Researchers Use Group Theory to Speed Up Algorithms - Introduction to Groups

[@Nemean]

Here I'll present the solution to my challenges. Because KZbin doesn't have spoiler tags, I'll leave them as a comment to myself.

[@Nemean]

First the second challenge at 22:34 The only interesting example with groups from this video is in Q*. It generates the elements 1 and -1. Less interesting examples are in Q* or in Z and Q+. They just generate 1 and 0 respectively. Now the first challenge at 17:57 This one you can just google haha. I recommend this thread: math.stackexchange.com/questions/616577/any-set-with-associativity-left-identity-left-inverse-is-a-group-fraleigh And by the way, the two graphs at the beginning are actually the same. They are both instances of the so-called Petersen graph.

[@hadinossanosam4459]

You might want to pin this :)

[@msq7041]

@@Nemean if you had a group of real numbers and modulo 1 addition would every generated subgroup already implicitly include 0 and the inverse element, or would that only hold for rationals?

[@Nemean]

@@msq7041 You're correct, if I read your comment correctly. Proof for the sake of completeness: If x is rational with denominator d, then adding x to itself d times gives you a whole number, i.e. something = 0 mod 1. If x is irrational, no multiple of x ever gives an integer and so is always ≠ 0 mod 1. This means you have to include 0 (and inverses) explicitly.

[@omerd602]

For the second challenge, would, say, a 60 degree rotation in the group of all rotations of a circle work?

[@samu_2822]

Bro casually created one of the best group theory intros out there, left a hangclift end and refused to elaborate further (at least a year after)

[@LabLeonOfficial]

agreed

[@thanyaniinnocent940]

Was thinking the same thing

[@wrog268]

@@thanyaniinnocent940 hope he didn't fall of a cliff

[@karig01]

Two years later we still have no follow-up

[@ramimohamed4255]

Bro said I will be back with part two then bailed for 7 months. I have seen this video first when it was 2 months old. Don't make us wait any longer, I BEG YOU!

[@swaree]

this channel is bound to become an example of high-quality, aesthetic, clear math/cs videos, keep it up

[@vivvpprof]

*Note to future self* For the record, I subscribed when the subscriber count was 61.9K.

[@ShauriePvs]

@@vivvpprof I did when it's 65.6k

[@vivvpprof]

@@ShauriePvs Good! This way we can track it if more people relay the number here.

[@user26912]

@@vivvpprof 69K (but was subscribed already since the Quake algorithm video, don't know how many subscribers were there then)

[@nickgardner5641]

74k

[@diegodoesstuff86]

Still waiting for part 2 of this amazing series

[@Grassmpl]

It's a power series

[@uhu2323]

This is some 3b1b level education. At some point this channel will blow up.

[@quantumgaming9180]

Indeed. I remember when he posted his first video about Quake 3 algorithm and I was left speachless at the algorithm itself and the amazing way he presented it. I rewatched the video everytime I got it recommend just to be impressed again.

[@jeper3460]

I disagree. This is much better than 3b1b’s stuff, at least his video on groups. I saw 3b1b’s video on groups, more specifically the monster group, and his approach to explaining groups was like “I’m not gonna give you the hard axioms because that’s so confusing, so I’m gonna give you this vague analogy about symmetries (which admittedly works for one type of group)”. When working with such a complex yet widely applicable concept like groups, a video like this is much better in my opinion; first giving the hard rules/axioms of groups, and then giving examples.

[@janmatula1534]

@@jeper3460 I was a but disappointed not to see the more intuitive cayley graph explanation of lagranges theorem

[@mitchtom1409]

not with this upload frequency it won’t

[@nictibbetts]

It’s already did blow up. Check his first video.

[@RB-fp8hn]

As a mathematics student who has since then become an AI researcher, I want to say that mathematicians have done the opposite of keeping things from others. It's just that every time a mathematician wants to tell people how amazing mathematics really is ... people run away screaming, "oh no, math ... algebra, eww, oh no!"

[@shenraine]

There are two types of people. For some of us math is a turn on.

[@johnbaxter7582]

Accurate

[@ezzouaouia.r1127]

so true.

[@kobtron]

Maths is one of those secrets that protect themselves.

[@benjaminmartin6294]

I'm a cs and linguistics student and the maths professors are doing you no favours. My algorithms class is basically applying all that we've learnt in graph and set theory and it's so much more interesting. The math classes are just so theory based and are interesting on the surface but the whole process and approach just aggghhh

[@Graham_Rule]

It is over 10 years since I looked at any group theory and even then it was only at a fairly basic level. I look forward to seeing your further videos as your style of presentation is great.

