the real turn is much more powerful than that, and it is the true climax of this episode.

Faced with this pile of proof that no one can check alone, the group theory community did not shrug and walk away. They did something almost unprecedented: they launched a new project specifically to rewrite the whole proof, not to prove anything new, only so that this proof could be checked.

This project is called the second-generation proof, led by three people, Gorenstein, Richard Lyons and Ronald Solomon, and known in the field by the initials of their surnames as GLS. Their goal, in Aschbacher’s words as rendered here, was to write the whole proof down clearly and carefully in one place, relying on only a few elementary textbooks, and to supply proofs for all those results in the original proof that were “well known” yet scattered everywhere or even never written down. The series has been published by the American Mathematical Society volume by volume since 1994, planned at about twelve volumes and five thousand pages, to compress, straighten out and fix in a definitive form the ten-thousand-page original proof, all in order to build a version that one person can really read straight through and is able to check. When Ronan wrote his book in 2006, 5 volumes had come out; by 2023 it had reached Volume 10 (from Volume 9 on, a co-author was added, Inna Capdeboscq), and it is still not finished. For a theorem declared “complete” in 1981, the task of “writing it clearly enough that one person is able to check it” has taken more than thirty years and a change of generation, and is still not done today: the progress bar of the second-generation proof is slower than that of the original proof.

It is worth stopping to think about the weight of this: a proof so long that it pushed humankind to build a new quality-control assembly line for “proof” itself. The proof itself is not wrong. The problem is only that it is too long and too scattered, so long that “whether it is actually right” needs a generation to spend decades reconfirming it with a new method. So what happened here is a rare kind of self-awareness: humankind is saying, “we must invent new methods to be worthy of what we have already proved.”

The four color theorem asks: “what we computed by machine, why should people believe it?” Its crux is whether a non-human verifier can be trusted. The classification of finite simple groups asks something else: “a proof of ten thousand pages that no one can read to the end: does it still count as proved?” Its crux is whether a proof still counts once it goes beyond the mental capacity of any single person. The two questions point to the same depth: when a proof exceeds the scale that any single human mind can hold, what does the word “proof” still mean? Facing this depth, the answer this episode gives is the second-generation proof: humankind put up a new scaffold for its own limits.

The machines take over: the first stone, checked

But GLS is people rewriting for people to read. Over these 20 years, a second kind of answer has also emerged, the one Ronan in 2006 was too early to write about: let machines do the checking.

First look at a line spanning a whole century. The odd order theorem was not a problem Feit and Thompson opened out of nowhere: it was a conjecture of Burnside (the Burnside who appeared in EP4) from 1911; they proved it only in 1963, in 255 pages; and in September 2012, after nearly another half century, Gonthier (Georges Gonthier, 1962– , French computer scientist) led a team of 15 (Microsoft Research–Inria, France) that used the proof assistant Coq to machine-verify this whole 255-page odd order theorem line by line: about 170,000 lines of Coq code and more than four thousand lemmas, six full years of work. The machine read it all and confirmed every step, and what it relied on was only the small piece of trusted logic in the Coq kernel, small enough to be audited again and again. One of the longest proofs written by people was, for the first time, checked from beginning to end by a machine that does not get tired, does not skip steps, and does not say “I think this is obvious”.

This stroke brings that earlier motif of the four color theorem right back to where it started. The four color theorem (1976) was the first great theorem that had to rely on a computer to be proved, and the question it raised was “what the machine computed, why should people believe it”. In 2005, it was this same Gonthier who used Coq to formalize the whole four color theorem as well, machine-checking it down to the level of the logical axioms, turning the answer to “why should people trust the machine” into “it is enough to trust that verifier, which is small enough to audit”. Seven years later came the 2012 verification of the odd order theorem in the previous paragraph: the same Gonthier carried the same method from the four colors over to the first cornerstone of the classification of finite simple groups. The same person, with the same key of “machine rechecking”, opened in turn a checkable door into each of these two mountains that “human effort cannot verify”.

