Simple Groups - Abstract Algebra

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
amzn.to/2oOBd5S
Milne, Algebra Course Notes (available free online)
www.jmilne.org/math/CourseNotes/index.html
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Teaching​ ​Assistant:​ ​​ ​Liliana​ ​de​ ​Castro
Written​ ​&​ ​Directed​ ​by​ ​Michael​ ​Harrison
Produced​ ​by​ ​Kimberly​ ​Hatch​ ​Harrison
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[@bckzilla]

Thank you for making KZbin a better place to pass time.

[@SalDin_]

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

[@mksarav75]

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

[@scipionedelferro]

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!

[@mortervolk6676]

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!)

[@daca8395]

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

[@raunitsingh676]

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

[@PunmasterSTP]

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

[@raunitsingh676]

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

[@kennedyada1117]

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.

[@janko4765]

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!!

[@eleazaralmazan4089]

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

[@herbertpalines3443]

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

[@jaimelima2420]

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

[@Socratica]

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

[@adeelali8417]

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]

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

[@noellundstrom7447]

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

[@Socratica]

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.

[@arpanbhattacharjee470]

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...

[@CSAN33]

These are actually really nice videos, I'm impressed!

[@gauravsinha6060]

I love this channel. Thanks for the great video.

[@ashwanirao7354]

Your way of explanation is wonderful

[@kamyarghandi9995]

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.

[@rajendralekhwar4131]

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 ...❤️

[@luyombojonathan6688]

Thank you alot for these series

[@NH-zh8mp]

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

[@Jung850]

This is really awesome! Great work. 😍🤗🤗

[@bobsagget9212]

I study business but I really like these videos

[@My_oxytocin]

Love your indetails information on group.❤️❤️❤️

[@derciferreira2523]

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

[@ruiyingwu893]

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]

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

[@jonmolina948]

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.

[@moaadmaaroufiii2057]

amazing work!! keep up

[@kresimir1965]

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

[@Grassmpl]

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

[@priyanka-samal.]

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

[@saurabhsingh-ow7ue]

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

[@Socratica]

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

[@gylje-9905]

I am just amazed! I only can thank you..

[@upendraagnihotri2686]

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

[@noraalbogami586]

unique presentation information ... thanks a lot ..

[@guru6644]

Very clear explain... Thanks.

[@christianorlandosilvaforer3451]

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!

[@ChaudharyAteeq440]

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

[@anyachan567]

Briliant work!

[@vaishaliitkan7443]

Nice explanation

[@gharsepadhonasantoshkumarj9154]

Great lecture

[@RalphDratman]

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

[@Socratica]

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

[@mohdfarhan8562]

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

[@alvaroquispe-unsa]

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

[@HikingWithRiley]

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

[@bat_man1089]

Thank you teacher 😊

[@ladmondraxngfuskiii9426]

视频讲的很清楚，获益匪浅！

[@MarvellousMartha]

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

[@mohdfarhan8562]

It's well undertaken video

[@information2949]

Ma'am plz make one vedio on P group, sylow p-subgroups and related theorem

[@AHeil1963]

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

[@bhavyaanbarasan7919]

Hi mam Can u please do a video on decomposition of graph

[@mueezadam8438]

A masterclass presentation.

[@woahdaggies]

Please do a video on the Monster!

[@davidpal1378]

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 .

[@rodneytopor1846]

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.

[@michaelren4845]

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]

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

[@muhammadafzaalkhan9277]

at which author of abstract algebra book i select for himself to read.

[@sebastiananaya25]

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

[@IjazKhan-fm4si]

Great Socratica❤❤❤

[@aniketchandak7426]

Please upload video on Riemann integral...

[@yenmejos3162]

hi, can you also make videos about non-cyclic groups :)

[@colaurier2594]

Thanks a lot.

[@user-kw4er9un3e]

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

[@paramanandadas1319]

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

[@RurczakKurczak]

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]

seems like a mistake?

[@ahmedengineer5778]

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

[@information2949]

Thank you ma'am :)

[@fengzm]

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

[@heliumfrancium8403]

3 millions to watch abstract algebra is tough nowadays especially after the corona virus. Corona hit the education quality and will take time for that quality to recover.

[@qaziarshad939]

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

[@eliasgarciaclaro6136]

Este canal es increíblemente bueno

[@chindinaresh7305]

Vf

[@keepmoving5453]

thank you so much mam (from India)

[@FranFerioli]

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...

[@filipve73]

1) Time will tell ?? (abstract) Perhaps there is a group between the "Happy Family" and the "Pariahs" 2) For patreon support do you accept also Bitcoins ?

[@Socratica]

1) People are researching ways to unify the sporadic groups. I'll need to check on the latest research to see what progress has been made. 2) We *do* accept bitcoins! :) You can find our address on our "About" page: kzbin.infoabout Thank you so much for considering supporting us!!

[@himanshugarg6062]

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..

[@Drtsaga]

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]

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.

[@MrMojo0417]

Let H be a subgroup of G. You can make cosets of any H within G. But cosets only quality as factor groups if H = N. Is this correct?

[@MrMojo0417]

This video was packed with information!

[@mariojonaban1139]

Nice!

[@ashishpathak2947]

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]

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.

[@governmentday3511]

thank you

[@House_ssb]

More about calculus please

[@cameronspalding9792]

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

[@algonte]

Amazing...

[@duckymomo7935]

Abel-Ruffini theorem?

[@jamesshin4901]

Great!!

[@thavibu]

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

[@mohitsaini2114]

Nice

[@Henry-yr2hn]

A13 is a huge group !

[@AkiraNakamoto]

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

[@Grassmpl]

Why A13 in particular?

[@spotlight9027]

I need your lectures about real analysis and topology

[@giorapeniakov3153]

A factor group is also known as a quotient group

[@basudebmondal954]

Group is very interesting chapter in abstract algebra

[@eduardogss9535]

me gustaría ver este mismo vídeo en español por favor

[@AzrgExplorers]

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

[@manuelroman815]

Muy buenos sus videos pero si colocan los subtitulos para que el publico en español pueda seguir disfrutando de sus videos seria genial!

[@nowornever5598]

Thanks