Why Do Sporadic Groups Exist?

⬣ LINKS ⬣
⬡ PATREON: / anotherroof
⬡ CHANNEL: / anotherroof
⬡ WEBSITE: anotherroof.top
⬡ SUBREDDIT: / anotherroof
⬡ TWITCH: / anotherroof
Permutation Group Visualiser by Tesseralis:
permutation-groups.glitch.me/
⬣ 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
[6] mathshistory.st-andrews.ac.uk...
[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
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Periodic table
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Tesseract
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Rings
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Rack
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Magma
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Lattice
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Knot (quandle)
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Ernst Witt
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Bernd Fischer
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Robert Griess
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Fisher-Price Monster
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E8 Diagram
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Circular arrows
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G2 Diagram
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F4 Diagram
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E6 Diagram
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E7 Diagram
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William Burnside
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Platonic Solids
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Explosion
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Explosion
• Explosion croma key gr...
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: www.reddit.com/r/anotherroof/comments/158a5he/31623_subscriber_qa_ask_your_questions_here/ And check out Tesseralis's permutation visualiser here! permutation-groups.glitch.me/

[@proloycodes]

thanks for the video! feedback: the bit about the definition of normal subgroups was a bit fast-paced for me.

[@AnotherRoof]

​@@proloycodes Thanks for letting me know! I deliberately sped through a few things because I just wanted to offer an intuition for certain ideas that weren't really essential for the main punchline of the video. I might make a more in-depth group theory intro at some stage!

[@kmjohnny]

@@AnotherRoof I discovered this channel because of group theory. 3b1b and Nemean really piqued my interest on this topic, but it feels like all series stopped on this, while there's much to discover. I watched this video multiple times now, and still finding more details about groups that I find exciting.

[@AnotherRoof]

@@kmjohnny Thanks! My PhD was in group theory so it's a topic I plan to dive into many more times in the future, so look forward to that!

[@jazzman616]

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.

[@viniciusmiradouro1606]

Wait until you get to category theory. Every once in a while after I started studying it, a profound insight about mathematical structures that I didn't get before just clicks. And that is happening to me, someone who graduated in philosophy and self studied maths, skiping calculus completely. I don't even image how mind blowing category theory is for someone with a more formal math background.

[@lorenzodiambra5210]

ring theory: 🍷🗿

[@PamellaCardoso-pp5tr]

​@@viniciusmiradouro1606 category theory was the thing that connected all the dots in mathematics for me. Categories being so abstract reveal such beautiful insights about many topics in mathematics like how inside the category of all spaces with base set X, there must be a morphism from a topological space in X to a measure space in X or from a measure space in X to a topological space in X. And turns up it not only does exists, but that morphism is THE MOST IMPORTANT SIGMA-ALGEBRA IN X, which is the the Borel sigma-algebra or the sigma-algebra built from a topology in X. category theory is mind blowing.

[@johnjeffreys6440]

Saw 3 videos on the monster groups, and still no clue what they are or what is their origin.

[@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.

[@debblez]

I always thought they were called rings because the simplest example of a ring is the complex numbers with modulus 1, which is a ring shape

[@drdca8263]

@@debblezthe unit circle isn’t a ring though? Rings are closed under addition.

[@debblez]

@@drdca8263 wait true. its funny I remember the definition but I still always think of a circle in my head

[@rogerlie4176]

In Swedish the name for field is kropp, meaning just that, body.

[@Antrascyte]

In french the name for field is "corps", meaning body too

[@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).

[@AnotherRoof]

Totally agree -- it never ceases to amaze me! Thanks in advance for sharing my videos, and thanks for all your work on GAP!

[@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.

[@fakezpred]

Dummit and Foote unfortunately uses the inferior notation.

[@MasterHigure]

@@fakezpred I suppose they use S24 to denote the full symmetry group on 4 points as well, then. No, wait, that would actually be consistent and sensible. We can't have that. *Everyone* uses S4 for the full symmetry group. A4 for the alternating group. C4 for the cyclic group. Who in their right mind would use D8 for the dihedral group? It makes no sense.

[@fakezpred]

@@MasterHigure That's the hypocrisy. D_2n is horrendous notation.

[@TankorSmash]

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

[@sillysad3198]

i subbed for the props

[@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. 🙂

[@smiley_1000]

We can't directly conclude f^n = f, only that there are n and m such that f^(m + n) = f^m. If we only work with the knowledge that the set of powers of f is finite, f^a could for example also stabilize at some large a (of course this doesn't actually happen).

