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

This is the most information-dense introduction to group theory you'll see on this website. If you're a computer scientist like me and have always wondered what group theory is useful for and why it even exists and furthermore don't want to bother spending hours learning the basics, this is the video for you. We cover everything from the basic history of group theory, over how and why subgroups partition groups, to the classification of all groups of prime order.

0:00 Intro
1:42 Abstract Algebra
4:28 Group Theory
8:01 Z Q Zn Dn
14:29 Proofs
18:58 Subgroups & Cosets
25:31 The Theorem
29:11 Classification of Groups of Prime Order

#SoME2

[Nemean]

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

[samu_]

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)

[swaree]

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

[triggerhappy]

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

[dewaard]

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.

[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.

[uhu]

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

[diegodoesstuff]

Still waiting for part 2 of this amazing series

[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 ?

[cubething-x]

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.

[Golden]

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.

[augusto]

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 YouTube.

[nickm]

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.

[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.

[PixelSergey]

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

[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

[rizalpurnawan]

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!

[lorenzoaste]

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.

[nolanalexander]

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.

[edvogel]

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!