[@byronwilliams7977]

Hasn't been 10 years for me, but it might as well be. I don't remember much, and I second that. I think this video is excellent.

[@mujtabaalam5907]

This is a fairly basic level

[@AtticusDenzil]

he never gets to the point though ...

[@nuke_parasitine]

5 years for m

[@dieSpinnt]

Please calm down and just don't do that shit what Évariste Galois has done. ( Yeah, the 31 May 1832 is more than 10 years ago, but there aren't so much black humorous group theory jokes available, yet ) Anyway, for some reality connections, maybe? A book tip: "Group Theory in Physics. An Introduction", by J.F. Cornwell "Introduction to Symmetry and Group Theory for Chemists", Springer, Arthur M. Lesk "Symmetries and Group Theory in Particle Physics: An Introduction to Space-Time and Internal Symmetries", Springer, Giovanni Costa, Gianluigi Fogli "Matrix Groups: An Introduction to Lie Group Theory", Springer, Andrew Baker

[@dewaard3301]

I've been looking my whole life for a series on Group Theory, ever since I guess I heard about 'The Monster'. And now it seems I finally found one that starts from zero, is narrated by a pleasant voice, and has high-quality visuals to illustrate the concepts. Really looking forward to this entire series.

[@hughcaldwell1034]

I believe Socratica has some good beginner videos on the subject - though I'm also looking forward to this series. NJ Wildberger might have a lecture series on it too. His videos are here on KZbin and, while he has some funny ideas about infinity, he's a very engaging and clear-spoken teacher.

[@Mr.Nichan]

I was familiar with a lot of what he talked about because I somehow acquired this little book called "Teach Yourself Mathematical Groups" (Bernard, Tony, Neil, Hugh) from my mom (I think a library was getting rid of it?), and worked through something like half of it. You can get it for < $6 on Abe Books, not that it's necessarily the best one. It has a bunch of practice problems with a lot of focus on proofs from what I remember.

[@mihailmilev9909]

exactly! Thanks for reminding me of that. 3b1b?

[@mihailmilev9909]

@@hughcaldwell1034 thanks

[@mihailmilev9909]

@@hughcaldwell1034 wym by funny lol

[@PixelSergey]

This is the most beautifully animated intro to group theory i've ever seen!

[@augusto256]

I have to say that it's unbelievable that you have only four videos and yet I consider this one of the best channels on KZbin.

[@juanignaciosilvestri6782]

checking every few weeks to see if second part is published!!! looking forward to it!

[@minerharry]

This is by far the best group theory video I have seen. This is the first one that let me truly understand it from a casual perspective - even 3b1bs videos seemed opaque about the mechanisms of groups. This lays it out so nicely and concretely it’s hard to get lost at all! I cannot WAIT for more. Instant sub

[@cubething-x64]

As a math major who now works as a programmer, it's really cool to see how my favorite subject relates to the work I do now. Thanks for making this, looking forward to seeing more.

[@mango-strawberry]

Are you working as a SWE?

[@cubething-x64]

@@mango-strawberry yes

[@Socsob]

Hey I just graduated in theoretical math, and now I'm going into coding. Exactly why I clicked on this video

[@mango-strawberry]

@@cubething-x64 cool

[@Iogoslavia]

It's kind of ironic that in the end they didn't circle back to show how groups are used for graphs

[@kummer45]

This man made a clean introduction to abstract algebra. It makes perfect sense.

[@Simchen]

What a fantastic, well structured, visually pleasing introduction to group theory. This video deserves the highest levels of praise. Regarding the question at 0:36 - I know a Petersen-Graph when I see one! For my Diploma-thesis in mathematics I developed a program that uses spring-force algorithms to calculate different stable versions of how to draw a graph. The Petersen-Graph was one of my test cases. So that is how I know that these two graphs are in fact isomorphic just by looking at the shapes.

[@simpleffective186]

Sounds cool! Is it published?

[@skewer45]

As an aside, you might be interested in these works as well! They also use a simulation-like method to simplify graphs, but apply it to 2D- and 3D surfaces instead. This allows one to unravel shapes such as complex knots and twisted, nesting tori, and identify the isotopies between them. kzbin.info/www/bejne/Y6a7d4ymi9qgnpY kzbin.info/www/bejne/qXvKfGOgn8lppac

[@ClaudioParraGonzalez]

@@simpleffective186 you should never ask a doctor whether their thesis is published. For mont people it brings PTSD... [please read this comment as an inocent joke]

[@Copperhell144]

Is it enough to say that two shapes are isomorphic if they have equal amounts of nodes that have equal amounts of connections?