Today, what has been verified by machine is only

the odd order theorem, “the first piece at the starting point”

. The whole ten-thousand-page classification theorem, together with the ten volumes of GLS,

has to this day not been completely verified by any proof assistant

. The formalization of the odd order theorem is a

proof of concept

: it proves that “it is possible for a machine to verify a behemoth of this kind”, and is still very, very far from “a machine has verified the whole classification”. The mountain that no one can climb to the top alone: humankind is, on one hand, redrawing it with GLS into a map that can be read, and, on the other, teaching machines to recheck it one piece at a time;

the first stone, 2012, checked.

A chapter still being written today: when “no one can read it to the end” becomes the new normal

The machine verification of the odd order theorem was a one-off feat in 2012. But “a proof so big that no one can carry it alone”, which was first forced into view by the classification of finite simple groups, is today slowly turning from “the exception of a century’s project” into the everyday of mathematics. One of the most active mathematicians today, Terence Tao(Terence Tao), put it very clearly: what formalization really changes is not only “whether it is right”, but also “how many people can prove one theorem together”. In his words, “a traditional mathematics collaboration rarely involves more than five or so coauthors, in part due to the need for each coauthor to trust and verify the work of every other; but formalization projects routinely involve scores of people who may have had no prior interaction, precisely because the formal proof assistant allows for individual subtasks in the project to be precisely defined and verified independently of the other subtasks.” Put this back into this episode: this is exactly the situation of those hundred authors of the classification; no one understands the whole ten-thousand-page map. The classification theorem was, back then, a conspicuous precedent on this road; what formalization offers is a way to make it no longer fragile: replacing “trusting those hundred people” with “trusting that verifier, which is small enough to audit”.

Figure 14: Terence Tao, one of the most active mathematicians today. He put the situation of the classification of finite simple groups very clearly: what formalization changes is not only “whether it is right”, but also “how many people can prove one theorem together”. Source: IPAM, CC BY 4.0

Tao also pointed out which kind of proof formalization is worth the most for, almost as if he were writing about the classification theorem: in his account it is especially valuable for proofs that are particularly long, and for which the field lacks referees willing to check them line by line. Ten thousand pages, scattered across 500 papers, no one able to referee it from beginning to end: it would be hard to find a more fitting footnote than the classification of finite simple groups. So is the word “proof” changing? Looking out from these few remarks of his, the picture is not the science-fiction one of “a superintelligence solving everything alone”; it is more that proof is becoming an industrial-scale collaboration, guaranteed line by line by machines. The classification of finite simple groups sits exactly on the threshold between old and new: it is the first proof that the old rules (one person carries it, a group of people referees it line by line) could not hold up, and also the first mountain the new methods have to catch.

Epilogue · Twenty-six zero remainders

The classification was “completed”: announced in 1981, with the last piece really filled in only in 2004. The final list has four classes in all: the cyclic groups of prime order, the alternating groups, the $16$ families of groups of Lie type, plus $26$ sporadic groups.

The atoms of the first three classes all follow a pattern and line up in infinite families: you give the conditions, it gives the group (and the $16$ families of Lie type are exactly what the method of §1 built in one go); this also includes the 5 exceptional Lie types of §1 ($G_2,F_4,E_6,E_7,E_8$): although they are called “exceptions”, they still belong to families; behind them stands the unified pattern of the octonions, and they can still be generated infinitely, family by family, as $q$ varies. In other words, those 5 are “the oddest members of the families”. But these 26 sporadic groups are a deeper layer of “exception”: they belong to no family at all and cannot be lined up in any infinite family; they are not “odd members of a family” but orphans with no family at all, the zero remainders outside the $18$ infinite families, lodged all alone in the corner of the list. The counting of these thirty years, in the end, swept the whole world of finite symmetry perfectly clean, and yet swept out, in the corner, $26$ specks of dust that would not go into any dustpan.

Figure 15: The complete list of finite simple groups: on the left the three classes (cyclic, alternating, Lie type) line up in $18$ infinite corridors (you give the conditions, it gives the group, on and on); on the right the $26$ sporadic groups, which fit into none of the corridors.

And among these $26$ crouches the largest one, the one that least follows the rules: of order about $8\times10^{53}$, a $54$-digit number (for comparison: all the grains of sand on Earth put together come to a number of only about $20$ digits). It cannot be built by any method, and yet it definitely exists. This Monster group is exactly where Symmetry and the Monster has been pointing since the first episode: it is not inside this regular classification machine at all.