[@cannot-handle-handles]

@@smiley_1000 That's true, I cut some corners.

[@felipedutra5276]

Oh no! Not the monster 😵

[@MatthewBouyack]

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

[@soninhodev7851]

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!

[@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

[@AnotherRoof]

That's correct yeah! Thanks for watching :)

[@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.

[@dranorter]

I was able to devise an M11 Rubik-like puzzle too, but quite a bit less elegant. The Fano plane symmetries one uses two generators that are very similar to one another and feels like a natural sort of puzzle (I based it on permuting 8 elements, which then end up like cube corners). The M11 puzzle just kind of implemented two very different generating elements (though I came closer to a physically realizable mechanism with that one).

[@plesleron]

I ended up doing something somewhat similar! I came up with a weird non-euclidean space consisting of 7 cubic rooms (corresponding to the points of the fano plane) where traveling along any single coordinate axis would cycle you through the three rooms on a given line. I'd have to find my old notes but IIRC, the orientation of each room would be different depending on which room you entered it from so your normal sense of direction would be basically useless. I drew a diagram of how the rooms would be connected and I wanted to try simulating it but didn't have the skill to do so at the time. I thought it might make for a novel VR experience like the VR simulations of hyperbolic space or 4 dimensions.

[@dranorter]

@@plesleron What got you into it? For me, it was a numerical coincidence: The first two conjugacy classes combined are of size 22, matching the Major Arcana; then if you throw in the third conjugacy class you get 78 elements, the size of a full Tarot deck. So I was trying to understand the group structure better while envisioning a 168-element card deck.

[@dranorter]

Ah I found my old notes. The Rubik's cube with Fano plane symmetries is a square antiprism, proportioned so that the triangular faces are equilateral, and only the triangular faces able to rotate, not the square ones. Additionally, opposite triangular faces (well, near-opposite... since it's an antiprism) rotate together, somehow connected through the center. The eight corners can be numbered so that the "upper" set of triangular faces are 168, 123, 345, 578. Then we obtain two generators for the Fano plane symmetries: (168)(345) and (123)(587). The bottom triangles provide two more generators, which make the puzzle easier if they're allowed, but aren't necessary in a solve. Note that I'm treating entire corners as my permutation elements, not paying attention to the way their orientation would change when exchanged. Note also that I'm permuting eight corners! This is the representation of the group via permuting eight elements, rather than seven; which I simply found online.

[@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!

[@AnotherRoof]

Thanks for watching and welcome to the channel!

[@TacioMedeiros0]

Non-mathematics inclined people: "things started to get complicated at school when they added letters to maths". Mathematicians: "What if I take the numbers 1 to 24 and study all the ways I can choose 8 of them?"

[@ionarevamp]

Letters are just the beginning... before too long you'll be performing abstract nonstandard operations on weird symbols

[@caiodavi9829]

@@ionarevampi think you missed the point of the comment

[@ionarevamp]

@@caiodavi9829 Oh. I thought I was making a humorous observation

[@ionarevamp]

@@caiodavi9829 I obviously don't think non-mathematics people are actually going to go that far, it's more saying that they're ignorant of how irrelevant the existence of letters is

[@elshadshirinov1633]

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

[@benstucky8414]

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.

[@oncedidactic]

major shoutout to combo class whiteboard fall

[@Lilly-Lilac]

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

[@AnotherRoof]

I hope the subject brings you as much joy as it has for me!

[@tanchienhao]

@@AnotherRoofyour videos are awesome in introducing finite group theory for beginners thank u so much!

[@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.

[@nahblue]

I started watching this but couldn't continue until I'd listened to Finite Simple Group (Of order Two) first. Couldn't resist. What a banger.

[@movax20h]

9:50 - Small correction or G is one of the 27 sporadic groups (26 usual sporadics, plus a Tits group). Excellent video!

[@AnotherRoof]

Yeah I cut a few sentences that clarified this where I kind of lumped the Tits group into the groups of Lie-Type. Not strictly accurate but for the purposes of the video and someone's first exposure to the topic I thought was an acceptable amount of handwaving!

[@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!

[@anatheistsopinion9974]

This channel is criminally undersubscribed. Let's spread the good news folks! 3blue1brown should be shaking in his animations.

