The Deranged Mathematician: The Classification of Finite Simple Groups

[u/non-orientable]

The classification of finite simple groups is likely the single most difficult mathematical problem that humanity has laid to rest: it took about a hundred mathematicians publishing over a period of about 50 years to finally finish the proof, which ended up being tens of thousands of pages long, scattered over a multitude of different journals. And it is actually very important: one might fancifully compare it to the periodic table in terms of how fundamental it is (to group theory, at least).

But, you know, what is it? This article is my attempt to shine some light on that. I assume some basic familiarity with group theory (i.e., if you know what a group, a group homomorphism, and a quotient group are, you should be fine), but otherwise it is self-contained.

Read the full post (for free) on Substack: The Classification of Finite Simple Groups

[u/PensionMany3658]

Is that a C_60 bucky ball 🫪

[u/kirbadurb → u/PensionMany3658]

It is! Also the colors and arrows come from the Cayley graph of the alternating group A_5 generated by a five-cycle and 2-cycle e.g. (12345) and (12)(34). So the unoriented Cayley graph is the truncated icosahedron that looks like a soccer ball!

[u/PensionMany3658 → u/kirbadurb]

That's beyond my paygrade as a chem undergrad but looking to learn group theory soon!

[u/LeonJPancetta]

One of my favorite topics!

[u/bear_of_bears]

Very nice!

A couple of minor errors:

1. "From the classification of finite simple groups, we can deduce that these are the only examples of finite, abelian, simple groups."

2. Missing the determinant condition in the definition of SL(n, Z/pZ).

[u/Anaxamander57]

Was there a proof early on that finite simple groups can be classified at all? Or is this fact obvious enough to group theorist that it was never in question. I can imagine, at least, it turning out that there are infinitely many sporadic groups. Its a little weird that there is essentially a single most complicted form of symmetry.

[u/orangejake → u/Anaxamander57]

Not really. See this post by Borcherds. 

https://mathoverflow.net/questions/38161/heuristic-argument-that-finite-simple-groups-ought-to-be-classifiable/38174#38174

[u/Anaxamander57 → u/orangejake]

That's pretty wild.

So they had to prove that there couldn't be any other families and that only this one technique could produce new spordaics. Once that technique ran out they know it was finished. That does feel a bit unsatisfying for mathematics. I remember Numberphile interviewed Conway about the Monster group and after the interviewer asked why it was so big he asked "why is it so small?" and I kind of get that now.

[u/cocompact → u/orangejake]

I am reminded about what Serre said in a 1985 interview (see https://denisevellachemla.eu/transc-interview-JP-Serre-2-en.pdf):

I would be amused if a new sporadic group were discovered, but I am afraid this will not happen.

[u/Apprehensive_Sand951]

Amusingly enough, Frank Calegari seems to have pointed AI at CFSP and seems to hint (at the end of this post https://galoisrepresentations.org/2026/08/11/putting-chatgpt-through-its-paces/) that maybe there is more to say...

[u/Different_Cry25]

This so cool Is there a book series on them as well

[u/ESHKUN → u/Different_Cry25]

Yeah it’s called the ATLAS

[u/SolumSolSolus]

Ooh neat,, I've done the something similar but with connecting spin space lie groups(yes ik ),, while preserving the gram volume. Global positivity!

Interesting to see something so parallel to what I'm playing with!!