Cosets and Lagrange’s Theorem - The Size of Subgroups (Abstract Algebra)

Подписаться 891 тыс.

Просмотров 620 тыс.

50% 1

Lagrange’s Theorem places a strong restriction on the size of subgroups. By using a device called “cosets,” we will prove Lagrange’s Theorem and give some examples of its power.
Be sure to subscribe so you don't miss new lessons from Socratica:
bit.ly/1ixuu9W
♦♦♦♦♦♦♦♦♦♦
We recommend the following textbooks:
Dummit & Foote, Abstract Algebra 3rd Edition
amzn.to/2oOBd5S
Milne, Algebra Course Notes (available free online)
www.jmilne.org/...
♦♦♦♦♦♦♦♦♦♦
Ways to support our channel:
► Join our Patreon : / socratica
► Make a one-time PayPal donation: www.paypal.me/...
► We also accept Bitcoin @ 1EttYyGwJmpy9bLY2UcmEqMJuBfaZ1HdG9
Thank you!
♦♦♦♦♦♦♦♦♦♦
Connect with us!
Facebook: / socraticastudios
Instagram: / socraticastudios
Twitter: / socratica
♦♦♦♦♦♦♦♦♦♦
Teaching Assistant: Liliana de Castro
Written & Directed by Michael Harrison
Produced by Kimberly Hatch Harrison
♦♦♦♦♦♦♦♦♦♦

Опубликовано:

30 сен 2024

Ссылка:

Скачать:

Готовим ссылку...

Добавить в:

Мой плейлист

Посмотреть позже

Комментарии : 627

@Socratica 10 месяцев назад

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

@GlenMacDonald 7 лет назад

While Mathologer, Numberphile and other RU-vid channels present entertaining aspects of math, this channel -- IMHO -- does the best jobs of *teaching* math, while still doing so in an entertaining way. Thanks, and keep up the great work!

@adeljasmin4214 7 лет назад

I can't agree more, it is indeed a very professional, talented, and entertaining way, ...

@geogeo14000 3 года назад

That's what I was going to write, you cited great channels but here it was much more clear and easy to understand, very well explained

@RadicalCaveman 3 года назад

This video (I haven't seen others on this channel yet) does PURE teaching. Mathologer is very educational, while also paying attention to being entertaining. Numberphile is lighter on the education side, but still informative. You can learn a lot from Mathologer videos. But it's true that, taken together, they don't make a course.

@fahrenheit2101 11 месяцев назад

@@RadicalCavemanmathologer videos are nice, but they're less approachable

@Dr.JudeAEMasonMD 8 месяцев назад

Absolutely. I devour alphanumeric puzzles and came to learn the proper fundamentals of discrete mathematics .

@rmb706 2 года назад

"The proof of Lagrange's Theorem is definitely going to be on your next exam" [which is tomorrow]. thank you!

@someone7752 Год назад

[In 1 hour]

@Ali-1_0_1 7 месяцев назад

how can you know

@walidnouh1747 7 лет назад

she deserves an oscar for her wonderful and refreshing rendering of dry and abstract topic

@viveknsharma 6 лет назад

Couldn't agree more...

@ccgarciab 5 лет назад

More like "for proving that abstract algebra isn't as dry as some professors"

@luyombojonathan7715 2 года назад

True

@Shubham-ic5tx 2 года назад

Ohhh what? I found abstract algebra intimating

@handledav Год назад

intimating@@Shubham-ic5tx

@tharagleb 7 лет назад

Q; What's purple and commutes? A: An Abelian grape.

@obinnanwakwue5735 7 лет назад

You don't get the point, do you?

@randomdude9135 5 лет назад

It's a joke

@wdai03 5 лет назад

I thought it was bonzai buddy

@emmymatt 4 года назад

I'm only 19 but I laughed at this so I figured that I have fatherly instincts 😂😂

@chynxyune3546 3 года назад

Purple?