[@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!

[@luxiel00]

so glad to finally get these sporadics group explained in simpler terms- i feel like i finally understand what's going on!

[@shashvatshukla]

I think this argues convincingly why there are only a small number of these transitive groups, but why are they not part of any other family, this part is still a mystery for me. In any case, this was an excellent video and I have learnt so much, thank you!

[@immortale4643]

Thank you so much for this hell of a thumbnail

[@DeclanMBrennan]

Bravo. You certainly got through a lot there. I'm glad I had a rewind button available a couple of times but it was really enjoyable. I always assumed they named things in Abstract Algebra by blowing up a dictionary and then randomly picking the words as they fluttered to the ground.

[@rielco8442]

2:36 As a Physicist I took that personally (/s, love your videos ❤)

[@ericvilas]

Finally, someone who gets me

[@watcher8582]

It's a bit weird because I'm not sure I've ever met someone not liking SO(2), SO(3) or SU(2).

[@a52productions]

Oof, something about discrete mathematics hurts my brain. Give me an infinite group any day over one where I have to deal with integers! Great video though, even if I struggled to follow it. This kind of stuff is really impressive to me for that reason.

[@bogdanlevi]

The group Z of all integers is infinite.

[@thomaskaldahl196]

I really appreciate you starting with the permutation context and building everynhing as subgroups of permutation groups! It's no replacement for the axioms, but it's a wonderfurly concise way to define all finite groups in a super tangible way. No group theorist could get mad at that!

[@josh34578]

Sporadic groups are weird and that's the fun. Any thoughts on the first Janko group J1? It's not even part of the monster, so it's an oddball, but not so big as to be unwieldly.

[@AnotherRoof]

I have ambitions to revisit sporadic groups in the future, making this video the first in a series covering various constructions. No promises, but hopefully! Also, you might know this already but those "oddballs" go by the name "Pariahs" in group theory; there are six of them.

[@mouduge]

Just discovered your channel through this video, it's amazing, please keep doing what you're doing! ❤

[@dcterr1]

Wow, fascinating video! For many years, I've also been mystified by sporadic simple groups and have wondered why they exist. Your explanation of Mathieu groups and why they are sporadic makes a lot of sense! At some point, I'll have to try to see if I can understand why the other sporadic groups are sporadic as well. Great job!

[@erikross-rnnow5517]

Found you through the test you did with Tom. You have such a natural and exciting way of communicating some of the hardest math I've seen, cheers!

[@przemekmajewski1]

your trilogy on golay codes, symmetries and Mathieu groups is amazing. Simple, yet without any real holes. as a fellow mathematician I can only give a big thumbs up

[@malignusvonbottershnike563]

Every group-related video I've seen on this channel does an excellent job of presenting the material clearly and intuitively. Having recently completed the first year of my mathematics undergraduate, and not having a clue what was going on with groups despite the best efforts of my lecturer and supervisor, these videos really do something for me, and I find myself being able to say 'oh, so that's what [insert thing] was doing all this time, that makes much more sense' haha. Thank you

[@Macieks300]

Very good video. Looking forward to the continuation!

[@themathguy3149]

You sieve such complicated topics into nicely self-contained and reasonably passed videos, is always very clarifying even if we brushed up a little bit of the topic beforehand, keep with the good work !

[@profezmo]

This is so well done, thanks for putting it together, subscribed

[@AnotherRoof]

Thanks for watching, and welcome!

[@DarioSterzi]

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

[@tangentfox4677]

I really love you keeping the mistakes in. Especially cause we get to see that the board is supported by a broken brick.

[@guyedwards22]

It may not turn out to be this profound, but I get the feeling that someday, the complete understanding of The Monster and other Sporadic Groups will reveal something astonishing about the "programing" of reality itself, and will probably end up being another example of mathematical developments preceding deep truths in physics.

[@Arnaz87]

Slowly understanding groups and their structures, and their weird self grouping, you could tease us with the monster for many more group theory videos 😆 I'm really curious with the payoff. I think I will somehow be disappointed yet amazed at its arbitrariness.

[@nice3294]

I was so hyped to see this video, this is such an interesting topic

[@disasterarea9341]

love the group theory content. i got a masters degree in maths and group theory was my fave/best part, rly wanted to learn more abt the classification of finite simple groups but hadnt seen any youtube content on it and i work professionally w databases now so

[@dickybannister5192]

thanks. great video. remember reading Finding Moonshine by Marcus Du Sautoy years ago and thinking I really need to get more rigorously into the derivation of Group classification. there are some great stories about some really interesting people. Conway, obviously, but people like Norton too.

