Simple Groups - Abstract Algebra

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Simple groups are the building blocks of finite groups. After decades of hard work, mathematicians have finally classified all finite simple groups. Today we talk about why simple groups are so important, and then cover the four main classes of simple groups: cyclic groups of prime order, alternating groups An (n bigger than 4), groups of Lie type, and the 26 sporadic groups, including the Monster Group.
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We recommend the following textbooks:
Dummit & Foote, Abstract Algebra 3rd Edition
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Milne, Algebra Course Notes (available free online)
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Teaching Assistant: Liliana de Castro
Written & Directed by Michael Harrison
Produced by Kimberly Hatch Harrison
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Опубликовано:

30 сен 2024

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Комментарии : 178

@孙林可 3 года назад

I've heard that there is only ONE mathematician alive now who understands the whole 10000 pages of simple groups. S a d.

@dekippiesip 3 месяца назад

And his name is?

@SalDin_ 6 лет назад

literally have an abstract algebra exam tomorrow. videos are undoubtedly a great help!

@bckzilla 6 лет назад

Thank you for making RU-vid a better place to pass time.

@mksarav75 6 лет назад

Thank you to the entire team who worked hard to produce this great video series.

@scipionedelferro 3 года назад

This is the most packed video of the series, content-wise. So many interesting and fascinating stuff mentioned, too quickly! It would be fantastic to have more videos on simple finite groups. You guys are the best!

@ruiyingwu893 6 лет назад

I am pretty new at group theory, so I did some 'research' (aka me typing it onto Google ) ... |A_13|= 13!/2= 3 113 510 400 Thats... a lot of subscribers you are asking for...

@randomdude9135 5 лет назад

T series has surpassed both pewds and Music to become no 1. But even they've got apprx 107M subs.

@jonmolina948 5 лет назад

You could've simply taken the cardinality of S_13, 13!, and divided that by 2. The cardinality of even permutations in S_n is always the same as the number of odd (If n >= 2). You can prove it by defining a bijection between the two sets.

@RalphDratman 6 лет назад

This is a wonderful presentation -- thank you! What exactly is Socratica?

@Socratica 6 лет назад

Thank you for your kind comment! We're a small team of educators who make videos for RU-vid! You can read more about us here: www.patreon.com/socratica

@daca8395 6 лет назад

"releace a video every hour" Nooo, I will witerally spand my life watching your videos...

@mortervolk6676 6 лет назад

I can't believe you guys have covered so much info in less than 10 minutes! That's really great, Socratica. Keep up the good work. (From Syria with love!)

@sambravers 6 лет назад

You guys want 3+ billion subscribers?

@eleazaralmazan4089 6 лет назад

Thank you so much Socratica! You make mathematics very intriguing!

@Socratica 2 года назад

Sign up to our email list to be notified when we release more Abstract Algebra content: snu.socratica.com/abstract-algebra

@michaelren4845 3 года назад

I cannot believe I have followed from episode 1 to 22 and intend to keep going. You explain these abstract and difficult ideas in a much clear way than my any of my professor. Thank you so much! [I might find a small typo in episode 22 for Simple Groups at 03:04 in the second line (title not included) "Quotient groups are simple: (N_1/1), (N_1/N_2), (N_3/N_2)..." Is it intended to be (N_2/N_1)?]

@huttarl Год назад

I wondered about that (N_1/N_2) as well. Glad it's not just me.

@adeelali8417 4 года назад

This is where my journey with your series ends, you have been a great help! This video in particular is very comprehensive! :D Thank you thank you thank you!

@Socratica 4 года назад

We're so glad you found our videos helpful! Thank you so much for watching. Please share with anyone you think we could help! 💜🦉

@raunitsingh676 3 года назад

It's so sad that they have stopped making videos, now who will teach me more of such awesome things

@PunmasterSTP 3 года назад

That zooming-in and Monster group music! So suddenly intense, just like the rate of my learning after finding Socratica several days ago!

@jaisanatanrashtra7035 4 года назад

I hate Maths but John Conway video got me here 😂😂😂😂😂😂😂😂

@oldPrince22 2 года назад

To be honest, this video has a much higher requirements for the audience. Hence is not that consistent with the previous videos about abstract algebra. And the topics covered in this video is seldom used for a beginner of abstract algebra.