Figure 16: The order of the Monster is about $8\times10^{53}$, a $54$-digit number; by comparison, all the grains of sand on Earth put together come to a number of only about $20$ digits.

In the next episode, we open that box.

Appendix · Standing on EP4, learning this episode step by step

First, an honest account of the ceiling: the building, local analysis, and the full picture of the classification

do not

have short proofs that can be squeezed into an appendix (that is a matter of ten thousand pages). But standing on the foundation of EP4, there are still quite a few points in this episode that “you truly understand once they are explained thoroughly” and that can be walked through step by step. Below are ten pieces; the first nine are

arranged in the order of the main text, §1→§2→§3, with a bonus J at the end that returns to the exceptions of §1. Within a group, the premises come first and then the conclusions that use them; each piece (or its review piece) comes with an example you can compute by hand and one symbolic check. The difficulty has three levels: [light] can be read with pure high-school or first-year-university linear algebra; unmarked means advanced (one level deeper);

(optional)

is a bonus that can be skipped. Readers who only want the intuition can skip the whole section. Notation: $\mathbb{F}_q$ is the

finite field

with only $q$ elements ($q$ is necessarily a power of a prime), a number system in which you can freely add, subtract, multiply and divide. Permutations use

cycle notation: $(1\,2)(3\,4)$ means ”$1\!\leftrightarrow\!2$, $3\!\leftrightarrow\!4$, the rest fixed”; concrete elements always use

numbers, and letters $(a\,b)(c\,d)$ are used only to refer to the

general shape

of a certain class.

Appendices A–D | Supporting §1 of the main text, “After the war: one method that builds the atoms in one go”

This group of four appendices fills in the computations passed over in a single stroke in §1: how the order formula is counted out (A, B), why the Suzuki groups are not divisible by $3$ (C), and what the conjugate-transpose constraint of the “twisted groups” looks like (D).

Appendix A · The order formula of $\mathrm{PSL}(2,q)$ [light]

Below we push forward step by step, using only “counting” all the way:

1. The general linear group $\mathrm{GL}(2,q)$ = the invertible matrices over $\mathbb{F}_q$ of size $2\times2$. The first column must be a nonzero vector: $q^2-1$ choices; the second column only needs not to be collinear with the first column, so excluding the $q$ multiples, $q^2-q$ choices. Hence

2. The special linear group $\mathrm{SL}(2,q)$ = those with determinant $1$. The determinant is a “score” that maps onto $\mathbb{F}_q^\times$ (the $q-1$ nonzero elements), and each score corresponds to equally many matrices, so

3. Projectivizing: then divide out the center $\{\pm I\}$. When $q$ is odd, $I\ne -I$ and the center has $2$ elements; when $q$ is even (characteristic $2$), $+1=-1$ and the center has only $1$ element left. The two cases are written uniformly as $\gcd(2,q-1)$. Hence

Work an example by hand: - $q=7$: $\frac{7\cdot 48}{2}=168$. - $q=5$: $\frac{5\cdot 24}{2}=60=|A_5|$ (hence $\mathrm{PSL}(2,5)\cong A_5$). - $q=4$: $\frac{4\cdot 15}{1}=60$ once again ($\mathrm{PSL}(2,4)\cong A_5$; $q$ even, center trivial).

Honest small note: substituting $q=2,3$ gives $6$ and $12$, and the corresponding $\mathrm{PSL}(2,2)\cong S_3$ and $\mathrm{PSL}(2,3)\cong A_4$ are neither of them simple groups. $\mathrm{PSL}(2,q)$ only starts being a simple group at $q\ge 4$: the formula computes as usual, but the property of being “simple” has a small threshold.

Appendix B · The general order formula: $\mathrm{GL}(n,q)$, $\mathrm{SL}(n,q)$, $\mathrm{PSL}(n,q)$

Appendix A computed only the $2\times2$ $\mathrm{PSL}(2,q)$. The method of Chevalley in §1 builds matrix groups of any dimension, so we generalize the “counting” of A unchanged to $n\times n$; you will see that the order of $\mathrm{PSL}(3,2)$ needed in §2 is exactly what this general formula spits out.