[@mbdtsmo]

Very instructive. Are you planning a video on the pariahs as well? Those seem even more mysterious to me…

[@decare696]

31:10 lmao "3 1 blue brown" is so funny for some reason

[@greymonwar9906]

The best video on finite simple group ever

[@pmadjidi]

I enjoyed this one, specially the connection to Convey groups and the Monster... very well explained. Group theory is a jewel of mathematics... Good old Pascal triangel and the complexities emerging from it....

[@Nylspider]

31:10 "In this 3 1 blue brown video..."

[@Wielorybkek]

crazy stuff! loved it!

[@e8root]

My brain melted somewhere near beginning of this video.

[@generalyoutubewatching5286]

I am so glad this video exists!

[@SWebster10]

One of the most interesting groups is the M62, in which all the elements prefer Rugby League to Union

[@drorfrid]

2:35 Hey! No dissing infinite groups on my watch! They are symmetrical and pretty with the main two examples (in my head) are lie groups (groups of matrices) and the free group. I study Geometric Group Theory, were a good way to think of the free group is as the fundamental group of this shape: ∞.

[@sumdumbmick]

lamenting the loss of Mathieu specifically, due to disinterest among the mainstream at the time, is odd, because it's something that plagues all mathematics and sciences all the time. why was Australopithecus, originally discovered in 1924, ignored for about 3 decades before being accepted by mainstream science? because the mainstream believed Piltdown was genuine, and they believed it so hard that they refused to even chemically date Piltdown for 4 decades after its first presentation in 1912. for context, the chemical dating technique used was invented in the 1860s, and was used on the Calaveras Skull in 1879, so it was absolutely a deliberate choice not to verify the age of Piltdown. a little closer to home, Evariste Galois was initially ignored by French mathematicians whose ability to use subtraction is now rivaled by second graders in the US, on the grounds that Galois was not sufficiently rigorous. Logicism itself was more or less officially seeded within a year of Galois' death when George Peacock called out French mathematicians for their absurd misunderstanding of how subtraction works. and ironically, even this account of subtraction continued to misconstrue it as a non-commutative binary operation, failing to recognize that it's simply a special case of addition, and thus fully commutative when handled correctly. but when I say this, I get told that I don't understand... carrying on the tradition of ignorant, dogmatic people constantly winning out simply due to being bullies.

[@jakeehrlich8113]

This is for sure one of the best series on group theory on KZbin that I’ve thus far found.

[@Adityarm.08]

Kisi incredible explanations. Thank you.

[@MrMctastics]

I did some scratch work a couple years ago and there was some kind of relationship between adding up the first few numbers of n choose i != a power of two => something about the 3x+1 problem. Dang I should look through my notes now

[@erikvanderplas4175]

Very clear explanations here, really helped me get some more intuition into the sporadic groups while writing my math bachelor thesis. Thanks a lot! Would also be really cool to see a series on the different sporadic groups :) not just covering Mathieu groups, but also Conway, Janko, Monster, etc.

[@AnotherRoof]

I have ambitions to dip back into various sporadic groups in the future so stay tuned!

[@dylanrambow2704]

Brilliant video!

[@EpicMathTime]

Oh boy I can't wait to hear this one

[@user-gi7ru7mv2f]

Nicely explained!! Thanks

[@avramlevitter6150]

Group theory was one of the most fascinating courses in my degree. I'm very sad that I wasn't better at it.

[@mr.inhuman7932]

Amazing Video!

[@ditch3827]

Great video - I do hope you do do a video on Lie type groups

[@eumorpha876]

you mean like politicians?

[@ditch3827]

@@eumorpha876 🤣🤣🤣

[@NotSomeJustinWithoutAMoustache]

This video is way, way over my head and it doesn't help that it's late right now. I'll try watching this when I'm not sleepy so I actually get more than 10% of what you're saying.

