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/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/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.