[@Szyman91]

@@Copperhell144 I had the same question in mind, please let me know once you know

[@Golden63]

I remember I had to learn group theory as a chemistry major to understand molecular symmetry and the nature of chemical bonds. Your video is so well made that I would recommend it to any chemistry major.

[@bdnugget]

Same, just wish I had this video back then lmao

[@mihailmilev9909]

50th like

[@mihailmilev9909]

wow I didn't know that that was required. Thanks!

[@lorenzoaste9173]

Beautifully made video, it’s been years since I’ve found a video so captivating. Not only is the editing top notch, the math is explained well too.

[@SlingerDomb]

You did summarize the abstract algebra course i took for 1 semester really elegantly. I really greatly appreciate your work.

[@sandybeach95]

This is one of the most eloquently made and beautifully explained math videos I've seen. Great job! I'm looking forward to future videos

[@nolanalexander8696]

This channel (and video specifically) prove that there are people that are good both in explanation and visualizing them with animation. I learn group theory back then at uni and sadly not gave them too much attention because the lecturer is boring. Listening to this channel explain the characteristics of group it suddenly make sense, especially when you applied them through Integers, Rationals, Cyclic, etc. The animation style is smooth and comforting. Hope you post other video about topics, especially for prospective computer scientists.

[@philkaw]

22:00 I believe the generator of an element is defined usually to be the intersection of all subgroups that contain a. And kn general is the intersection of all subgroups that contain S where S is a subset of the group. This takes care of the infinite groups.

[@nickm3694]

Before this video, the only other video I had seen on group theory was the one 3b1b made about the monster group, and when I first watched that I was mind boggled how we could even go about beginning to prove things about such abstract concepts like symmetry. After watching this video though, I feel like I got a new insight on how proofs could be derived and built on each other that I was really looking for a while ago! I do hope this series continues, as this whole subject seems really captivating, but the internet seems to be sorely lacking in digestible content about it.

[@tomaso0]

Your channel is absolutely going to blow up. Great content and gorgeous presentation man

[@moonyeclipsey]

This is really good, I didn’t even realize it was 30 min until i finished the vid. Good job!

[@jaopredoramires]

I am absolutely amazed and stunned by both the visuals and the didactics. Incredible material, godlike stuff really. Congratulations man, you've got my maximum respect

[@AB-gf4ue]

I hope you finish this series! I've been at the edge of my seat for months!

[@tastygold]

I have long considered your video on the fast inverse square root to be one of the very best mathematical videos on all of KZbin. This video definitely lived up to that legacy. Excellent work :)

[@Dom89_]

Im loving these soME2 videos. cant stop watching them

[@chriskevini]

You did an amazing job with the animations! Thank you KZbin gods for showing me this gem of a channel

[@pawebielinski4903]

I really admire the way you use those gorgeous visuals to aid understanding.

[@louisauffret]

Something really fun about group theory is that it shows up where you don't expect it, the picture usually used to describe group theory is a Rubik's cube : - each action (combination of rotations) is an element of the Rubik's group, composition is applying actions one after the other, so it's closed under composition - the action "doing nothing" is the neutral - associativity checks out because applying (A then B) then C is the same as A then (B then C) - each action has an inverse I don't know exactly how many actions/configurations are possible on a Rubik's cube, but if you take all the configurations where only 2 opposite sides are being rotated, you notice it's a subgroup containing 16 actions, and you can just tell it's a multiple of 16 using Lagrange's theorem. Isn't that crazy ?

[@pwnmeisterage]

A standard 3x3x3 "Rubik's" cube has over 43 quintillion permutations. And almost 500 billion quintillion "illegal" permutations - arrangements which cannot occur during normal rotations (and which will not result in a "solved" state) - the sort of thing which happens when cheaters physically deconstruct the cube to move pieces or stickers. There are many algorithms to solve the puzzle. Some are incredibly fast and efficient, but they're all plodding brute-force sorts of approaches, they largely ignore the state of the cubelets and methodically rearrange all the pieces from top to bottom. No algorithm exists (yet) which can assess the state of all movable cubelets then immediately devise the minimal path towards solution. Likewise, no algorithm exists (yet) which can devise the maximum possible "mixed" state.

[@techstuff9198]

My heavily exhausted mind asked "Does rotating the cube without shifting the pieces count as an action?" like it wasn't obvious. I'm going to take a nap.