[@yto6095]

i checked out codes in other bases, it looks like the only other possible perfect code (where the radius is at least 2 and isn't n) corresponds to (11 choose 0)+2*(11 choose 1)+2^2*(11 choose 2) being a power of 3 (specifically, it's 1+2*11+4*55=3^5). but i'm not sure if there actually is such a perfect code, and i don't have the time to look for it right now. if it exists, then maybe some of the other sporadic groups arise from it

[@AnotherRoof]

So there is actually a perfect ternary Golay code I didn't mention in the video because I didn't want to introduce another big structure, but its automorphism group is actually M11 (and M12 for the extended ternary Golay code). So they are other ways of defining M12 and M11 instead of the dodecad way I used in the video. Thanks for watching!

[@000bHd000]

Literally was looking for an easier to digest introduction to the Matthieu groups/ other sporadic groups and then this video pops up in my feed

[@personperson278]

29:50 I think the codes corresponding to the first half of a row of the Triangle actually are the simple repetition codes with two codewords, and the Hamming codes correspond to the cases where the first two numbers in a row add up to a power of two. (Hamming 7,4 corresponds to 1+7=8 on row 7…)

[@saltyowl3229]

I wonder if these groups’ numbers could be used to prove strange things with M-theory multidimensional interactions, maybe even something testable

[@Holsp]

You said every shuffle has to have an inverse. Does the inverse also have to be one operation, or can it get to it's starting state as multiple operations?

[@not_David]

7:38 I don't know if i've ever encountered a field of mathematics with as much notational inconsistency as group theory (and more specifically discrete groups). I've always thought this is partly due to how applicable and important they are in chemistry, quantum mechanics, and solid state physics, and everyone has just kind of evolved to using their own notation. It was perhaps the most annoying part for me when I was learning group theory. I think continuous groups are primarily used in my theoretical settings and so the notation hasn't diverged as much, though given most of my experience is with discrete groups I could be wrong in this.

[@AnotherRoof]

Thanks for watching, good to see you as always! It's awful. My original draft of this video had a three-paragraph rant about notation >_

[@not_David]

@@AnotherRoof I always tune in to see where your prop game is at and I feel like this is the best one yet in terms of that

[@AnotherRoof]

@@not_David Thanks, I appreciate that! Looking forward to your next video :)

[@angelmendez-rivera351]

@@AnotherRoofCan we get an outtake video with the rant? I'd love to listen

[@AnotherRoof]

@@angelmendez-rivera351 Haha it was in the script but I didn't shoot it unfortunately! I'll share what I had in my second draft: Annoyingly, some authors use D8 or Dih(8) because that’s the size of the group. Already we’ve hit upon the worst aspect of group theory and that’s its infuriatingly non-standard notation. It’s bad, and makes self-study very difficult, with many authors just stating things like Dih(8) without clarifying and expecting the reader to keep up. It’s bad practice at best, and kind of arrogant at worst, and why journals stand for it I have no idea. It would be like every chemist using different symbols for the elements and just expecting readers to know what they mean.

[@SwordQuake2]

Yeah, I've studied some part of group theory. It was horrifying.

[@tom7]

Very good!

[@josh34578]

Is there a nice way to generalize transitivity to General Linear groups? Maybe looking for subgroups of GL_n(F) that act transitively on m dimensional subspaces?

[@rogierbrussee3460]

This is really enlightening.

[@chiefsnoz]

"Don't get me started on notational inconsistency" ❤😂

[@binathiessen4920]

This is exactly the subject I was wanting to learn about.

[@dhonantarogundul1737]

I really love the intellect face diagram lmao. I would put my pfp far on the left.

[@beamathematician2487]

Thank you so much for this beautiful video. You are group theory expert person. I have an idea on representing number using different digits, for example, any integer n can be written using cubes as a digits where represention is not unique but + - pattern preserve. 78 = -1³×10³+2³×10²+3³×10+2³. In similar fashion we can express all integer but the Question is, how to collect all the base set that spans set of integers. This topic pretty much included group theory. I'm looking forward to someone for collaboration. What should I do?

[@caspermadlener4191]

The question of when the sum of the first three terms of a row of Pascal's triangle is a power of 2 is actually a famous problem, with its own wikipedia page, equivalent to the diophantine equation x²+7=2ⁿ. Ramanujan conjectured it, and it took half a century for it to be solved. What I find interesting about this problem is the simplicity of its proof. It took me a few hours to proof, entirely via elementary methods available in the time of Gauss. But I only worked on it because I was told it was doable for me.

