How We Got to the Classification of Finite Groups | Group Theory

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Finite Simple Groups
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Bryce Goodwin Paper
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Classification of Quasithin Groups
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[@oflameo8927]

Having a periodic table group of finite simple groups would make constructing datatypes for code challenges much easier.

[@dibeos]

@@oflameo8927 you’re right. The only problem is that the list is too long and the level of detail necessary to describe each group is way too high

[@WillJohnathan]

I'd definitely love to see you talk about groups of Lie type. But I did find this video... weird, so to speak. I don't see what the intended target audience is. If it were the general public, then you explain way too fast, at least that's what I feel. Still, the other aspects of the video is great, and the animation is also very nice. Keep up your work!

[@dibeos]

@@WillJohnathan thanks for the advice! We will definitely slower the pace so that it is more accessible for everyone interested in learning math. Also, we will make a video about groups of Lie type, just as you asked 😎

[@ValidatingUsername]

@@dibeosLie groups are fantastic but definitely, for me at least, needed some extra time to fully grasp it the first time around - a slower pace would definitely be helpful!

[@dibeos]

@ValidatingUsername thanks for the tip!!!

[@joeeeee8738]

I've found the same. But I wish they go deeper into the topics with more details and examples. Don't make it slower just for more audience (same type of surface level content is all over the place) !

[@dibeos]

@@joeeeee8738 yeah, it’s just that there is a looooot to talk about. So I guess we will try to pick an even more specific topic in group theory and give a bunch of examples

[@expchrist]

This video is good. I like anything on the simple finite groups and the unexpected connection between the monster group M and modular functions also known as monstrous moonshine

[@dibeos]

@@expchrist thank you so much for your support!!! We will definitely post much more often about these subjects 😎

[@OpPhilo03]

Please make video About groups ,field and rings. I want to know more and more about groups. Every hidden things.

[@dibeos]

@@OpPhilo03 hi! Thanks for the suggestion. Yeah, we are starting to notice that many people are interested in these subjects, which is great because we love them as well!!! We will definitely post more videos on these things 😉 actually, today we are publishing about an example of Lie group: SE(2)

[@ramaronin]

aeeeee!!!! algebra abstrataaaaa!!!! 🎉🎉🎉🎉

[@theograice8080]

I would love to see more about the Extension Problem of Group Theory. I have been noodling about an object in my head that seems to invoke properties of the natural Exponential function over the integers, but with an infinitely stretchy band being wrapped around an infinitely large spindle torus. The modular nature of the extension problem intrigues me for the purposes of my mental play toy.

[@victork8708]

I NEEED not to see about the last one…. Combining simple groups to get complex groups and all bout it! It could be actually related to what i am tackling now

[@fullfungo]

I love your new style of videos

[@dibeos]

@@fullfungo thanks!!!! Let us know what kind of content you are interested in! 😎

[@nycholasgr8112]

Yess please make a video about the open problem at the end

[@dibeos]

@@nycholasgr8112 yessss we will 😎

[@MetaBuddha]

I'm curious 😮

[@josephmellor7641]

What exactly does it mean for groups to be the "building blocks" of other groups? For example, what does a group built from the alternating group with five elements and a cyclic group of order 17 look like?

[@dibeos]

When we refer to simpler groups as 'building blocks', we touch on the idea that more complex groups can often be constructed or understood through combinations of simpler ones. As an example, every finite group can theoretically be broken down into a series of simple groups through a process called composition series. The specific groups you mentioned can be combined in a few ways. The most straightforward method is through the direct product, where you pair each element of A_5 with each element of {Z}_17, resulting in a new group where the operations are done separately within each component of the pair. There's another method called the semidirect product, which allows one group to dictate some of the structure of the other, possibly creating a non-trivial interaction between them. This can only happen if there's a suitable way (defined by group actions) for one group to influence the group structure of the other. So, constructing new groups from simpler 'building blocks' helps us understand possible group structures and their properties.

[@josephmellor7641]

@@dibeos Would it be possible for you to make a video on the semi-direct product?

[@dibeos]

@@josephmellor7641 of course, we will include it on our list right now!! 😎

[@logosecho8530]

The alternating group on five elements has order 60, which is coprime to 17, so they can only be combined trivially via a direct product.

[@SobTim-eu3xu]

Interesting video!)

[@jakeaustria5445]

Thank You

[@JosBergervoet]

Why do you say (at 0:06) that mathematicians are still unable to "describe them all"? Isn't that contradicting that the classification is now complete? (O, and the Monster should have been explicitly mentioned of course!)

[@dibeos]

The classification theorem deals specifically with finite simple groups, not all finite groups. Emphasis on simple here. The simple groups have been completely classified, but finite groups in general (which can be built from these simple groups in more complex ways) are not fully classified. Hope that makes sense! :)

[@JosBergervoet]

@@dibeos Not really, to "describe them all" we just construct all products of simple groups. We cannot easily see which one of those are isomorphic, be in that way we do describe them all! We just may get duplicates. So we could perhaps say that we are missing a unique standard way to describe each finite group as a product of simple groups in just one preferred way. (But I think the statement in the video suggests that it's worse than that... that's why I asked. 😇)

[@logosecho8530]

@@JosBergervoet > to "describe them all" we just construct all products of simple groups No, you can glue finite simple groups together in many different ways. A product is just one way of gluing groups together. It's very difficult to figure out what all the ways of gluing two groups together are.

[@JosBergervoet]

@@logosecho8530 Then to avoid calling it a "product" of groups, let's say that every finite group has a composition series, en.wikipedia.org/wiki/Composition_series#For_groups . That's still gluing finite simple groups together, as you call it, so it still leaves unclear what we are missing: is there fear that we don't get all finite groups in that way? We could even go completely back to basics: every finite group can be "described" by its multiplication table. So some (tedious) procedure could just generate them all. As viewers of the video we are of course curious to know: what are we missing? What more would mathematicians desire after the classification? It sounds like something is finished (nicely explained in the video!) and something is still missing, but there we are left in the dark, which of course makes this second point unbearably intriguing... You will need to make a follow-up video!

[@bjornfeuerbacher5514]

Could one compare the sitation to atomic physics and chemistry? Being able to describe the behaviour of all types of single atoms does not mean that one can also describe the behaviour of all molecules.

[@mohammedfarhaan9410]

I! LOVE! GROUPTHEORY!

[@Prof_Michael]

Can you do a video on Lebesgue Measure or Any Real Analysis topic

[@dibeos]

@@Prof_Michael yes, I’m actually working on a video that will be titled: All Types of Integrals in Analysis (or something like that). So as the title shows, we will cover a brief explanation of all integrals, including Lebesgue Integral and Lebesgue measure, of course 😎

[@ravikantpatil3398]

❤

[@expchrist]

This video is good. I like anything on the simple finite groups and the unexpected connection between the monster group M and modular functions also known as monstrous moonshine