[@GrGalan6464]

@@pwnmeisterage There actually is a way to get an extremely move efficient method, through computers. We do actually know actually know the most scrambled state, with the number of rotations of the cube being known as “Gods Number”, 20. The minimum value was discovered to be 20, since 20 moves are required to make a “superflip” pattern on the cube, where all edges are flipped in their place. Then, a massive sum of computers checked through all the possible scrambles to confirm that there was no other higher move count state. Although, we still don’t know what percentage of scrambles require an x number of rotations (since the computers code stops looking for a move efficient solution after it gets one in 20 moves, for time efficiency reasons). Although, it is predicted that most scrambles have at least a movecount of 17-18. There is no such thing as a “perfect method”, you’d need to be a god to be able to figure that out. Computers, however, can get very close, using a very efficient method. I wouldn’t call the method that we use to speed cube “brute-forcing”. Algorithms for speedcubing are never usually intuitive (except for ~400 3-style algorithms used for extremely advanced blindfolded solving, or when you are inventing new algorithms, you should check out some of the logic for how they work if you are interested). Instead, we rely on muscle memory, otherwise we would have to mentally memorise algorithm in cube notation and convert it to actual rotations. Memorising 43 quintillion 18-19 move algorithms is impossible. Instead, by learning smaller sets, we only need to feasibly 1 algorithm for our first time solve, to maybe close to 150-200 move sets for advanced solvers. Not only that, but you also have to plan out the cross (which you always do intuitively, never with algorithms), and usually predict the next steps in advance before even beginning to turn.

[@yashaswikulshreshtha1588]

@@techstuff9198 No one replied to your comment weird.

[@techstuff9198]

@@yashaswikulshreshtha1588 The answer is "yes", because it counts as a transformation for group theory's purposes.

[@ooogabooga5111]

NOOOOOOOOO the video is over, I want more. This is sooo good wtff

[@L.Mandrake]

I literally cannot wait for the next part of this video. I've watched it days ago and I can't stop thinking about it! Amazing exposition. Thank you.

[@alexenriquecrispim830]

I was really expecting more videos, thanks! Going to watch right now.

[@camilocagliolo]

Beautiful visuals! There's a lot of love and effort put into this. Very clear explanations as well. I'll be waiting for the next one.

[@nomukun1138]

This was really cool and I'd love to see your continuation of Group Theory topics and eventually the graph isomorphism result. Thanks for what you have now. Life must have gotten in the way of continuing your project. Hope you're doing very well and staying safe from any violent Greek heroes!

[@Nemean]

All youtube comments should be written like this

[@rsv9999]

i second this

[@schenzur]

I feel this video is why mathematicians get so self-absorbed with math itself. In quest to find the solution of the graph isomorphism, I think all of us found great pleasure in the iterative process of finding strange theorems, and in doing so forgot the aim, and in the future videos I hope you keep on doing this.

[@suicraft8395]

Omg, you're back, i watched you video about the fast inverse square root algorithm and i never expected to find this channel again until now. Good job and good luck

[@Quaquaquaqua]

As a programmer interested in math this is best introduction to group theory I’ve seen

[@SlackwareNVM]

I love how the video only began to tackle the idea posed in the title. Makes me excited for future videos.

[@lexinwonderland5741]

as someone from the world of abstract algebra and group/ring theory, i thought your explanation was BRILLIANTLY accessible and you managed to plant the seeds for figuring out the Sylow theorems without feeling forced. but i wanna know the graph problem now, damn it!

[@redrikh_putiata]

it's the best popular introduction to the group theory I've ever seen. Thanks a million. Looking forward to the next video.

[@angel-ig]

Probably the best introductory explanation of group theory I've seen. You made several things click for me. Hoping to see more of the series!

[@mikkolukas]

Damn, this is a good video and explanation - and the animations support it *SO* well!

[@rainrunya]

5 months ago, you said you will be uploading "A piece, the likes of which have never been seen before." . 5 months of anticipation, with the only clue being a GIF of a spinning icosahedron. 5 months later, you did not disappoint. Fantastic video!

[@nitsanbh]

you summed up two semesters of "advanced algebra" into one very reachable video

[@shortc1rcuit326]

Genuinely the best explanation of group theroy I've seen. Bravo!

[@limitless-codes]

Aside from everything, lets appreciate the content quality of this channel!