@TheNomadic69 6 лет назад

"Don't get overly excited about LaGrange's Theorem..."

@theproofessayist8441 5 лет назад

BECAUSE IT IS NOT A BICONDITIONAL!!! It's not that powerful folks!

@gatoradeee 4 года назад

It'll be hard bit I'll do my best

@lucasm4299 3 года назад

If you want an important result about the converse, then look at the First Sylow Theorem

@RadicalCaveman 3 года назад

Damn! And here I was all set to go out on the town, womanizing and carousing with LaGrange's Theorem...

@Onoesmahpie 3 года назад

She didn't even mention how it could be used to derive Fermat's little theorem and Euler's theorem

@gghelis 4 года назад

Took me a couple of minutes to figure out who this Cosets guy is.

@sb-jo2ch 4 года назад

It's pronounced _ko-say_

@CorbsVids 3 года назад

@@sb-jo2ch yeah its french!

@theboombody 2 года назад

Even as a math major I think abstract algebra may be the most boring subject in the world. SO many definitions before you can do anything at all with them. What the heck do we have to learn a magma for? I don't plan on using anything without an inverse function. That said, I sure appreciate this channel for uploading content about it. Because it is WAY less dry than books on the subject would be.

@radfordmcawesome7947 23 дня назад

abstract algebra is so powerful in software engineering. when you need to model a problem, if you can define your "elements" and "operators" in the right way, then you get (without having to reason about it) all the theorems that tell you how your constructs behave when combined in complex ways. likewise relational algebra for databases, lambda calculus for computation, category theory for types. AA may be boring for people doing pure math, but as a SW engineer, im thankful that y'all have done all the tedious work so i can just use it

@andrewmoeller4538 7 лет назад

These videos are absolutely fantastic. FINALLY a series of serious mathematics videos that TEACHES material well. Please keep making great upper-level math videos such as these! SO much better at explaining concepts/proofs than my professors.

@melm5189 7 лет назад

WHY AREN'T YOU MY TEACHER?!?!? This is amazing, you present this in a way that someone who's never taken higher level math might even be able to understand. You're also very engaging, clear and your thought process is very organized. Will be watching more of your videos tonight to help me with my Algebra final tomorrow afternoon :) Thanks so much for doing these videos!!!!!

@Socratica 7 лет назад

We're so glad you've found us!! Good luck on your Final tomorrow!! We're rooting for you!! :) :)

@pqb0 2 года назад

how was the final?

@AsaNole 5 лет назад

That sound effect when we hit a contradiction at 5:31 haha

@pathagas 2 года назад

I love good pedagogy that get's deep into the subject! Great video!

@terryendicott2939 7 лет назад

How far down the rabbit hole are you going to go? Sylow theorems , normal series ....? Thank you.

@escobasingracia962 7 лет назад

Topology please! I love this kind of videos.

@vecter 7 лет назад

Yes, I would love to see topology. The way they break things down is so clear.

@michaelnovak9412 6 лет назад

and also analysis

@genericperson8238 6 лет назад

It's been a year. I'd still love to see Tpology

@weinihao3632 4 года назад

It's been another year. And I'd still love to see Topology, too!

@gregoriousmaths266 4 года назад

It’s been a month and I’d love to see topology

@Socratica 2 года назад

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 with Kindle Unlimited: amzn.to/3atr8TJ

@nolwazizakwe9231 7 лет назад

Thank you so much. can you please make videos for real and complex analysis

@autodidactusplaysjrpgs7614 7 лет назад

Thank you for not dumbing your content down.

@dragonslayer31415900 4 года назад

THANK YOU!!!! The way you present such abstract topics make them easy to digest and understand. I'll have to give you credit on my coming exam period!

@pranjalshastry9113 7 лет назад

u r the best teacher u help me a lot in my studies thanx

@barend4285 5 лет назад

I love your videos, thanks to you I got 78% on the last Abstract Algebra test I wrote! Keep up the good work.

