Normal Subgroups and Quotient Groups (aka Factor Groups) - Abstract Algebra

Normal subgroups are a powerful tool for creating factor groups (also called quotient groups). In this video we introduce the concept of a coset, talk about which subgroups are “normal” subgroups, and show when the collection of cosets can be treated as a group of their own. As a motivation, we will begin by discussing congruences.
Our Abstract Algebra playlist is here:
bit.ly/AbstractAlgebraSocratica
Be sure to subscribe so you don't miss new lessons from Socratica:
bit.ly/1ixuu9W
♦♦♦♦♦♦♦♦♦♦
Ways to support our channel:
► Join our Patreon : / socratica
► Make a one-time PayPal donation: www.paypal.me/socratica
► We also accept Bitcoin @ 1EttYyGwJmpy9bLY2UcmEqMJuBfaZ1HdG9
Thank you!
♦♦♦♦♦♦♦♦♦♦
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/CourseNote...
♦♦♦♦♦♦♦♦♦♦
Connect with us!
Facebook: / socraticastudios
Instagram: / socraticastudios
Twitter: / socratica
♦♦♦♦♦♦♦♦♦♦
Teaching​ ​Assistant:​ ​​ ​Liliana​ ​de​ ​Castro
Written​ ​&​ ​Directed​ ​by​ ​Michael​ ​Harrison
Produced​ ​by​ ​Kimberly​ ​Hatch​ ​Harrison
♦♦♦♦♦♦♦♦♦♦

[@Socratica]

If you'd like to learn more, we have a free course on Group Theory! www.socratica.com/courses/group-theory

[@siddharthprakash8942]

The most useful series of mathematics videos I have encountered since 3blue1 brown

[@randomdude9135]

Yup. If you know any other awesome series like this, then it'll help me a lot.

[@manuthebroker5598]

I agree

[@vibodhj349]

Check out Faculty of Khan as well.

[@mychannelofawesome]

@@randomdude9135 please check Mathdoctorbob's series on abstract algebra... It's really great, really intuitive, and goes into phenomenal depth.

[@effy1219]

@@mychannelofawesome thanks!

[@howmathematicianscreatemat9226]

Everything they do here concerning teaching is badass., meaning they look "bad" in front of most professors because their biggest fear is - paradoxically- to be understood while the greatest mission of Socratica is to appear understandable. And hardly a one does a better job. Because maths is first and foremost supposed to be one thing: intuitive and fun. and ONLY THEN to be formal but only AFTER one has established and examined the concrete cases. Maths then appears to be a collection and characterization of those examples and not a collection of dead and unmotivated formal arguments, definitions and theorems. Formal symbols do have phenomenal value but only if one has gotten and intuitive understanding of the theorems and definitions first. Socratica does exactly this. That's why she should be nominated the Oscar prize for teaching mathematics.

[@winstonjiang3621]

“A living Socrates”

[@howmathematicianscreatemat9226]

Winston Jiang yeah,she kinda is :)

[@theboombody]

I blame the subject more than the teaching. It's very difficult for me to relate abstract algebra to anything I've seen in the past. The only interest I have for it now is it appears to be central in understanding why there is no general solution in radicals to the quintic equation. Which is interesting, but man, do we have to learn ALL this stuff just to understand that one problem?

[@alxjones]