@kresimir1965 6 лет назад

I got goosebumps when I saw Monster group :O And the music was whaaat

@ChanawerebiChanawerebi 6 месяцев назад

hello! why do we get R^(n^2) ? why is n^2 a dimension?

@thavibu 5 лет назад

Interesting that two of the concepts in the video are named after 19th century Norwegian mathematicians, Abel and Lie

@rajendralekhwar4131 5 лет назад

First of all thanks for your all videos.. I don’t get time to comment, on every video, but let me tel u , Your explanation is just awesome ..👍👍keep it up 👍👍 Please every time keep trying to make abstract mathematics as a layman language subject as long as possible I know it’s hard to do every time , but that’s the only way we can convert maximum individuals to love higher mathematics ...❤️

@bobsagget9212 4 года назад

I study business but I really like these videos

@RurczakKurczak 3 года назад

3:04 can we take N1/N2, where N2 is bigger than N1? I think not, since N2 has to be a normal subgroup of N1 to be able to take a quotient group.

@giorapeniakov3153 2 года назад

seems like a mistake?

@paramanandadas1319 4 года назад

At 3:05 there is a mistake. I think that is not N1/N2 but N2/N1

@upendraagnihotri2686 4 года назад

Thanks for making me understand a bit in the ocean. I am struggling very hard to get the essence of it.

@raunitsingh676 2 года назад

I can only imagine how much time and effort and knowledge is required to put out a video like this.

@Drtsaga 4 года назад

Hi guys! Can anyone explain how the monster group can contain quotient groups? I thought that in order for a group to contain a quotient groups, it needs to contain normal subgroups. (simple groups do not contain normal subgroups, and the monster is a simple group) Thank you.

@MuffinsAPlenty 4 года назад

@jaimelima2420 3 года назад

I am afraid to going to sleep today and have bad dreams because of this monster group. Thanks making us understand these concepts.

@sebastiananaya25 6 лет назад

Hola Muy buenos videos, excelente calidad Me gustaría que volvieran en español

@himanshugarg6062 5 лет назад

Is this connected to M theory in physics (because Monster group) and 26 dimensions that were needed before modern string theory allowed for 10 (before moving on to 11)..? P.S.: Very pop sciency.. I know..

@davidpal1378 6 лет назад

I like your videos on abstract algebra , but can you make videos on real sequences. Like bounced and unbound sequences , least upper bound greatest lower bound , infima , Suprema etc. if you do so then , It would be a great help .

@saurabhsingh-ow7ue 4 года назад

well this 8 mins video is the best investment of my life till now....thank you madam.....

@Socratica 4 года назад

That is so nice of you to say, thank you! We're so glad we could help. 💜🦉

@derciferreira2523 8 месяцев назад

You resumed 300 years of mathematics in just 8:52 minutes. Thank you.

@Ziplock9000 5 лет назад

ML will make this all obsolete.

@kennedyada1117 5 лет назад

Man, I have an exam tomorrow and I was looking for slow easy to understand videos with examples that drive the points home, but you're just as fast as my lecturer assuming that I already knew everything about math when I was born.

@Grassmpl 2 года назад

This lady knows so much. How about a video on cohomology groups?

@arpanbhattacharjee470 6 лет назад

Wonderful presentation!!! The videos are a great resource to understand the basics as well as some of the advanced concepts of Abstract Algebra neatly, quickly and efficiently... I'm a researcher in Applied Mathematics and the videos helped me a lot to revise my algebra concepts in a gist... Thanks a lot... Waiting for more topics on Advanced Mathematics to come...

@ahmedengineer5778 6 лет назад

I like your enthusiam ..... you sure have passion for the subject you are discussing ..... but I think that you need to add more examples .... and more important real life applications ..... the problem that makes alot of people hate math is that they feel it is irrelivant to thier every day life ..... one of the merets of educational videos on youtube is the appility to show people how science really affects thier life

@Henry-yr2hn 4 года назад

A13 is a huge group !

@kamyarghandi9995 6 лет назад

Would love for this series to eventually get to an explanation of what E8 is and why it is considered such a beautiful mathematical object.

@AHeil1963 5 месяцев назад

There is a typo at 3:10: (N1/N2) should probably be (N2/N1). Congratulations and many thanks for the excellent video!

