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

[@Socratica]

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

[@GlenMacDonald]

While Mathologer, Numberphile and other KZbin 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]

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

[@geogeo14000]

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]

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]

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

[@Dr.JudeAEMasonMD]

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

[@walidnouh1747]

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

[@viveknsharma]

Couldn't agree more...

[@ccgarciab]

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

[@luyombojonathan7715]

True

[@Shubham-ic5tx]

Ohhh what? I found abstract algebra intimating

[@handledav]

intimating@@Shubham-ic5tx

[@rmb706]

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

[@someone7752]

[In 1 hour]

[@Ali-1_0_1]

how can you know

[@TheNomadic69]

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

[@theproofessayist8441]

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

[@gatoradeee]

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

[@lucasm4299]

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

[@RadicalCaveman]

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

[@Onoesmahpie]

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

[@andrewmoeller4538]

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]

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]

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

[@pqb0]

how was the final?

[@tharagleb]

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

[@obinnanwakwue5735]

You don't get the point, do you?

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It's a joke

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I thought it was bonzai buddy

[@emmymatt]

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

[@chynxyune3546]

Purple?

[@autodidactusplaysjrpgs7614]

Thank you for not dumbing your content down.

[@pathagas]

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

[@claudefazio]

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!

[@gghelis]

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

[@sb-jo2ch]

It's pronounced _ko-say_

[@Nonyabisness-12]

@@sb-jo2ch yeah its french!

[@MrOvipare]

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!

[@dragonslayer31415900]

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!

[@hftconsultancy]

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

[@AsaNole]

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

[@naturesbliss2456]

3:35 : "This is the cautionary TALE on the limitation of Lagrange's Theorem" , n the background music, giving that wild west vibe. Just mesmerizing. Thank you for this amazing lecture.

[@barend4285]

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

[@alexman97412]

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

[@viveknsharma]

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

[@Uejji]

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

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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!

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Topology please! I love this kind of videos.

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

[@michaelnovak9412]

and also analysis

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

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

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

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

[@terryendicott2939]

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

[@evansokosodo2791]

I learned something from your teaching on Cosets. Brilliant lady, you are the queen of Abstract Algebra!

[@emersonrico6061]

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

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

[@a.b.c.d.e...]

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

[@Beingaspirant-h4r]

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

[@kipkoechchepkwony6948]

You are a cool teacher who does not wish to present abstract algebra in an abstract way like some professors here.They come to class without a book or any reference material and pour their content on us.Teaching and learning doesn't have to be arrogant.If I must trail without loss of direction algebra routes then i must follow you shoulder by shoulder.Thank you so much.

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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.

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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. :)

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I'm studying English and I like how clear your enunciation is.

[@samuelbenson201]

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

[@pauleugenio5914]

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.

[@danievanwyk3975]

I think you also need to to confirm that k

[@antonionanni6683]

I think it follows from the rule of the construction -- you construct the coset one element of G\H [G\H is the set of the elements of G not in H] at a time. Let |G\H| = m. I believe it follows from the construction that m < k. BTW, this is valid also for trivial groups. If H == {e}, m = n-1 < n; if H == H, m = 0 < n [any group has at least one element {e}]. Again in the case of finite groups/sets!

[@antonionanni6683]

Sorry, you also need to observe that |Hg1| >= 1, which follows again from the fact that any sub-group will have 1 or more elements

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

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Honorable madam, Socratica; first of all I really appreciate you indeed with a little bit comments that it is very important if your videos are in series or consecutive with identification number means that may be 1, 2, 3 .... or what else as you want just to follow your tutorials accordingly.

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Never disappoints.

[@noorameera26]

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

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literally you made me fall in love with the concept . Thx to u for helping in this crucial exam time by keeping video short and effective .

[@nolwazizakwe9231]

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

[@carlosmendes7]

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!

[@vrendus522]

I wish you existed when I was in school. They just threw algebra at us and expected us to get it. Your'e a god-send and very much apricated. Thanks xo

[@michaelc.4321]

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]

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

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Socratica makes the best videos ever in the history of the internet. True fact.

[@Socratica]

You're the nicest. Thanks for this today. 💜🦉

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Your illustrations make it very clear, thanks.

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

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Your presentation level so amazing and interesting.

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Abstract algebra was not easy like this for me. Thanks for making abstract algebra as easy as possible.

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YOU'RE SAVING MY CAREER, THANK YOU SO MUCH FOR EXISTING !

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

[@ankitpandeynitdurgapur9813]

Very Helpful Video for cosets and langrange theorem. Cleared my doubts. Thank you

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i want to cry , you save my life with this amazing video

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You're honestly hilarious. Your comments at the end of videos get me every time 😂😂😂

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I am from India.This KZbin channel is very helpful for Bsc Mathematics students.

[@Socratica]

Hello to our friends in India! Wishing you safe and well. 💜🦉

[@daisyorcine856]

Thank you so much for making these videos on Abstract Algebra they helped me a lot in my studies.

[@ChaudharyAteeq440]

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

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You're one of the best hv ever seen in life... Don't stop..