[@dcterr1]

I also understand that simple groups play an imporant role in the theory of solvable and unsolvable Galois groups, both of polynomials (whose Galois groups are finite and thus involve various sequences of finite simple groups) and of nonintegrable functions, like e^x^2, to which a version of Galois theory can also be applied to show that they are nonintegrable for the same reason that the roots of general polynomials of degree 5 or larger cannot be solved in terms of radical expressions. I wonder what roles if any sporadic simple groups play in either of these versions of Galois theory. I'd greatly appreciate any info you can provide on this topic.

[@quarkonium3795]

It would so awesome if this could be the first part in a four-part series about the sporadic simple groups, with the first three covering the 3 generations of sporadic simple groups and the last video talking about the Pariahs. The other generations seem complicated but I've been looking for a reasonable-ish explanation of their properties (especially the monster). The pariahs seem to have even less easily available information on them

[@AnotherRoof]

I intend for this too be the first in a series about the sporadic groups, but it'll be more than four videos long! Unfortunately the others are more complicated and so far the easiest construction of the monster is still pretty... monstrous. I'll visit the topic one day though for sure!

[@quarkonium3795]

@@AnotherRoofGlad to hear it and glad it's going to be even more than four videos! I kind of figured it would take a while given that the paper cataloging and proving the completeness of the set of finite simple groups is one of the largest mathematical projects in history. I would much prefer a well-researched, detailed explanation of the subject that takes many months to years to complete than an oversimplified and unsatisfying video series rushed out in a month or two. In the meantime, I'll watch your other content since I only recently discovered your channel and it seems like you have a lot of great stuff here-keep up the good work!

[@Criz454]

goddamn i love freaky maths, i should definitely get into group theory, sometimes i hesitate but this channel is an instant follow

[@iamtraditi4075]

Excellent!!

[@shingsey8349]

Not gonna lie : I'm a student who is doing sort of a extra scolar work on redundancy and theory of information, especially on Hamilton Codes. I did not expect skipping the intro because i know well how groups work, and just randomly having a group theorist talking about perfect codes xD. Really cool video tho

[@vrixphillips]

really? you studied E8(2)? :O that's so exciting! have you made a video on it? I'd love to hear about your research!!!!! It's such a beautiful group, just aesthetically speaking (and that's just in 2d and 3d representations!)

[@polyexp]

Your channel is really underrated.

[@reecec626]

I could watch this guy all day long.

[@alexpotts6520]

The classification of finite simple groups is one of the most impressive things humanity has ever done. It's on a level with sending astronauts to the moon.

[@ictogon]

wrong. Lil B's discography is the most impressive thing humanity has ever made

[@emmabellhelium]

8:11 Would it be accurate to say two permutations are in the same class if both have the same number of objects of each period when the permutation is repeatedly applied to itself? Identity: four period 1 objects Swaps: two p1, two p2 Double Swaps: four p2 3-Cycles: one p1, three p3 4-Cycles: four p4 Does this hold for permutations of n>4 objects?

[@AnotherRoof]

Permutations in the *symmetric group* are in the same class (known in group theory as a "conjugacy class") if and only if they have the same "cycle type" which is as you described. It's a little more complicated for other groups. Because it isn't that relevant to the rest of the video, I skipped past the detail here but I do go into a bit more detail in the extended cut available to Patrons. Hope that helps! 🙂

[@sonarbangla8711]

Very well explained. Why do you resort to 'octonions'? What is so special about number 24?

[@tr0m]

another banger 🔥🔥🔥🧌🔥

[@duhfish6030]

I know you slandered infinite groups, but can we get a geometric group theory video? It's a really cool topic

[@stichomythia3435]

oh yes please! My PhD was on Thompson's group V (a finitely generated infinite group that contains every finite group as a subgroup); it 'feels like' an infinite symmetric group so would fit in. I don't know near enough geometric group theory so would love that

[@duhfish6030]

@@stichomythia3435 Oh wow, could you send me a link to it? I learned a bit about Thompson's groups from of my professors. There are lots of really cool ways to represent them. I'm surprised you don't know geometric group theory, as I don't know where else they would be relevant. Algebraic topology? Small cancellation theory?

[@myca9322]

the gagging at infinite groups is sad :( lie group theory is beautiful!

[@Achrononmaster]

Excellent. Ding dong, I bid you Abel.

[@exponentmantissa5598]

I liked this. I am a retired electrical engineer/physicist and math is my passion.