@rodneytopor1846 8 месяцев назад

Nice summary. I think it would be helpful to elaborate the correspondence between prime numbers and simple groups as follows: Every finite group (positive integer) can be expressed as a product of a unique set of simple groups (prime numbers) by the Jordan-Hoelder Theorem (Fundamental Theorem of Arithmetic). But a given set of simple groups can be multiplied in different ways to give different product groups (the extension problem you mentioned), whereas a given set of prime numbers can be multiplied in only one way to give a unique composite number. I guess the reason for this is that arithmetic multiplication is commutative but group multiplication is not.

@SSJProgramming 2 года назад

Great video, But slightly misleading at 5:14 There is no general formula for degree 5 and higher *** IF *** you consider only using BASIC operations like +,-,*,/, roots, powers, exp(x), log(x), sin(x), cos(x) etc. Its a common misunderstanding that this hold for ALL types of multivalued functions you can consider. And in fact, there are GENERAL solutions for degree 5 and higher. Using elliptic functions, or jacobi theta functions, some others I can't even recall, hypergeometric etc.

@herbertpalines3443 3 года назад

This is a nice introduction to finite simple groups! Thank you, Socratica!

@HikingWithRiley 5 лет назад

Slide at 6:27, “intervertible” is written, “invertible” was spoken

@万物之父 Год назад

God the order of A13 is 3,113,510,400 nearly half of the world population😂

@alvaroquispe-unsa 10 месяцев назад

Thanks for the video series, although I don't speal English, there are so useful for me. My best greetings from Arequipa - Peru

@gajananvanjari322 5 лет назад

I wish I could subscribe it |An| times

@fengzm 7 месяцев назад

|A13| is approximately 3 billion. Way to go, Socratica!😃

@mehdinadjafikhah7314 4 года назад

S_n is not simple for any n>1 because A_n is a normal subgroup with index 2 of it! But, A_n is simple for n>4.

@cameronspalding9792 3 года назад

@8:23 The number she’s aiming for is half of 13 factorial which is 3.1 *10^9

@Ishhann22 4 месяца назад

Is anyone here watching 2x speed right before the exam day🫥?

@Jung850 6 лет назад

This is really awesome! Great work. 😍🤗🤗

@chounoki 3 года назад

Proof of 10 thousands of pages. I wonder how many people have actually read it and verified that there is no error or miss in the proof.

@whalingwithishmael7751 5 лет назад

Can you do a video on the monster group? John Conway thinks that he’s going to go to his grave without having learned why it’s there and that would be tragic

@alfredbeadman7114 3 года назад

Would love that. Want to learn more about it!

@qaziarshad939 3 года назад

Hausdorff Space and T2 space is also T3 space ? is it Right?

@cotasamnemano366 3 года назад

|A₁₃| = 3 113 510 400 = 2⁹×3⁵×5²×7×11×13 A big number indeed.

@eagerdip8086 Год назад

very insighrful dear🇳🇵🇳🇵🇳🇵🇳🇵😍😍😍😘😘😘

@johndwolynetz6495 2 года назад

the cool thing about this is that a n y nontrivial quotient group of a simple group is t r i v i a l. took me a while to realize this. flimsy little things.

@AkiraNakamoto Год назад

3:05 There is a typo/error. N2/N1, not N1/N2. The latter doesn't make sense.

@LovingFriend614 Месяц назад

5:32 when you pause at just the right moment...

@gauravsinha6060 6 лет назад

I love this channel. Thanks for the great video.

@priyanka-samal. 4 месяца назад

Thank you in these 9 min video u explained a lot and in a simple way

@basudebmondal954 2 года назад

Group is very interesting chapter in abstract algebra

@ashishpathak2947 9 месяцев назад

At 7:27 the instructor mentions that monster group contains 20/26 sporadic groups as either subgroups or quotient groups. But as monster group is a simple group, then it shouldn't have any normal subgroups right? And hence we shouldn't be able to form any quotient groups? Can someone please comment on what I'm missing here.

@MuffinsAPlenty 8 месяцев назад

I looked it up. Apparently the happy family are all _subquotients_ of the Monster Group. Given a group G, a group K is a _subquotient_ of G if K is isomorphic to a quotient of a subgroup of G. In other words, K is a subquotient of G if there is some subgroup H ≤ G and some normal subgroup N ⊴ H so that K ≅ H/N. This does not contradict the simplicity of the monster group because H will not be normal in G there. This is not a failure on your part, though, because the video didn't say this.