Figure 17: Counting the order of $\mathrm{GL}(n,q)$: choose column by column (the $k$-th column has $q^n-q^{k-1}$ choices) and multiply to get $|\mathrm{GL}|$; then “peel” twice: divide by $(q-1)$ to get $\mathrm{SL}$, divide by $\gcd(n,q-1)$ to get $\mathrm{PSL}$. $n=2,q=7$: $2016\to336\to168$.

Explicit object. The general linear group $\mathrm{GL}(n,q)$ = all invertible matrices over $\mathbb{F}_q$ of size $n\times n$ (a group under multiplication). A matrix is invertible if and only if its $n$ columns are a set of linearly independent vectors in $\mathbb{F}_q^n$. So “counting matrices” = “counting ordered independent sets of vectors”.

Step by step (pick column by column; each step only needs to exclude the space spanned by the earlier columns):

Multiply the numbers of choices for the columns together:

The special linear group $\mathrm{SL}(n,q)$ = those matrices with determinant $1$. The determinant is a group homomorphism onto $\mathbb{F}_q^\times$ (the $q-1$ nonzero elements), $\det:\mathrm{GL}(n,q)\to\mathbb{F}_q^\times$; each “score” corresponds to equally many matrices (every fiber of the homomorphism has the same size), so divide by $q-1$:

The projective special linear group $\mathrm{PSL}(n,q)$ = divide out the center as well (the matrices that commute with all elements). The center of $\mathrm{SL}(n,q)$ is exactly the scalar matrices $\lambda I$ with $\lambda^n=1$, $\lambda\in\mathbb{F}_q^\times$. And $\mathbb{F}_q^\times$ is a cyclic group of order $q-1$, in which the number of $n$-th roots of unity is exactly $\gcd(n,\,q-1)$. Hence

Substitute $n=2$, and this expression reduces back to $\dfrac{q(q^2-1)}{\gcd(2,q-1)}$ of Appendix A: the general formula and the light version agree.

Work an example by hand: $|\mathrm{PSL}(3,2)|=168$ ($n=3,\ q=2$, needed in §2).

When $q=2$, $q-1=1$, and the three groups are essentially one (there is no nontrivial determinant and no distinction of $\pm I$): $\mathrm{GL}(3,2)=\mathrm{SL}(3,2)=\mathrm{PSL}(3,2)$, all of order $168$. Remember this $168$: in Appendix F (and also in §2 of the main text) it will collide with a $168$ that comes from a completely different road.

Symbolic check: each step above has been independently rechecked by symbolic computation: the general formula reproduces Appendix A for $q=2..11$; $7\cdot6\cdot4=168$; and along the way $|\mathrm{PSL}(3,4)|=20160$ and $|\mathrm{PSL}(4,2)|=20160$ were pinned down (used for the uniqueness in Appendix F).

Appendix C · The order of the Suzuki groups is not divisible by $3$

§1 of the main text says that the Suzuki groups $^2B_2(q)$ are the only family of non-abelian finite simple groups with no factor $3$ in the order. This sounds mysterious, but in fact one line of congruences sees right through it.

Explicit object. The Suzuki groups exist only over fields of characteristic $2$, with the parameter $q=2^{2n+1}$ (an odd power of $2$: $q=2,8,32,128,\dots$). Its order has the closed form

Work an example by hand: $q=8$.

There is no $3$ in the factorization: this is what the main text means by “the smallest Suzuki simple group has $29120$ elements, and its order contains no $3$”.

Step by step (valid for all $q=2^{2n+1}$). We only need to show that $3$ divides none of the three factors. The key is $2\equiv -1\pmod 3$, so the powers of $2$ modulo $3$ only swing between the values $1,2$:

The three factors modulo $3$ are $1,\,2,\,1$ respectively; none of them is $0$, and their product is $\equiv 2\pmod 3$. So

Honest small note: substituting $q=2$ gives $|{}^2B_2(2)|=4\cdot5\cdot1=20$, and this group is solvable, not a simple group (a small threshold of the same kind as ”$\mathrm{PSL}(2,q)$ is simple only for $q\ge4$”). The Suzuki simple groups start at $q=8$ (order $29120$). The reason “containing no $3$” is so rare for the Suzuki groups is that every other family of non-abelian finite simple groups cannot avoid $3$ in its order (by deeper structure theorems, only cited here): $^2B_2$ is the only exception.