@michaelc.4321 4 года назад

I got super confused by the proof for the size of the cosets. But then I just remembered that each coset is basically just a shifted version of H that can’t have duplicates because of that proof

@dynamo58 3 года назад

Thank you so much! Your comment helped me, I got stuck on that part aswell and I couldn't figure it out.

@richardm5916 4 года назад

I have learing dificulties and find it hard to understand dry complex books so this is a blessing, Thank you

@MrOvipare 3 года назад

I have to admit that I often need to replay some bits in this abstract algebra series... However, it's hard to imagine that the subject could be presented even more clearly. really high quality content and excellent teacher/host! Brilliant!

@rikenm 7 лет назад

Never disappoints.

@noorameera26 7 лет назад

It will be great if you guys have video on normal subgroup and quotient group. Your videos have been very helpful so far:)

@claudefazio 3 года назад

I wish I had had professors when I attended college who can explain an abstract theory in such an entertaining yet rigorous way. You are a great teacher!

@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

@kirtankoro 7 лет назад

Thank you very much for your work! Even though there're not so many subscribers yet, it helps a lot!

@Socratica 7 лет назад

Thank you for your kind comment! We'll keep making more videos! It would be a huge help if you would share our videos with your friends or on Twitter/ Reddit etc. That will really help us grow! :)

@GyanendroLoitongbamgyanendrol9 7 лет назад

Nice lecture... @3:33 I'm confused on how O(6) get 0... How is alternating group calculated? Please post video on alternating group too...

@kevinjyh 5 лет назад

I had same question about this part. But seems nobody knows?

@miguelcerna7406 3 года назад

3:25 please explain how you calculated your orders. I am having trouble understanding why the Order of 6 is zero.

@roomcard1989 3 года назад

Even I was confused. Later I understood that it is a whole different topic. Here is it. If you open the video below link then it explains how. m.ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-6sOAYTQnKLM.html

@omnibrain8 5 лет назад

Please with the number of subgroups, the values that you have are not clear for me to understand. How did you come by the numbers 1,3,4,1,0 and 1?Thank you.

@G5349 5 лет назад

Those are just orders that were selected to ilustrate (i.e examples based on the divisors of 12) that not always a subgroup of a certain order exists. In other words, that the converse of the Lagrange theorem does not always hold.

@kevinjyh 5 лет назад

More confused about G5349's reply. I had same question that Joe said.

@G5349 5 лет назад

@@kevinjyh The numbers (orders) are just random. They are just examples.

@juanreza4500 Год назад

I am learning about the many classifications offered by abstract algebra. However, I am waiting for an application of any of the theorems and classifications to something, other than more abstract algebra. For example, calculus is applicable to many problems such as calculating varying velocities, distances, mass, and energies of actual *things" in the real world. Other than the rotations of repeating flower petals and molecules, which are easily done without recourse to abstract algebra.

@RahulKumar-dy2pk 6 лет назад

You're best teacher in mathematics.. So good mam

@viveknsharma 6 лет назад

I don't remember when I enjoyed Algebra last time... You all are REALLY REALLY SOCRATIC...

@partialorder5596 5 лет назад

These videos are so clear, thorough, and concise! I'm taking an algebra course right now, and although the lectures and textbook are quite good, I get an even better understanding of the material after watching your videos. Thank you!

@a.b.c.d.e... 2 года назад

Best explanation I found online (and also much better than what my university showed us)

@carlosmendes7 4 года назад

In 6:51, the "n" in g(_n)H doesn't refer to the order of G, though they used the same symbol (like in 3:45). Probably, using g(_k-1)H would be better, as later they define k as the number of cosets!

@luisirisarri1085 4 года назад

Thank u for this wonderful course on Abstract Algebra, super intuitive and well explained! Just a little coment, in order to be consistent with the notation shouldn't you write at 6:53 g_k (where g_nH) because later at 7:51 you use g_k. Also you used |G|=n , and if then you use g_n it may be confusing... Thanks a lot

@Uejji 5 лет назад

Yup, this is starting to remind me of my Modern Algebra class several years ago.