@mohitsaini2114 3 года назад

Nice

@solewalk 3 года назад

3:04 Looks like ( N1 / N2 ) should have been ( N2 / N1)

@ajs1998 Год назад

John Conway stood behind the proof that classified all simple groups, but said probably nobody has read the whole thing. How can the mathematics community be so confident in such a proof? I feel like the proof that the monster group is a simple group should tell you exactly WHY it's a simple group, I guess not though? What else is there to know about the monster group, why is it so mysterious if we've already proved that it is simple?

@dekippiesip 3 месяца назад

So a few people have read one part, a bunch of others another, etc, etc. All very competent people. As long as these proofs stand on their own merit and don't rely on the other proofs it is a perfectly valid situation. You may as well say multiple proofs where checked independently from one another.

@sarojpadhy5535 3 года назад

Mam I assure u subscriber must cross over order of A thirteen

@cameronspalding9792 3 года назад

Any finite group of prime order p is isomorphic to Zp

@himanshugarg6062 5 лет назад

Stick to one of the names : like factor group or quotient group.. And similarly in other situations.. Maybe show an asterisk comment at the bottom of the video.. I'm a fan.. Trying to help..

@1337w0n 3 года назад

7:33 How is it that the monster group contains any of the other members of the happy family as quotient groups? I'm under the impression that simple groups can't have quotient groups.

@MuffinsAPlenty 2 года назад

@tharagleb 6 лет назад

Order of A13 is 3,113,510,400

@FranFerioli 4 года назад

The gist of Galois theory in under 10 min! The groups might be simple, but this video is certainly not. Outstanding work, as usual Socratica...

@wliaputs 5 лет назад

I just realize we have happy family and pariahs in math lol

@zokalyx 3 года назад

I love those names

@moaadmaaroufiii2057 3 года назад

amazing work!! keep up

@noellundstrom7447 6 лет назад

I love seeing you go a little deeper into abstract algebra, nice job you earned a donation!

@Socratica 6 лет назад

We're so glad you are enjoying our videos! Your donation is SO appreciated. It will help us make more of these videos!! Thanks so much for your kind words and support.

@AzrgExplorers 4 года назад

Wait, how can the monster group have quotients if it's simple? Don't quotient groups and normal subgroups go together?

@Grassmpl 2 года назад

Why A13 in particular?

@luyombojonathan6688 8 месяцев назад

Thank you alot for these series

@69erthx1138 4 года назад

It'd be cool if she turned into Trinity @6:05 and said, "Lem'me show you some of the physics."

@MarvellousMartha 4 года назад

you say "remember manifolds?" but i am not able to find a video of yours covering manifolds. which one is it? thx

@mohdfarhan8562 6 лет назад

Plz give video's on some examples on abstract algebra like inverse , order of an element..etc.

@christianorlandosilvaforer3451 5 лет назад

wow at least i came to this video.. finally i can understand why pol eq. of 5 or more grade have not a general formula as solution!!! thank you socratica team!

@PeterManger 5 лет назад

I thought I was a simple group of 1. Now it appears to be too complex... so I'll stick to being i.

@NH-zh8mp 2 года назад

Bravo, I love this video, it’s so fascinating and helpful

@My_oxytocin 2 года назад

Love your indetails information on group.❤️❤️❤️

@po91914 4 года назад

Actually there are general solutions to quintic equations, you just need to use infinite series. But yeah there is no solution in terms of a finite number of combinations of radicals,

@ChaudharyAteeq440 6 лет назад

Great...Please upload more videos on abstract Algebra...also in linear algebra and real Analysis

@janko4765 6 лет назад

So, I am learning quantum mechanics and the abstract algebra is like a language you're using to talk about it. Although these lectures don't cover representations of groups and Lie's groups which are also needed for my quantum mechanics classes, I must say I'm in love! The concepts you're covering seem like they come in a natural way one after another and you want to know everything about every single concept. They don't seem just like a random topics you need to understand as fast as you can! The enthusiasm and the sort of an adventurous vibe I'm getting from the way you're talking is making me feel like I'm watching a movie! Thank you!!