@@theboombody Linear algebra is abstract algebra, as a vector space is an abelian group with a compatible field action (scalar multiplication). So in a sense, anything you can use linear algebra for is an application of abstract algebra. That aside, the slight generalization of vector spaces (where the field may be weakened to a ring), called modules, appear in calculus on manifolds: the set of vector fields on a (real) manifold M forms a C^r(M,R)-module. A formal theory of polynomials and rational functions also falls under abstract algebra, in the form of rings and fields. Polynomials are more than just "which ones can be solved via explicit formula" though; for example, differential equations such as y'' + 2y' + y = 0 can be studied as polynomial differential operators e.g. D^2 + 2D + 1. This is, of course, trivial for the constant coefficient case, but when the coefficients are polynomials, you end up with a polynomial ring which is not commutative, and so different techniques need to be developed. Groups themselves also find a good amount of importance in calculus and differential equations on manifolds, in the form of Lie groups. Lie groups are groups, first and foremost, which also have some kind of (smooth) manifold structure. Their related objects, the Lie algebras, are vector spaces with a certain kind of vector product (for example, R^3 with cross product is a Lie algebra). It is precisely the properties of groups that make Lie groups so useful, either as a manifold of study or as the typical fiber in a principal bundle structure. One last thing, the quotients that are being developed in this very video are the basis for the major tensor algebras, including the exterior (Grassman) algebra and the symmetric algebra. The tensor product of vector spaces itself is constructed by taking the vector space whose basis is indexed by pairs of vectors, then taking the quotient by the ideal generated by the properties we wish to hold. From the complete tensor algebra, the exterior and symmetric algebras are achieved by taking the quotient by the ideal generated by skew-symmetric and symmetric multiplication, respectively. Ideals are simply the equivalent of normal subgroups for rings and similar contexts, basically those substructures which allow quotients to have the desired structure. This is just a small sample of the use of abstract algebra in other areas of mathematics, obviously localized to my particular area of study. I hope you can come to realize that abstract algebra is not as self-contained as it seems, and the techniques and language learned from studying the subject is of great importance even in the relatively grounded subjects of calculus and differential equations.

[@itszeen7855]

@@alxjones what is your area of study/research?

[@whypeoplehategemini]

I had to play the video multiple times with several pauses along the way for me to grasp the concept.

[@christopherkemsley4758]

Somehow I stumbled upon your channel while searching for the Primer Vector Theory a couple days ago, and then watched your entire astronomy series ... and here I am watching the entire Abstract Algebra series. One thing that has always frustrated me trying to learn these things from, say, Wikipedia is that they're always written by people who fully understand the subject FOR people who fully understand the subject and and are quite difficult to understand until you understand it - even in cases where the concepts are quite simple. I'm so glad to have found your channel where you explain things so simply and so clearly. Thank you so much!

[@ericvosselmans5657]

I am already quite old and trying to learn abstract algebra. Sometimes I just need a very clear and down to earth description of a mathematical object which can be quite hard to teach yourself from a book This channel provides an excellent tool in that regard. Thank you!

[@ThefamousMrcroissant]

Now try Analysis or Calculus III and absolutely tear those last remaining hairs on your head out. I took a second master in Electrical engineering when I was 32 and I felt like a fucking grandpa already.

[@samiaario8291]

I find it helps to have different source material for the same subject, and to skip back and forth between sources. These videos are great for that purpose.

[@alexamadori9884]

Can watch this almost effortlessly in the evening, trying to read the same theory from a book took almost a week of studying every morning and led to a more superficial understanding than this video. You guys are geniuses when it comes to presenting ideas, you're definitely on the list of channels I'd like to support when I'll be able to.

[@homiramanuj]

In Motivating Example, How do we get remainder 1 if we divide -14, -9, -4 etc. by 5? Please reply i am so confused 😢 integer mod 5 is confusing me

[@godspower_eze]

@@homiramanuj The smallest number closest to -4 that is divisible by 5 is -5 so -4 - (-5) is 1. Same goes for -9, -14 and so on.

[@erfanmirzaei705]

@@homiramanuj In the division quotient can be negative numbers. Thus, by dividing -14 by 5 we get -3 as quotient and -14-(-15)= 1. The point here that the quotient times divisor should be less than or equal to dividend.

[@IBMboy]

my head hurts :( but i will try to watch it again later :)

[@randomdude9135]

Me too

[@ScilexGuitar]

lmao same

[@randomdude9135]

This time I understood atleast 50% I think. Time to ponder on my own and scribble around in a book.

[@adeelali8417]

SAME! I'll come back to the ending later....

[@yeast4529]

You know it's a good video when the content seems simple and is really easy to comprehend. Sometimes I lose myself in all of the new definitions etc. in my Algebra course, but these videos pull everything together and help greatly with the motivation behind everything you learn.

[@souravacharya]

The way she teaches and explains , totally incredible...!

[@sadied0g]

This playlist is awesome! TOPOLOGY WHEN?!?!? 😁👍🏻

[@bakkamydestination]

S... waiting

[@ThePimp4dawin]

What an amazing series, this is a goldmine! Perfect depth and speed.