@ChaudharyAteeq440 7 лет назад

your style of explanation is great...Please made More Videos on Pure mathematics

@nievsbest 5 лет назад

Wow why did I not know this channel when I was an undergrad. Very well put compact lessons.

@joshmart2569 5 лет назад

Socratica...please teach on centre of a group

@Daniel-ng8fi 6 лет назад

omg, this channel is amazing

@alexman97412 3 года назад

I'm french, and this video helps me a lot to get the big picture. Now i can understand my maths course in french. Thx alot

@115-ananda4 10 месяцев назад

Your presentation level so amazing and interesting.

@adarshnarayan 5 лет назад

How did you get to know the orders and number of subgroups? Can you please explain. Also any example of the group A4, if possible.

@emersonrico6061 6 лет назад

Very well explained! This channel deserves my tuition fee than the course I currently enrolled.

@arnabacharya349 7 лет назад

Fan of your videos. Very helpful stuff. Not many interesting videos on advanced courses. One comment, in the proof, checking cases 1 and 2 was redundant I think, as the existence of subgroups of sizes 1 and n does nothing to prove or disprove Lagrange's Theorem. Am I right? Though it's probably put there for pedagogical purposes.

@jorgekennedy3241 5 лет назад

She needs to proof for H=G because in case 3 she uses H is a proper subgroup of G. But never uses H is not the trivial subgroup, so that proof is not necessary.

@medicallifewithjohn 5 лет назад

Mashallah

@pauleugenio5914 3 года назад

Thank you! I found the pictorials rather helpful. Once I realized we were only using elements of G not in H, and likewise elements of G not in subsequent cosets of H, then all the pieces started to fall in place.

@samuelbenson201 6 лет назад

I appreciate this effort socratica. Its really educating, and simple to understand.

@ibraheemmoosa 5 лет назад

I do not understand how you go from "a coset does not have any duplicates" to "each coset having the same size as H".

@nahblue 5 лет назад

Let's go to 04:58. You have the first coset g1H = { g1 × h for all h in H } This means we have the set of all g1 × something in H, so the coset can be at most exactly the size of H. It can be smaller if any of the g1 × h combinations map to the same elements, so only if there are any duplicates. Because all g1 × h for all h in H are distinct, we have that there is exactly as many elements in g1H as in H. For each h in H, there is g1 × h in g1H.

@MD-kk9mq 4 года назад

please tell the background music!!!!!!!!!!!!!!!!

@theflaggeddragon9472 7 лет назад

YES TOPOLOGY!! Thank you so much for these videos but please upload more often!!

@jerinfatima4999 5 лет назад

Wonderful video but why 6 doesn't form a subgroup.i got confused

@achievers2001 4 года назад

Best explanation of cosets and lagrange's theorem ever LOVE FROM INDIA❤️❤️❤️

@bishaesmukhiya7623 5 лет назад

Respected,team SOCRATIA please make some videos for complex analysis I....

@nate4511 7 лет назад

I love your videos. I am currently in my final year of my undergrad and your videos have helped me understand the material I am being introduced too. Thank you so much for the work youre doing. I wish i could donate money to show my support.

@Socratica 7 лет назад

Nate, we are so happy you are finding our videos helpful! This makes us really excited to make more videos. :) Please don't worry that you can't donate right now. Focus on school and work hard and we will help as much as we can! If you share our videos with your friends and on Twitter and Reddit and other places, that is a bigger help than you can even imagine. We hear from a lot of viewers that they are just now finding our channel - so the more you can get the word out the better! Thanks again for your inspiring comment. :)

@keroroxx520 День назад

AWESOME! It's my first time understand "coset".

@shubhamk840 5 лет назад

Thank you very much lagranges would be very happy after listening this lecture.