[@triggerhappy899]

I’m a math major and when I heard “there’s a field called abstract algebra that no one has ever heard of”, my only thought was “ yeah and you want to keep it that way”. My god that class was hard

[@DuckGia]

Not that hard

[@Loots1]

Im in it right now,theres a lot of remember but its really cool because you can see the power pf these tools

[@DuckGia]

@@Loots1 Remember?

[@tmann986]

Would you put it above real analysis?

[@Namegoeshere-op9hg]

@@tmann986yes. Real and complex analysis was easy. I lost it at the Sylow theorems

[@matteopiccioni196]

The quality of this video is 10/10

[@daniilgavrilikhin34]

You are finally back! Really love your videos. It was a pain to know that there is only three of them :(

[@kgangadhar5389]

Thank you for the video, waiting for your upcoming group theory videos!!

[@DamageIncBHS]

Amazing description and teaching style. If I had this resource in my first abstract algebra course, my junior/senior years of undergrad would have been so much easier. Thank you!

[@lih3391]

Best explanation I've heard! Thank you for not dumbing it down or overcomplicating so that we actually progress at a decent pace!

[@kono152]

I hope you'll continue this series, I really wanna learn and understand the algorithm now that i know about graphs

[@projectmanager2878]

Extraordinary! Well done. So many KZbinrs try to teach math but none of them even come close to your video. And I love the other, mostly helpful, comments.

[@animowany111]

This is great. I completely forgot about the graph isomorphism at the beginning until you mentioned it at the end. I hope you go into some of the scarier groups and at least mention the monsters. I tried looking into truly understanding group theory in the past, but the text I found that enumerated all the groups was incredibly dense. Would be nice to have a link to the lecture mentioned at the start, too!

[@Nemean]

The lecture is in the description. And yes, my goal is to increase the difficulty in groups as the series goes on. I can mention the monster if you want, but don't expect too many finite simple groups in this series. Personally I'm still working through Wilson's book on them and my god, are they complicated.

[@110jmartin011]

Well, I would of really enjoyed having matlab up in the brackground, or what ever tool he used for graphs lmao , but it was not the less a good lecture

[@numbers3268]

@@Nemean i'd personally like to see some of the simple Lie groups, like U(n), SU(n), and/or O(n), mainly because of how they come up in quantum mechanics. a prime example being the gluons, chromodynamics, and SU(3), in that from what i understand each of the eight gluon types corresponds to one of eight generators for the group, but i haven't yet found a good visualization for what the group is doing past 'the generators are made up of a triplet of commutative ones, another of anticommutative ones, and two seperate ones that are unchanged for either 2-cycling the colors or 3-cycling the colors'.

[@Nemean]

@@numbers3268 I'm no physicist, but doesn't this correspondence come from the irreducible representations of SU(3)? Because apart from the fact that I genuinely don't know QFT or gauge theory or whatever theory this belongs to, computer scientists use representation theory more like number theorists and less like physicists, so I'd first have to get familiar with the ways of the physicist. What I'm thinking of doing though is covering the affine group, which is used a lot in relativity, but no promises. Would this maybe interest you? I'm genuinely curious, because I have no idea what physicists are up to these days.

[@numbers3268]

@@Nemean im not a physicist _or_ a computer scientist (and i've only dabbled in number theory so far), so what i know is effectively grasping at straws (maybe sometime i'll find a textbook i can get into and find my way from there). i'll still watch the next video(s) though, this one was certainly interesting

[@WorldsBestGuys]

At 18:50 you say what’s the point of all this abstract stuff if we can’t do anything concrete. I 100% agree as an applied mathematician, but I am so thankful there are super nerds who don’t care about the concrete in the present and make up all these insane theories that seem useless at the time and then end up working out for our benefit. Hail the super nerds

[@zenzizenzic]

this video is really good! its got a nice quality and visual aesthetic to it, that I rarely see in most group theory videos, that matches how intuitive the idea of groups ought to be.

[@kmjohnny]

This video made me very interested in Group Theory, and making it's sequel my most anticipated YT video as of today.

[@michaell01]

One of the greatest mathmatical videos I have ever watched on youtube! You are doing an amazing job of introducing true mathematics to the broad audience, and to refresh our knowlegde of it. Keep on the great work! 👍

[@argandzero0]

Very well explained, I'm kind of envious.

[@nice3294]

I love the visuals to this video, and the concept was explained well

[@shashvatshukla]

I finally understood Lagrange's theorem thank you so much.

[@titusjung3023]

It's been more than 10 years since I studied group theory and I find your video to be highly engaging. Looking forward to the next one!

