Comments by "Gavin Tillman" (@gavintillman1884) on "The Most Powerful Diagram in Mathematics" video.

I wasn't expecting to find this on YouTube! If it's what I think it is (I've only seen the first minute so far, but it looks like it ma be) then it takes me back to my Cambridge post grad dissertation in 87/88
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It's all coming flooding back. Miracles? Steiner systems? Remembering the heartache of trying to render MOG in Plain TeX (I never learned LaTeX at the time though it seems a no brainer in hindsight)
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@AnotherRoof i certainly wasn’t expecting to see it described as the most important diagram in maths, although I was aware of its importance in coding, sphere packing, simple groups etc
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Just got to the end - was so please to stumble across this, especially as a) I wasn't particularly looking for it, YouTube's algorithm obviously deemed I might be interested and b) it looks like it's only been up a few hours!
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Pretty standard in the UK I think
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It relates to how group M24 is constructed. One of the constructions takes the finite field F23 and extends it with infinity. I can’t remember the detail, it was so long ago. Something to do with Möbius maps and quadratic residues.
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The labelling is arbitrary but over the years, different constructions are discovered, and the labelling is meaningful within particular constructions. The classic presentation involving Möbius maps on a finite field is going to involve elements of that field (numbers from 0 to some prime minus 1 such as 22) together with infinity. A different construction might have different labels but give you the same group.
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@AnotherRoof I'll probably rewatch this tonight. Thinking about this video prompted me to do some searches on YouTube. Started by looking for content on the Mathieu groups. I found videos on the sporadic groups and the Monster, but (with the exception of what appear to be spoken versions of the relevant Wikipedia pages)nothing specific on the Mathieu groups. That's a quite a gap to plug, don't know if that might be something that would interest you?? I guess this video is the closest thing on YouTube to the Mathieu groups but there is a fair bit more ontent that could be included, such as construction and highy transitive groups. Likewise (though not as relevant to the present video) a shout out for classical/Chevalley groups - I looked at some in my part III year at Cambridge (GL/SL/PGL/PSL, U/SU/PU/PSU. O/SO/PO/PSP, Sp all over finite fields) but would love to have gone deeper.
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@AnotherRoof ah great - just watched the second part, hadn't seen that before. And as I said, I rewatched part 1 last night. This could well be my favourite YT channel. Certainly my favourite pure maths channel! Part 3 after breakfast.
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Just finished the whole playlist. Brilliant stuff. Might have to watch the extended versions!
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Some other areas I suspect we might find if mutual interest: Sym6 and Alt6 and why they buck the trend. And Coxeter/reflection groups.
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