@obinnanwakwue5735 7 лет назад

8:38-9:07- Now that was really abstract there.

@headie947 4 года назад

god bless you

@julioezequiel8935 7 лет назад

Yes,Topology !!

@sabrishalalfeh 4 года назад

Julio Ezequiel what? 😂

@nahidfoysal9252 7 лет назад

thanks mam..your tutorials helps me a lot..please make more videos..respect from bangladesh which is a south asian country..

@107ashrafulmasum5 6 лет назад

Light house Amazing teacher...respect a lot l am also Bangladeshi

@meivelanmurugesan1095 9 месяцев назад

By this theorem can we conclude, a group with order of prime number only have standard subgroups (itself and trivial) . This implies z2 should not be subgroup of z3 ? And coset 2+H = {0,2}, but 0 belongs to both H(z2) and G(z3)? 2+H and H are overlapped?

@NarutoUzumaki-t2v 9 месяцев назад

@anamkhuram9596 5 лет назад

A question: If we use all the elements in G to form each coset, then shouldn't the number of cosets be n too? Why did we take k instead?

@khalidkasab4010 3 года назад

Kindly teach langrange theorem for functions of two variables with one constraint from book of Edwin K. P Title "an optimization problems".

@jeyaneepan 7 лет назад

hard to understand sub group (3.34 mint) could you explain more

@pulkitnijhawan1059 5 лет назад

too impressed to comment with words.Watching your videos is an overwhelming experience.

@bhaskarsingh9115 5 лет назад

Upload the video on normal subgroup , quotient group, field and ring

@GeoffryGifari 2 года назад

lets say we have a group G of order 323 = 17x19, as in the example. if we have a subgroup H of order 17, can we say that its the only subgroup of order 17 (uniqueness)?

@harpreetdhillon2993 5 лет назад

If left cost is not equal to right Coset (aH=\Ha) Can we say that aH is a coset Ha is also a coset

@b43xoit 5 лет назад

She talks about left cosets and right cosets, if I recall correctly.

@calebhale9865 Год назад

Can someone explain the end of the proof at 5:30? Why does hj*hi^-1 belong to H? Does having the right-hand side belong to a group (or subgroup) mean that the whole expression (including everything to the left) belongs to a group too? Or just under certain circumstances?

@fahrenheit2101 Год назад

It's weirdly effective to get an actress to play the role of a teacher, rather than using a plain old teacher. Is she just following a script a lot of the time, or does she actually know the math, too. Her wiki just credits her as an actress, and 'teaching assistant at Socratica', and this video was 'written and directed by Michael Harrison' Do y'all just get clueless actors to do this lol? Not complaining btw - it WORKS. Anyways, another very clear video - thanks to whoever's responsible for coming up with these nice explanations, and to Liliana for conveying the math in a friendly fashion.

@goumuk 4 года назад

I have a doubt. Since, each of the H sub group and k cosets are mutually exclusive, then the relation of their orders and group G order should be |H| + k*d = n or d*(k+1) = n . Thus, d | n . Please correct me if I am wrong in my approach.

@johnwalker6318 Год назад

Is there a globular -40 C probababilitys of H2O in H??? Is it a matter of Percentage in H???

@Santoshmylapilli 3 года назад

Who the hell discovered these and why....😤 By the way, the explanation is smooth and understandable.

@connorclark8945 4 года назад

I got lost on the "cosets must be equal size" bit. This reddit explanation really helped: www.reddit.com/r/explainlikeimfive/comments/47wvps/eli5_langranges_theorem/d0h0lvl/?context=3

@ybc8495 3 года назад

not very clean; especially suddenly come out the "d*k = n"; this part should explain more detail, would be better with sample. And come out the A4 from no where without explain what is A4, every tutorial should assume reader has no experience and only depends on your serial of vedio. In the end I figure out myself, but these things need improve.