Symbolic check: $64\cdot65\cdot7=29120$ and the factorization $2^6\cdot5\cdot7\cdot13$ are exact; and for $q=2^{2n+1}\,(n=0..4)$, printing the three factors modulo $3$ one by one shows all of them nonzero, with the total order modulo $3$ always $2$.

Appendix D (optional) · Twist it: the “conjugate transpose” constraint of the unitary group $^2A_2=\mathrm{PSU}(3,q)$

§1 says that Steinberg’s “twisted groups” added a gear to Chevalley’s method, and the most familiar branch is the unitary group $^2A_2(q)=\mathrm{PSU}(3,q)$. The sentence in the main text, “a conjugate transpose constraint is added”, is made concrete here: a concrete $3\times3$ matrix makes the “twist” something you can touch.

Explicit object. The untwisted $A_2(q)=\mathrm{PSL}(3,q)$ is defined with ordinary matrices. To “twist”, we need an involution like complex conjugation. A finite field comes with one: on $\mathbb{F}_{q^2}$, the Frobenius map $\ \bar{x}:=x^{q}\$ is a field automorphism of order $2$, and it fixes exactly the subfield $\mathbb{F}_q$: it is the finite-field version of “conjugation”. For a matrix $A$ (with entries from $\mathbb{F}_{q^2}$), define the conjugate transpose $A^{*}:=\overline{A}^{\,\mathsf T}$ (first apply $x\mapsto x^q$ to every entry, then transpose). The general unitary group consists of the matrices that preserve a Hermitian form: take the simplest Hermitian form $=$ the identity matrix; then

The “conjugate transpose constraint” $A^{*}A=I$ is just the ordinary orthogonality condition $A^{\mathsf T}A=I$ with the transpose replaced by the conjugate transpose: the twist lies precisely in this $\overline{(\cdot)}=(\cdot)^q$.

Work an example by hand: $q=2$, checking a concrete matrix over $\mathbb{F}_4$. $\mathbb{F}_4=\{0,1,\omega,\omega^2\}$, where $\omega^2=\omega+1$ and $\omega^3=1$. The conjugate is $\bar x=x^{2}$, so $\bar1=1,\ \bar\omega=\omega^2,\ \overline{\omega^2}=\omega^4=\omega$. Take the diagonal matrix

Check the constraint:

So $A$ is unitary ($A^*A=I$) and has determinant $1$: it is an element of $\mathrm{SU}(3,2)$. Looking at the diagonal entries one by one, the constraint comes down to the single sentence “the norm is $1$”: $\lambda\bar\lambda=\lambda^{\,q+1}=\lambda^{3}=1$. This is the “conjugate transpose constraint” in its barest form.

Order formula and honest small note.

Work an example by hand, $q=3$: $\dfrac{27\cdot 28\cdot 8}{\gcd(3,4)}=\dfrac{6048}{1}=6048=2^{5}\cdot3^{3}\cdot7$; this is the smallest simple unitary group, $\mathrm{PSU}(3,3)$. And for $q=2$ above, $\mathrm{PSU}(3,2)$ has order $=\dfrac{8\cdot9\cdot3}{3}=72$, solvable, not simple (another example of “small $q$ not over the threshold”, like $\mathrm{PSL}(2,q)$ with its $q\ge4$ and Suzuki with its $q\ge8$). We use $q=2$ only to get a clear feel for the conjugate transpose; the genuinely simple unitary groups start at $q=3$.

Symbolic check: over $\mathbb{F}_4$ ($\omega^2=\omega+1$), checked $A^{*}A=I$ and $\det A=1$, and that the norms of the three $\lambda$ are $\lambda^{q+1}=1$; $|\mathrm{PSU}(3,3)|=27\cdot28\cdot8=6048=2^5\cdot3^3\cdot7$.

Appendices E–F | Supporting §2 of the main text, “The man from Uccle: seeing a group as a building”

This group of two appendices makes the “building” metaphor of §2 concrete: the smallest building, the Fano plane and the Heawood graph (E), and the fact that the two groups that both equal $168$ are actually the same one (F).