[@achievers2001]

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

[@keroroxx520]

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

[@GyanendroLoitongbamgyanendrol9]

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]

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

[@miguelcerna7406]

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

[@roomcard1989]

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.kzbin.info/www/bejne/bKSycoyKhtN-gq8

[@lukaszr8834]

Very interesting and helpful video! I've been confused of Lagrange's theorem until this clever explication

[@rishant5645]

We wish we had professors like you

[@Socratica]

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

[@VocalBeast]

Thank you so much! I have abstract algebra in my second year and this really helped solidify lagrange's theorem:)

[@Socratica]

That's so great to hear!! We wish you good luck. 💜🦉

[@athame7206]

This was very helpful! Thank you!!!

[@Tyler-rh8mu]

Your channel is giving me a fighting change on my final today thank you so much.

[@Socratica]

Good luck-we're rooting for you! 💜🦉

[@obinnanwakwue5735]

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

[@kemsekov6331]

I did not understand it untill proved it by myself following your video. After I did it myself everything seemed to make sense. Yup. We need to do some work in order to understand it. I love your videos!

[@juanreza4500]

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.

[@totoplopp6630]

Super pedagogical!! helped me out a lot with the discrete math course I'm doing

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Hello, just wanted to say how perfectly aligned your videos in Abstract Algebra are for the High school International Baccalaureate Higher level math optional topic. Unfortunately, this is the last year they are offering the topic, as their curriculum changed. Boo! I've really enjoyed teaching the topic, and I sure appreciate the effort and level of sophistication. It is SO hard to find material that is appropriate for willing high school students.

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This was wonderful to read - thank you so much for sharing our videos with your students! We hope it planted some seeds so we'll see them again when they're in college! 💜🦉

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Thank You so much Socratica. Your tutorials were/are awesome!! Let me wait for the results now ☺

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Saving me in uni maths one video at a time, thank you

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The best explanation of groups theory. Thank you, Ms.Liliana

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omg, this channel is amazing

[@luisirisarri1085]

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

[@JoopWilkens]

Very well done! Superb exposition and explanation.

[@siyuanchen659]

Fix a typo: 5:48 Should be "pick g2∈G not in both H and g1H" (Note: this typo is only in slide). This explanation of cosets is more intuitive than the standard definition, since there are duplicate cosets in standard definition (Of course, duplicate cosets can be eliminated by set theory). Thanks.

[@32-rishavsharma39]

Best mathematical teacher on utube

[@infernal314]

Nice video, though there is one slightly problematic thing. The picture at about 6:58 illustrates a partition of a group G, which suggests there are n cosets (actually n+1 including H...), in place of k. I know how it is meant but it might be quite confusing. What I think is missing here is the equivalence relation, which might have been introduced. So when we construct the sets g_i*H, where i=1..n, we group them into k equivalence classes, the cosets, that form a disjoint and complete set on G. Loosely speaking, cosets are to groups what a base is to a linear space.

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LITERALLY YOU ARE ONE OF THE GREATEST ,KEEP GOING

[@ozzyfromspace]

She won me over. Now I understand cosets and Lagrange's Theorem for the order of subgroups ☺️

[@jessejordache1869]

I have an abstract algebra book that I was teaching myself from, and this was the part I kept slamming my head into. Missing the nuance that a coset is *not* a subgroup also did not help.

[@mueezadam8438]

Thank you so much for providing such concise summaries of Group Theory topics!

[@zorlo42]

These videos are fantastic! Thank you so much for helping me through my online Abstract Algebra course!

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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]

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

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[@theboombody]

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]

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

[@SIrLoneRabbit]

At 3:00, she says "In this case, it just so happens that G has subgroups of 17 and 19..." In which case? A hypothetical group G of with 323 undefined , hypothetical elements has also undefined hypothetical subgroups with 17 and 19 hypothetical, undefined elements?

[@moayadyaghi]

the same question :( even the following example is not clear to me at 03:30 why there is no subgroup of the order 6? and why there are subgroups of the other orders?

[@alexloyal3909]

This baffled me too. But as mathematicians we have to take it as we see it as simply a hypothetical example since we have no means of knowing whether it is a group or not.

[@bananarama6661]

sylov thms. it's a great question. just like there is only one group of order 2, there is only one group of order 323, and it must contain subgroups with those orders. This is because 17 (a prime) divides 323 but there is no a such that 17^a divides 323. Similarly, 19 (a prime) divides 323, but there is no such b where 19^b divides 323. (that is a=1, b=1 will work but any other natural number won't).

[@b43xoit]

I think the case she is referring to is that of a group having 323 elements. I don't know how many groups there are that have that many elements, but I interpret her talk as saying that everything that she says follows from "in this case" is true of any such group.

[@kquat7899]

Excellent concise treatment.

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I love this site because of this great teacher

[@tiowu]

That's some fine acting at the outro! Very cool.

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

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

[@rachanacrasta9352]

Wonderful!!! Beautifully explained. The concept is very clear. Thank you

[@shubhamk840]

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

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I like your videos..and i have watched all yours videos on abstract algebra many times. l love the way you explain the topics. Lots of love from India🇮🇳

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I love this channel because of this type of video