Why Do Sporadic Groups Exist?

⬣ LINKS ⬣

Permutation Group Visualiser by Tesseralis:

⬣ ABOUT ⬣
The concept of symmetry can be formalised through the study of groups. Groups can be constructed from simple groups, and during an incredible effort throughout the 20th century, all simple groups have been classified. But the classification is messy -- there are several infinite families and then 26 left-over groups which don't fit any of the patterns. In this video, we explore some of these so-called Sporadic Groups and why they exist.

⬣ TIMESTAMPS ⬣
00:00 - Intro and Housekeeping
01:56 - Introduction to Group Theory
05:13 - Classifying Groups
10:04 - Transitive Groups
15:06 - M24
19:20 - The Mathieu Family
26:29 - Why Sporadic Groups Exist

⬣ INVESTIGATORS ⬣
Nothing for you here. Sorry!

⬣ REFERENCES ⬣
[1] É Mathieu, Sur la fonction cinq fois transitive de 24 quantitiés. Journal de mathématiques pures et appliquées 18 (1873) pp. 25-46.
[2] E Witt, Die 5-fach transitiven Gruppen von Mathieu. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 12: 256–264
[3] P Cameron, Projective and polar spaces. (1992) University of London, Queen Mary and Westfield College.
[4] P Cameron, Permtation groups. (1999) London Mathematical Society Student Texts, Cambridge University Press.
[5] W Burnside, Theory of Groups of Finite Order. (1911) Cambridge University P
[7] P Duhem, Emile Mathieu, his life and works. Bull. New York Math. Soc. 1(7): 156-168 (April 1892).
[8] Aimo Tietäväinen, On the nonexistence of perfect codes over finite fields. SIAM Journal on Applied Mathematics, 24(1), 88–96.

⬣ CREDITS ⬣
Music by Danjel Zambo.
Intro music by Tobias Voigt.

Image Credits
Molecule
Periodic table
Tesseract
Rings
Rack
Magma
Lattice
Knot (quandle)
Ernst Witt
Bernd Fischer
Robert Griess
Fisher-Price Monster
E8 Diagram
Circular arrows
G2 Diagram
F4 Diagram
E6 Diagram
E7 Diagram
William Burnside
Platonic Solids
Explosion
Explosion
Saturn, Jupiter timelapse, Voyager 1
NASA Public Domain Image

[AnotherRoof]

Thanks for watching! Here's the link to my Q&A if you want to ask questions:
And check out Tesseralis's permutation visualiser here!

[jazzman]

Group theory is definitely the part of mathematics that has the highest ratio of excitement to knowledge for me. I would love more group theory videos; I hope you get great responses on all of them.

[mschoenert]

Thank you so much for your amazing video. I always loved group theory. And - to this day - are completely baffled by the amazing structures connected especially with the sporadic simple groups (Golay code, Leech lattice, Monstrous Moonshine, etc.) I always try to explain to others - especially people who tell me that they hate math - the wonderful beauty. And now I can also point them to your videos ("watch those - they explain it much better than I can ..." ;-). And thank you for mentioning GAP (I was one of its first initial developers).

[funktorial]

2:17 Ironically 'group' and 'ring' come from words meaning roughly the same thing, 'collection' or 'plurality'. The 'ring' in ring is as in smuggling ring, not diamond ring. Fields were first called 'Körper' in German (meaning 'body' to denote something like 'organically closed/whole thing'). So add all these to the pile with sets and classes and categories.

[TankorSmash]

This is really well produced, I like all the props you've made

[MasterHigure]

The mere fact that you use D4 rather than D8 for the symmetry group of the square makes me a fan. It is the only sensible convention.

[cannot-handle-handles]

Fun fact: For a finite collection S of permutations, closure implies the other two axioms/rules.

For any permutation f in the set, the subset of all powers of f must be finite. So for some natural number n>1, f^n = f, hence f^(n-1) = e and f^(n-2) = f^-1.

This uses the fact that, as a permutation, f has an inverse f^-1 (even if we don't know yet whether f^-1 is in S). We can thus multiply f^n = f by f^-1 to get f^(n-1) = e and again to get f^(n-2) = f^-1.

Of course it still makes sense to mention all three axioms. 🙂

[soninhodev]

i love it when some seeminly unrelated topics build up to a more complex one, like you did in this series with the MOG and the Golay code, when you first mentined the MOG in this video, i was like "yes!" later when the golay appeared there was another 'yes!', i love it when these things happen!

[MatthewBouyack]

"If you've never studied group theory what on earth have you been doing, it's the absolute best!" - Subscribed immediately!

[Lilly-Lilac]

You released this the day I started my first group theory class! What impeccable timing.

[quinterbeck]

24:44 does that mean there are no 6-transitive groups outside symmetric and alternating groups? Or any other higher degree of transitivity?

I love your enthusiasm for group theory! It was definitely the topic I enjoyed most at uni, so it's great to be led deeper into it. I'm so fascinated by the sporadic groups

[elshadshirinov]

Thanks so much for shedding some light on the sporadics. I feel like I'm a tiny step closer to understanding the Monster.

[dproduzioni]

I just stepped on this guy and THIS IS AMAZING!! I'm a math teacher and in the past years I tried so hard to formulate simple explanations to abstract algebra topics.. never reaching a fraction of his talent in clarity!
Really inspiring.
Wow, keep up the good work!

[shubhsharma]

31:11 "in this 3 1 blue brown video" made me laugh so hard

[dranorter]

The symmetries of the Fano plane make a great group. At one point I came up with a Rubik's cube like object that would exhibit those symmetries, although I didn't come up with a way to prototype it.

[DarioSterzi]

Always nice to see someone that suvived those chapters of Sphere packings, lattices and groups to see the beauty on the other side

[tubebrocoli]

This is the first time I've seen the coincidences that power families of sporadic groups explained in such an intuitive way thank you so much!

[benstucky]

As someone who studies infinite groups I have only one thing to say about finite groups: They are all quasi isometric to the trivial group, so what's more to understand? 😜

Great video, I will definitely watch your other ones. I love the combinatorics of finite geometric spaces such as the Fano plane and heard they were connected to Mathieu groups, but hadn't yet seen the specifics. This makes me want to explore that connection more.

[rickpgriffin]

(Forgive the imprecision, I'm not a mathematician) About why you're forced to select the final few numbers when re-ordering numbers, I think it might be helpful to go down to the most basic type of shuffle. Say you have three items in a set, A B and C, and you need to shuffle them so that none correspond to the previous order. Well, there's only two options: A goes to B, or A goes to C. If A>B then B>C and C>A, and if A>C then C>B and B>A. There's no other way to do this; at some point the choice you make must filter down to determining the remaining points. So despite there being three elements and you can choose for them to go ANYWHERE... you really only have one meaningful choice to make.

[uigrad]

Your expository skill here is amazing. I can't tell you how many hours I have spent trying to figure this out, when all I needed to do was find this relatively short video!

Most of what you covered is stuff I had already learned, but the connection to Pascal's triangle and the unsolved question concerning it was just magnificent, and completely new to me. Also, it is clear that this is your passion. Your enthusiasm really comes through!