[@Socratica]

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

[@jonathanpopham5483]

the most approachable abstract algebra course online. thank you so much for your hard work!

[@Anna-jy7cj]

This series is blowing my mind, your work is highly appreciated

[@homiramanuj]

In Motivating Example, How do we get remainder 1 if we divide -14, -9, -4 etc. by 5? Please reply i am so confused 😢 integer mod 5 is confusing me

[@ozzyfromspace]

This was my first time trying to learn and it didn't help. But I'm gonna try again, and again, and again until it makes sense. I'm committed to finishing your playlist with usable understanding. Keep up the amazing work, Socratica team!

[@luyombojonathan7715]

How did it go ??? Am starting on a similar journey

[@tekhiun]

One of the best math series on youtube. maybe the best if you already have enough background to understand this. Thank you for doing this !

[@joshuaronisjr]

From 6:00 onwards, although the real case is more general, the entire thing becomes a lot easier to understand if every time she says "times" or "multiply" you think "plus" or "add", every time she says "N" you replace it with "Modulo(someNumber)", "e" is "0Modulo(someNumber)", and "x" and "y" are just numbers that aren't a multiple of someNumber. Cheers!

[@darsh9156]

This really helped me understanding this topic. I was really confused and now all the confusion is gone. Thanks a lot .

[@merlijn1e]

I use this series to accompany my lectures on abstract algebra, it helps me so much to understand what is going on. Thankyou!

[@Arv.-]

The way you simplify a complex concept is great!

[@alayamaryim]

The most understandable videos of abstract algebra on KZbin.Very easy to understand

[@homiramanuj]

In Motivating Example, How do we get remainder 1 if we divide -14, -9, -4 etc. by 5? Please reply i am so confused 😢 integer mod 5 is confusing me

[@ClumpypooCP]

@@homiramanujbecause -4 = (-1)*5+1

[@raunitsingh676]

Trying to find a word that describes my gratefulness for such incredible explanative videos.

[@tomjoyce9401]

Excellent presentation: clear, to-the-point, fluid.

[@Bloodsaberxy]

Was self studying Galois Theory and this helped to recap a lot of forgotten theorems, thanks a lot!

[@ThePharphis]

Damn I really needed this video 4 days ago before my exam! (it went ok but factor groups was something that went over my head for most of the semester)

[@malenaalmasi1774]

In really like the presentation style. Everything is very clear and all the explanations are easy to follow. Thank you so much

[@user-gc4dx4nt2x]

she saved my whole fxxking life during the senior this fall

[@Gaspard832011]

Group is [ i (identity) , r1 (rotation 1/3), r2 (rotation 2/3), s1 (sym 1), s2 (sym 2) , s3 (sym3) ] (i,r1,r2) is a subgroup. This subgroup is normal because: s1* r1 *s1 =r2 s2* r1 *s2 =r2 s3* r1 *s3 =r2 (a symetry is its own inverse element)

[@sammie1824]

I got this too!

[@homiramanuj]

In Motivating Example, How do we get remainder 1 if we divide -14, -9, -4 etc. by 5? Please reply i am so confused 😢 integer mod 5 is confusing me

[@AdamDaouk-mb7ly]

eh meshe

[@Socratica]

Socratica Friends, we wrote a book for you! How To Be a Great Student ebook: amzn.to/2Lh3XSP Paperback: amzn.to/3t5jeH3 or read for free when you sign up for Kindle Unlimited: amzn.to/3atr8TJ

[@GGC728]

The best video i ever viewed on youtube about group theory. Thanks alot

[@quahntasy]

This was explained very amazingly. Thanks for this :)

[@sujoydey8359]

Too good explanation which covers most important part of normal subgroup. U are truly a good teacher.

[@Souley239]

I really like you because explain the subject in an easy and understandable way.

[@ajsdoa6282]

Thanks! Had to watch in 0.5x the speed to hang on, but very helpful! :)

[@192ali1]

Thank you. Excellent presentation.

