Group theory, abstraction, and the 196,883-dimensional monster

An introduction to group theory (Minor error corrections below)
An equally valuable form of support is to simply share some of the videos.

Timestamps:
0:00 - The size of the monster
0:50 - What is a group?
7:06 - What is an abstract group?
13:27 - Classifying groups
18:31 - About the monster

Errors:
*Typo on the "hard problem" at 14:11, it should be a/(b+c) + b/(a+c) + c/(a+b) = 4
*Typo-turned-speako: The classification of quasithin groups is 1221 pages long, not 12,000. The full collection of papers proving the CFSG theorem do comprise tens of thousands of pages, but no one paper was quite that crazy.

You may also enjoy this brief article giving an overview of this monster:

If you want to learn more about group theory, check out the expository papers here:

Videos with John Conway talking about the Monster:

More on Noether's Theorem:

The symmetry ambigram was designed by Punya Mishra:

The Monster image comes from the Noun Project, via Nicky Knicky

To join the gang, upload your own video on your own favorite number over 1,000,000 with the hashtag #MegaFavNumbers, and the word MegaFavNumbers in the title by September 2nd, 2020, and it'll be added to the playlist above.

Thanks to these viewers for their contributions to translations
German: dlatikaynen
Hebrew: Omer Tuchfeld
Italian: mulstato

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If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind.

Music by Vincent Rubinetti.
Download the music on Bandcamp:

Stream the music on Spotify:

If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.

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Various social media stuffs:

[austinmcconnell]

I'm a simple man. I see an outrageously large number in the video title, I click.

[sandearcubus]

He did the math, he did the monster math.

[mnym]

It feels like with "the monster" humanity has learned the answer to something without being able to even ask the question. Almost as if we've stepped into territory that isn't meant for us yet. Of course the given name adds to the lovecraftian feeling of it and I love stuff like this.

[lubbnetobb]

A quote from a Pratchett novel comes to mind, when a wizard tries to explain how a mysterious cabinet works.

'Yes. The box exists in ten or possibly eleven dimensions. Practically anything may be possible.'

'Why only eleven dimensions?'

'We don't know, ' said Ponder. 'It might be simply that more would be silly.'

[omnitroph]

the difference between fiction and reality is that fiction has to make sense.

[rexroyulada]

"We always consider the action of doing nothing to be part of the group" - 1:41

My favorite quote

[flatypus]

Honestly, there's something beautiful about the way 3b1b explains things. At around 4:28, he explains that the permutations of 101 different objects would amount to 9x10^159. However, instead of simply saying 'this is roughly the same as the number of atoms in the universe squared', he says 'if every atom in the universe had a mini universe inside of it, that would be how many sub-atoms there would be.' Take the time to appreciate the time he took to make these numbers just a bit more interesting!

[robinlinh]

the way you explain the concept of group as the concept of number "3" is really mind opening and important. A lot of people having trouble with math because it's seem so conceptual and they always try to link it to something more grounded, but to be good at math they need to approach math as how conceptual it is. Eventually of course math is used to help real life problem but it's not always straight forward, so you need to think about it in the world of math itself. It's like when I started coding and at first my mind will work only with what the final UI or graphic display on the screen, but slowly my mind would think purely of what happen data wise and not really the final representation.

[whynotanyting]

Other Mathematicians: "Polynomials, Permutations, Quaternions"
John Conway: "Monster, Baby Monster, Happy Family"

[papalouis]

Just got new earbuds and while trying them on, suddenly my google assistant starts reading my notifications. Of course, it had to read this never-ending title for me lmao

[cernsb]

I like your final quote "Fundamental objects are not necessarily simple. The universe doesn't really care if it's final answers look clean; they are what they are by logical necessity, with no concern over how easily we'll be able to understand them."

[adamhenriksson]

This is IMO the best video 3b1b has produced by far. Amazing explanations, visualizations, stories, everything. At first everything went over my head, but after learning a bit about group theory, this video is so cool.

[mulefatelluri]

15:54 reminds me of the "how to draw an owl" joke:
step 1: draw a circle
step 2: draw the rest of the owl

[BleuNoirProductions]

This was like watching anime without subtitles. I didn’t understand a thing but it was gorgeous.

[charlesrockafellor]

I remember reading "Symmetry and the monster" about 15 years ago, and fell in love with the monster group -- and due to one of the later chapters in it, the next video queued up after this one is now "Hamming codes and error correction
". Classic work. <3

[ChariotduNord]

This has got to be one of my favorite 3Blue1Brown videos. I love the way you present just how fundamental groups are. One line in particular I just love: "This is asking something more fundamental than 'what are all the symmetric things?' It's a way of asking, "what are all the ways that something can be symmetric?'".

[clayz]

Typical horror film technique: don’t show the monster to preserve suspense.

[saeedsn]

Someone should give him a medal for making such an abstract theory so beautiful and entertaining, and yet extremely educational! ❤️

[lourdesmartinaguilar]

I love your videos. I’ve always been a little intimidated for the level of abstraction of some mathematical concepts, but you can explain many of them more intuitively, with elegance and also generating more interest. Thank you and please keep doing this great work. :)

[forumbhanshali]

This is truly one of the best videos describing overview of group theory and its recent developments in the past few decades. Thank you!