[@epistemicompute]

Never prouder to be studying cs at UChicago until I heard Babai

[@arkasamajpati5319]

Bro This is the first time I heard about group and you already got me hooked, you are a great teacher.

[@mohammadresli6319]

Woooow, even though I've studied group theory back at university, I felt now that I'm learning a new freaking thing, you ACed it man, keep going.

[@trakksfendacre]

Man, this video is my favorite of yours. I'm so glad computer scientists and students can discover how general and powerful algebra can get. Plus, it is animated just right. Keep it up !

[@JohnAlamina]

You're a rockstar. I'm looking forward to future uploads

[@nekai4570]

Great beginner-friendly explanations, I dig the format and the animations. Looking forward to future videos/

[@mkuts120]

Extremely high quality. Very good video

[@trikeshed]

What a beautifully presented video. I am in awe of the graphics, and the explanations were so clear. I know this stuff from university (45 years ago) but it felt like you covered half a term's group theory lectures in half an hour!

[@CodyDanielson]

This must be the most intuitive explanation of subgroups and cosets I have ever seen! Can't wait for the next video!

[@arongil]

You have animated the most beautiful introduction to group theory. Well done. Your video could serve generations of students. I love the spunky color scheme, too! Simply excellent. Thank you for your contribution to SoME#2 !

[@danuso5172]

Super cool, man! Best wishes from Brazil.

[@michaelperrone3867]

Thanks so much for this description of group theory: framing it this way is perhaps the ideal way to introduce the subject.

[@notonlyit1361]

Today I gave my lecture on group theory and your video showed up in my feed. Glad to see a good explanation of this really beautiful field!

[@jadeglaze3390]

Nemean, this was fantastic. Easily the best group theory video I’ve seen. Can’t wait for the next video!

[@snowy0110]

If only I had had such beautiful intros when I was a student, I would have chosen a different pathway in my life. Truly, internet and smart people make our place better. Thank you, author!

[@Kralasaurusx]

Excellent explanation, and the animations are top notch. Good animations make a huge difference when explaining stuff like this. I lol'd at this part: "You already know that that'll cancel - I'm not even going to bother to animate the whole ordeal." I don't blame ya - the animations probably take a ton of time and effort. Anyways, keep up the good work!

[@EngineerNick]

Your introduction is brilliant :)

[@rizalpurnawan23]

Actually, group theory (as well as abstract algebra as a whole) is indeed the most beautiful math subject. Basically, I wasn't a mathematician, I have a bachelor's degree in civil engineering. I came to study math by self a few years ago, driven by curiosity. And I got interacted with abstract algebra 2 years ago. And I feel my mind blown by the beauty of group theory since my first interaction. It makes me cannot stop learning math. By the way, your presentation is awesome!

[@plasmarob741]

I am a computer scientist like you and these explanations are astonishingly well done.

[@아노미-g6f]

It is always refreshing and exciting to have an introduction to mathematics by someone from another field. Cool!

[@maximefournes9148]

Amazing introduction to group theory, I cannot wait for your next video!

[@arisweedler4703]

Awesome. Group theory is so satisfying to learn about. Great clarity with this video!

[@jamessinka]

Please never stop creating content, this is excellent

[@monsieuralexandergulbu3678]

Oh, your content is always so juicy and just shines with its high quality! Thanks, we appreciate the effort 🤝 Your work deserve much more appreciation!

[@mr2octavio]

Not to compare you with 3 blue 1 brown, but you're on the path of success. I love your videos

[@jadeyjung]

this is the very first video i watched on your channel and it's insane i bet you deserve a lot more subscribers thanks for your effort

[@metiscus]

Best introduction to cosets I've ever seen. Thank you!

[@peppy3519]

This video made me understand group theory, thank you!

[@kkski4817]

Beautiful animations and graphics

[@alexbos8211]

Very clear explanation - fascinating subject as well

[@edvogel56]

I now feel like I understand subgroup and coset to the degree that I can see their applications. Previously I was speeding through group theory to get some "aha" moment that illuminates all. It's "aha" all the way down now. Thanks!

[@Zer0CraftGaming]

I love that you touched on the 3-6-9 control theory without saying it. The subgroup portion really helped tie your justification for left sided neutrality - absolutely beautiful. This is light years beyond me but you made it make since- touche Sir.

[@soapmcsoaperson]

Can't wait for the next one. Thanks for this!

[@ianyap8941]

Hey Nemean, I took abstract algebra as a math undergrad. The first isomorphism theorem is my favorite theorem and its beauty belies in its simplicity and power.