Appendix E · The Fano plane and the Heawood graph: the building of $\mathrm{PSL}(3,2)$

§2 says that every group of Lie type is the symmetry group of some building. The smallest one can be laid out and drawn: it belongs to the group of order $168$ computed in Appendix B, $\mathrm{PSL}(3,2)$. This section lists it explicitly, turning “building” from a metaphor into seven lines you can count, and then counting all the way to that $168$.

Explicit object. Take $\mathbb{F}_2^3$ (three-dimensional, each coordinate either $0$ or $1$, $8$ vectors in all).

• Points = the one-dimensional subspaces of $\mathbb{F}_2^3$, $7$ of them. Over $\mathbb{F}_2$ a one-dimensional subspace is just $\{0,v\}$, and it can be represented directly by that nonzero vector $v$; so the $7$ points $\longleftrightarrow$ the $7$ nonzero vectors, labeled in binary as $1,\dots,7$ (e.g. $5=(1,0,1)$).
• Lines = the two-dimensional subspaces of $\mathbb{F}_2^3$, $7$ of them. A two-dimensional subspace contains $4$ vectors ($0$ plus $3$ nonzero ones), so each line passes through exactly $3$ points. Every two-dimensional subspace is, for some nonzero linear functional (an $\mathbb{F}_2$-linear map $\mathbb{F}_2^3\to\mathbb{F}_2$, written as a nonzero row vector $a$), its kernel $\{x: a\!\cdot\!x=0\}$; there are $7$ nonzero functionals, hence $7$ lines.

An equivalent description that is easier to remember: three points are collinear $\iff$ their binary labels XOR to zero ($a\oplus b\oplus c=0$), because the third nonzero vector in a two-dimensional subspace is exactly the sum of the first two, $a+b=c$. The two descriptions give the same set of $7$ lines.

Step by step / work an example by hand: list all $7$ lines.

Count them: $7$ lines with $3$ points each, $7\times3=21$ flags in all (one incidence of “a point lying on a line”).

Point–line duality: each point lies on exactly three lines. Conversely, rearrange the table above by points and count, point by point, which lines it lies on:

Each point lies on exactly $3$ lines (which can also be obtained in one step from $21\div7=3$). So ”$3$ points per line, $3$ lines per point” is perfectly symmetric left and right: this is the perfect self-duality of the Fano plane: swap the roles of “points” and “lines”, and the whole incidence pattern stays completely unchanged.

Counting out that $168$: picking frames one by one. How many symmetries does $\mathrm{PSL}(3,2)$ have ($=\mathrm{GL}(3,2)$; Appendix B has shown that at $q=2$ the three groups collapse into one)? There is no need to plug into the order formula; count directly on this diagram: count the smallest rigid skeleton it can move freely, an ordered frame, that is, a sequence of three ordered, non-collinear points $(P,Q,R)$ (equivalently: an ordered basis of $\mathbb{F}_2^3$). Pick them one at a time:

And $\mathrm{PSL}(3,2)$ acts on these frames simply transitively: given any two ordered frames, there is exactly one linear transformation sending the first to the second. “There is one” because the images of an ordered basis can be specified arbitrarily (a standard fact of linear algebra); “only one” because a linear transformation that fixes a basis point by point can only be the identity. As soon as an action is simply transitive, the elements of the group are in one-to-one correspondence with the objects acted on, so

This is the same number as what Appendix B obtained from the general order formula $(2^3{-}1)(2^3{-}2)(2^3{-}4)$, namely $168$, by two roads: one plugs into the formula, the other counts frames by hand on the Fano plane. (It also collides with the one in Appendix F that comes from $\mathrm{PSL}(2,7)$, that $168$: it is the same group.)