@Omnifarious0 3 года назад

Playlists are not subgroups. At least, if you consider the number of videos in the playlist to be its order. This video has the proof of that. By Lagranges theorem, the order of every subgroup must divide the order of the full group evenly. And this can't be true for any arbitrary playlist on this channel unless all playlists are only one video long. This because when you add a video, you add it to a playlist, and if x divides y, then x + 1 cannot divide y + 1. Now, there may be some playlists who's order does divide the total number of videos on this channel. And I submit that these are not subgroups either because ther set of all Socratica videos is not a group. There is no definition for a combining function in which any Socratica video may be composed in some way with any other and yield a third Socratica video. ;-)

@williamwarke8299 3 года назад

Could somebody help me understand how, in the proof for Lagrange's theorem, we know that hj*hi^-1 is in H?

@GeneralMathematics1 2 года назад

Superb but difficult to understand because I'm not English speaker.

@luyombojonathan7715 2 года назад

Great content !!! Cheers

@parthshukla7680 Год назад

Can you please name the background music? It is very soothing while learning Maths.

@helenshaji4617 4 года назад

Beauty with brain..thanku dear....helped a lot...u changed my attitude towards algebra...i wish i had a teacher like u..love from india..😍😍

@Socratica 4 года назад

We're so glad you are watching! It's so wonderful when we hear that we have helped someone. 💜🦉

@helenshaji4617 4 года назад

@@Socratica 💜💜😍😍

@sabira1889 3 года назад

I also want to teach math online n RU-vid. But I don't know, how to make effective videos. I love your kind of video presentation. Would you please tell me the process of this video making.

@ricardomarino8554 5 лет назад

Finaly a Higher math video without an indian explaining. I like indian people but the english is not my first language and i don't understand them for them english accent. Long live to this channel!!!! :)

@ajsdoa6282 5 лет назад

I really dont understand the part when showing that the orders of the cosets are equal... What does gh1=gh2 mean? Yes h1 and h2 is different.. But what is the statement when we put gh1=gh2? And how has that anything to do with the order? Im also confused over the conclusion of the contradiction there... Answer this please someone that knows!

@luishem 7 лет назад

Hello Socratica. I would really like to understand Polya's Counting Theory. I'm just not sure of what is the path to get there, and what are the previous requirements. For example, I want to know how to solve the 2 color necklace problems and really know what's going on, without just running the equations.

@seanziewonzie 7 лет назад

Atavachron Try the website of Richard P Stanley. His book, Algebraic Combinatorics, is free to read on there and one of the chapters goes quite deeply into Polya Enumeration Theory

@luishem 7 лет назад

Thanks Sean, this is why I love the internet.

@FishyTheFishy2763 7 месяцев назад

6:11 - 6:41 “Mhm…. Yea…. Yep.” *whispering to the persom beside me* “I have no idea what shes saying”

@eldduva Месяц назад

How can there be 4 subgroups of order 3 as in the example at 3:30, when there exists only one group of order 3?

@SadmanMufrad Год назад

If every playlists had the equal numbers of videos, then they would be cosets. Every cosets NEED to be of the same order

@gaaraofddarkness 4 года назад

7:57 shouldn't it be d *(k+1)=n as we have to consider H as well in addition to all the cosets??

@MuffinsAPlenty 4 года назад

H is, itself, a coset of H.

@nc7341 4 года назад

Yes, I too think so.

@MuffinsAPlenty 4 года назад

@@nc7341 H is a coset. Adding H in again is counting it twice.

@samiazaman5240 5 лет назад

Why are we able to multiply by hj^-1 at 6:18- that is- hj^-1 is not in g1H or g2H since hj does not exist in either, but we are still able to multiply by it? Can we multiply an element of a coset by any arbitrary element of the group it is contained in?

@syedibrahimahmad171 4 года назад

at 3:28 you say that the subgroup of A4 will not have the order 6, can you or someone explain why that is? or how can I know that when I am looking to find subgroups?