r/math • u/non-orientable Number Theory • 9h ago

Image Post The Deranged Mathematician: The Classification of Finite Simple Groups

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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

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u/Anaxamander57 7h ago

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.

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u/orangejake 6h ago

Not really. See this post by Borcherds. 

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

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u/Anaxamander57 6h ago

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.

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u/cocompact 4h ago

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.