[@cameronspalding9792]

@7:25 replace y with y^(-1)

[@cameronspalding9792]

@11:04 the set of permutations (123) (132) and the identity permutation form a normal subgroup of S3

[@Yougottacryforthis]

isnt it the only (non trivial) sub group as well as the only normal sub-group? any other basically fail to endure the closure property

[@cameronspalding9792]

@@Yougottacryforthis it’s the only non trivial normal subgroup, but not the only non trivial subgroup, just pick a set containing the identity and a 2 cycle

[@carlsagan9808]

These videos are so helpful it's unreal

[@saeedahmedhashmi9448]

Mam your way of delivering lecturer is amazing,outstanding.. God bless you

[@souravsingh3025]

Great explaination love it , makes the topic fun 💝💝

[@murielfang755]

So so great. So well delivered.

[@pubudunuwan1751]

your all videos are very descriptive .it helps to solve many troubles .i wish to do more and more videos. thank you

[@ijyoyo]

WOW, so nice and easier to understand. Beats the textbook 100%.

[@physicslover9227]

Thanks a lot this videos series is very useful. It explains everything in a very simple way🙂🙏🏻🙏🏻.

[@sutanuhait3283]

Excellent work. Students are recommended to watch this video. It will help to motivate you properly.

[@ashmytom8927]

You have made things simple!....Thankyou

[@EdgarMartinez-ws9ws]

Such a beautiful topic

[@LuizaDreamsDeutsch]

Thank you so much for the explanation!

[@ajaykanwar7223]

*Love the way you teach*

[@njahnavi7943]

Thanks a ton !!! Explained with such clarity. It was to the point , excellent explanation.❤️

[@sudarshann7194]

Is it nityananda who's in your profile?? 🤔

[@AdamDaouk-mb7ly]

@@sudarshann7194 eh ktir excellent

[@sarthakkrishna1737]

So well explained!!!! Thank you! I have an exam tomorrow. I have now more confidence than apprehension XD

[@rosakuli8375]

studying for my math classes is enjoyable with Socratica

[@fuckthishandlesystem]

THANK YOU SO MUCH FOR THIS

[@SuperAwesomeReasons]

Yes! More Socratica.

[@syamalchattopadhyay2893]

Outstanding video lecture. This video lecture is very helpful for self-study.

[@youtwothirtyfive]

One more bit of constructive feedback, the exercise at the end, "find a normal subgroup of S_3", assumes knowledge of what symmetric subgroup S_3 is --going by the Abstract Algebra playlist order, the concept of a symmetric subgroup hasn't been introduced yet.

[@RobotProctor]

I wish I could upvote this video 100 times.

[@halilibrahimcetin9448]

Happiness is nothing but understanding stg properly. These vids series are fuckin' awesome.

[@exx8eran]

wow you took such a complicated subject and make it so simple.

[@moularaoul643]

Thank you so much!!!

[@ramumaha2779]

took me watching it twice to understand perfectly(have to oil my brain)....awesome to the point explanation.

[@Socratica]

This is our favourite thing about KZbin compared to classes - you can just rewatch! Thanks for sticking it out with us! 💜🦉

[@rakoonberry7879]

You people are amazing!

[@kunslipper]

Very good. Thank you so much.

[@amberszulc197]

that was..... amazing. great job

[@gauravsahu2495]

I love the way you teach

[@SpiritVector]

Simple groups are the primes of group theory.

[@theboy4773]

saving my time & leisure time

[@turokg1578]

bro this video is amazing. i was like wtf is this quotient groups and cosets reading dummit&foote. came here and its all clear now. can continue reading. thanks

[@ishikasharma1000]

This is so helpful, I can now clearly visualise these concepts 😍🙌. Your videos are amazing, Thankyou Socratica✨

[@germanslobo]

Thanks. Great video. !!

[@keshabsingha7111]

it's very helpful to me . thank you .

[@ubersham]

Where were you in 2016 when I was taking Abstract Algebra??? 😝 Love the series. I’ll be going through each one several times until I understand your proofs and can duplicate them (again?).

[@KeystoneScience]

just in time for finals ;)

[@Socratica]

Hooray! That's what we were hoping. :D Good luck!!

[@CreolLanguag]

@@Socratica i have a question: since y^-1(N)y = N, if we multiply both sides by y in their left. y[y^-1(N)y] = yN Ny = yN so does this mean that cosets form a group only if left cosets is the same as their right cosets? is this always the case?

[@haroonahmad1850]

@@CreolLanguag good question. What's the answer of this question? Did you get it?

[@jamaluddin9158]

@@CreolLanguag Yes that is correct!