From the plane to the building: the Heawood graph. Draw the “points” and the “lines” each as one kind of vertex ($7+7=14$ of them), and whenever a point lies on a line, join the two corresponding vertices by an edge ($3$ points per line and $3$ lines per point, so $21$ flags $=21$ edges); the bipartite graph you get is the famous Heawood graph: $14$ vertices, $21$ edges, every vertex of degree $3$ ($3$-regular, exactly ”$3$ points per line / $3$ lines per point”), shortest cycle length $6$ (a bipartite graph has no odd cycles, and in the plane “two points determine a line, two lines meet in a point” blocks $4$-cycles). This graph is the building of $\mathrm{PSL}(3,2)$: the group acts on it and moves the vertices around, and each “apartment” that §2 talks about is exactly a hexagon ($6$-cycle) in the graph, corresponding to the smallest closed pattern of “three points and three lines” in the plane. $\mathrm{PSL}(3,2)$ (order $168$) is exactly the part of the symmetries of this graph that keeps the two kinds, points and lines, from mixing, that is, all the incidence-preserving transformations of the Fano plane; and if we also allow “points” and “lines” to be swapped as a whole (exactly the “flip” that the perfect self-duality above provides), the full automorphism group of the graph doubles to $336$. The building, at this point, is seven lines’ worth of edges you can draw with your own hand.

Symbolic check

(

_verify_appendix_je.py

Appendix E section, 54/54 PASS): $7$ lines with $3$ points each, $7$ points on $3$ lines each, $21$ flags, Heawood $14$ vertices, $21$ edges, $3$-regular, all checked; the two ways of building the lines, “XOR triples” and “two-dimensional subspaces as kernels of functionals”, give

exactly the same

$7$ lines; the number of ordered frames $=7\cdot6\cdot4=168$ (the point-by-point $7\to6\to4$ also checked), and $=|\mathrm{PSL}(3,2)|$ (a constructive group, $168=2^3\cdot3\cdot7$): the simply transitive action confirms ”$168$ = the number of frames”.

Appendix F · Two $168$s: $\mathrm{PSL}(2,7)\cong\mathrm{PSL}(3,2)$

This is the stitch that sews together §1 (building groups with a recipe) and §2 (looking at groups through geometry): the $\mathrm{PSL}(2,7)$ at the end of §1 and the $\mathrm{PSL}(3,2)$ behind the Fano plane in §2 are the same group. Two completely different roads lead to the same group of order $168$.

Explicit object: computing the two orders side by side. On the left use Appendix A ($n=2,q=7$), on the right Appendix B ($n=3,q=2$):

Equal orders are only a necessary condition, not an isomorphism. Add the second piece of fact:

Step by step (the honest route). Among simple groups (explained in the main text: groups that apart from the trivial ways can no longer be “quotiented” smaller by a normal subgroup, the atoms of the world of symmetry) there is a fact that is not obvious but can be verified:

Cited theorem (uniqueness of the simple group of order $168$): up to isomorphism, there is

only one

non-abelian simple group of order $168$.

$\mathrm{PSL}(2,7)$ and $\mathrm{PSL}(3,2)$ are both simple groups ($\mathrm{PSL}(n,q)$ is simple once past the small threshold, a textbook result cited in the main text), and their orders are both $168$. Since there is only one simple group of order $168$, these two can only be the same one:

Writing this isomorphism out completely by hand (matching the fractional linear transformations on $8$ points one by one with the matrix action on $7$ points) can be done, but it is rather tedious; in this episode we honestly stop at the step “uniqueness ⟹ isomorphism”.

How far from a matter of course this uniqueness is. Do not think that “simple groups of the same order must be isomorphic” comes for free: it only holds in a small range. The smallest counterexample is at order $20160$: there crouch two mutually non-isomorphic simple groups, $A_8\cong\mathrm{PSL}(4,2)$ and $\mathrm{PSL}(3,4)$ (compute with Appendix B: $|\mathrm{PSL}(4,2)|=|\mathrm{PSL}(3,4)|=20160$, the same order). So the uniqueness at $168$ is a small theorem with real content, not an empty remark, exactly to the taste of this episode’s “even how many rows the list has must be pressed”.

Symbolic check: the two groups were each constructed independently ($\mathrm{PSL}(2,7)$ acting on the projective line $\mathrm{P}^1(\mathbb{F}_7)$, on its $8$ points; $\mathrm{PSL}(3,2)$ acting on $\mathbb{F}_2^3$, on its $7$ nonzero vectors), and both orders come out as $168$; and for

each conjugacy class representative

it was verified that its normal closure $=$ the whole group, thereby strictly establishing that

both are simple

, and hence they are isomorphic by uniqueness. (The two non-isomorphic simple groups of the same order $20160$ also had their orders pinned down in Appendix B, supporting that the uniqueness is not empty.)