[@rekarlopunzalan]

Watching this during finals

[@benjiasdf]

great video, thanks a lot!

[@ramzihedimay1179]

great and very well done. Congratulation.

[@mohammadamanalimyzada8332]

thank u i love the way u teach. I didnt understand my professor but here go everything I needed any videos on Mathematical Analysis?

[@maulikjain3894]

one can listen this forever/..

[@narendrakhadka9598]

You are my best teacher.

[@shipratiwari1533]

Very interesting explanation...!!!!

[@shockline1]

Very helpful videos. I had to pause a lot to understand but worth it.

[@malicksoumare370]

My favourite teacher

[@jeromejean-charles6163]

Very good work : still to give a constructive critic: 1) I think the argument for definition yN=Ny could be exposed without going down to elements and avoiding inverse as much as possible.2) The transition from Z,+ to multiplicative is not the best though I cannot think of a simple multiplicative example fro cosets.3) It is so nice to finally see questions being asked to motivate a definition. Still from a didactic point of view it could be worth repeating the question at end ( recap style).

[@simasimo9796]

Need to watch More videos 😃

[@trainingvlogs]

Pls make more videos on abstract algebra i love your explanation very much

[@profsciencia]

these videos are awesome!

[@LOL091027]

When we will have topology series like abstract algebra ?

[@DiegoGonzalez-xl9us]

i wish they do it

[@howmathematicianscreatemat9226]

Would you want me to ? Or in other words: would it still be useful for you ?

[@_qpdbdbqp_]

@@howmathematicianscreatemat9226 yes!!

[@howmathematicianscreatemat9226]

@@_qpdbdbqp_ okay, till when do you still need it? Tell me the date and also if you think good explanations could help your classmates too? If you tell me, then maybe I'm gonna start producing them when I'm back from holiday on the 25th of February. You would then view your first plesant set-topology video on the 27th of February. But if you want to me to start, confirm your request.

[@John-js2uj]

@@howmathematicianscreatemat9226 I'd also be grateful if you began posting on 27th Feb

[@ninnymonger]

New Video!!!

[@ronycb7168]

Indian here really enjoying this series happy republic day guys

[@homiramanuj]

In Motivating Example, How do we get remainder 1 if we divide -14, -9, -4 etc. by 5? Please reply i am so confused 😢 integer mod 5 is confusing me

[@ibrahimhamim3135]

Salute to you guys 👌

[@s.k.potdarpotdar8377]

bye socratica, i watched your video on normal subgroups& quotient groups. very beautiful presentation & interesting maths subject. it is my request tu pl relaese vedio on "boolean algebra & it's applications" thank u

[@youtwothirtyfive]

Wow, great video! I learned a lot. One thing that felt unexplained was this statement just before 10:00 about factor groups that "the inverse of x⋅N is x^(-1)⋅N". I can play around with the integers mod 5 as an example and see it's true, but I'm wondering how to convince myself it works in general. Thanks again for making these :)

[@vladislavnikolaev800]

N is invariant subgroup, it means that for any x xN=Nx. (xN)(x^(-1)N)=(Nx)(x^(-1)N)=N(xx^(-1))N=N1N=NN=N. In factor group N is 1.

[@aashishkumarmaurya55]

Thanks again

[@abelabel4151]

thank you. from marrakech morocco

[@renatoalencar4451]

That's great! I bet you can do one on type theory.

[@sokaynazayat6029]

you are amazing!!!

[@briancannard7335]

It's great how you show the truth of mathematicians. They love to multiply elements and cosets in the same statement jumping between levels of abstraction.

[@briancannard7335]

Why care that cosets form a group?

[@briancannard7335]

Mathematicians need to invent a disclaimer about generalization levels, and attach it to all concepts created. Each time I see a "Factor Group" now or a "Quotient Group" I'll think about, ah... it's a group of cosets, not of elements! Thank you @Socratica!

[@Mathematics-with-MohsinRaja]

Amazing

[@hemanthkumartirupati]

Do you plan to teach vector spaces in future? Your videos are incredibly helpful btw :) Thanks a lot

[@ooos2989]

They have a video on it from three years ago.

[@richardjeremias]

Right on time ahuuauuahhu Tô revisando algebra xD