Group Definition (expanded) - Abstract Algebra
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- เผยแพร่เมื่อ 16 มิ.ย. 2024
- The group is the most fundamental object you will study in abstract algebra. Groups generalize a wide variety of mathematical sets: the integers, symmetries of shapes, modular arithmetic, NxM matrices, and much more. After learning about groups in detail, you will then be ready to continue your study of abstract algebra be learning about rings, fields, modules and vector spaces.
Our sincere thanks go out to our VIP Patron, Matt Peters. Matt supported us on Patreon, and thanks to his generous donation, we were able to make this video. Thank you for helping make this video happen, Matt!
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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)
www.jmilne.org/math/CourseNote...
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If you'd like to learn more, we have a free course on Group Theory! www.socratica.com/courses/group-theory
A solid proof how important a good teacher is!
Exactly.
The video explains everything clearly!
SOLID PROOF!
Easy on that Qian Chen
I'd really agree with you on this. My professor literally just went on to blabber mathematical shit that I didn't really know of but this video and the teacher made all the difference. Hands up
yes truly indeed, for the first time ever I found studying abstract algebra can also be fun.
Me: It's time to go to bed.
TH-cam: _True but have you seen first "Group Definition (expanded) - Abstract Algebra"?_
Same thing here !
Same here
same lol and I enjoyed
Lol 2am here
Same here Xd
In college, I was amazed how one of the first major theorems you learn is the fundamental theorem of calculus, and how the last one you learn is the fundamental theorem of arithmetic.
I was talking to a math major the other day. I am just a lay man, but I could remember Logic and sets were taught way further down (or not taught at all) in Uni math classes. The complete reverse...no wonder math is so confusing to many.
Well, that's because the "Fundamental Theorem of Arithmetic" is really the fundamental theorem of number theory. On the other hand, they COULD teach number theory sooner. It's not that hard, but it's not requisite for everything else the way calculus is.
Because calculus requires you to build on top of your current understanding of math, while the the fundamental theorem of arithmetic requires you to rebuild your current understanding of math. the second one requires a much more advanced mindset to actually understand
Jjhkjo
@@NuisanceMan People complain now that "I never use math", can you imagine what those people would say if they taught *number theory* instead of (I dont know) algebra?
My algebra teacher made it seem completely random and arbitrary when talking about this subject. These videos made everything SO MUCH CLEARER
This is the most well made course on the TH-cam by a significant margin
I can't believe how this made hours of my confusion just vanish away in mere minutes. Truly thank you for helping me begin to see the beauty of this concept
I am a spanish teacher.
Sometimes
I also help students in High school with Math.
Thus, I am looking forward to improve my skill in this very important field.
Math and spanish, mi respeto para usted
@Rafael Arturo Mateo Núñez math is for anyone dude
@Rafael Arturo Mateo Núñez they litteraly did during the bourbaki period in France, 12-14 years kids confronted to basics of abstract an linear algebra..
It worked wonders on ~10% of them and it was an utter disaster for the rest.
Totally reversed now.
This blew my mind!
Precise and to the point. Plus also be able to give the general overview! Excellent.
This is a scintillating instance of great pedagogy! One just requires pretty basic conceptual background to grasp this video in its entirety. I'll support you... Please keep doing this. I'm grateful TH-cam put this in my list.I was missing out on such great content.
So nice to have something explained so clearly. I once went though a book on Group Theory where it failed to tell you what it was.
These videos rock! I’m crying, that’s how elegant, informative and funny this content is! Bless! ❤️
I have been blown away by every single one of your videos I have watched. This channel is definitely a hidden gem. Thank you for your content!
It's amazing how a new perspective can totally change the way you view and deal with a subject. For months I'm trying to make sense of how different groups and their operations really work, and this video has made a huuuge difference. I'm very grateful for your job.
Fantastic video. The structure and delivery of the material is great. It's incredible how intuitive this topic can be if it's taught properly.
Wow! Best complete and comprehensible definition of groups I've ever come across. Thanks for this video, but especially for your approach at being complete but simultaneously comprehensible and clear. Considering all the arcane confusion out there, yours seems to be a rare and important skill.
I've tried watching other group theory lectures and yours are by far the most engaging and understandable!
Extremely well crafted video; you're a great speaker!
Good lesson on higher Math.
I am here just to express my gratitude as already valuable and well structured comments have been made regarding how you have made easier to understand "Group Definition" for us, who're having difficulties to understand this.
Thank you very much and my sincere, well wishes to your channel!
Thank you so much for your kind message! Much appreciated!! 💜🦉
that was by far THE BEST explanation video that I've ever seen! Can't believe these sort of contents are free on youtube! Thank you so much!
Thanks for this. I vaguely remember taking Abstract Algebra back in the 60s. I could not however remember any detail. It all came flooding back when you defined a Group.
Please don't stop making this wonderfull videos i mean WOAH! never seen abstract algebra so interesting THANK YOU!
Your delivery is just...flawless.
You are incredible at explaining abstract. I learned more in this 11: 14-minute video than I did in two 50 minute lectures at my school. Thank you for making these types of videos. THANK YOU
I passed an exam on Modern Algebra thanks to your playlist Abstract Algebra, which helps a lot to imagine these boring theorems and formulas. Thank you and please continue doing so magnificent lectures!
You are an extremely great teacher! This channel needs to be recognized af
In less than 1 hour i have learn more from than ever on group theory. Thanks a lot .
Holly molly! I expected this playlist to be really hard but it turned out having the perfect level of difficulty! I love what Socratica is doing, really focusing on clarity above all, regardless the subject. Keep up the good work :)
What an amazing way to learn abstract algebra from an amazing teacher in TH-cam! Excellent job!
Been missing the abstract algebra videos, thanks :)
Great video, very clearly explained the subject.
Thanks, I learned something.
You're an excellent teacher! The clarity of your explanations is astounding.
The presenter was so calm and composed. Watching this video felt like meditation to me.
I don't know how I ended up here after watching a Miley Cirus video, but it feels like landing on a fresh, clean, wholesome, normal planet .... Awesome ! Keep it up.
"wholesome".. lol... millenials!
You came in like a wrecking ball. We Can't Stop judging you.
love the Miley Cyrus part
I love you. You made algebra sound so easy to me and Ive been crying for two years to undestand it
Socratia, you don*t really imagine how much I love your clear explanations, you are an excellent teacher, love you for this : )
Just Found this Channel... I'm Hooked! Great Quality Content !
This is really nice to watch. I was always drawn towards math. I studied engineering in university which is very practical, but then when it comes to work it's mostly basic arithmatic and algebra. I miss mathematics as a pure subject 😒.
I'm so thankfull for this material! I'm freshmen in uni and currently we're studying group theory, and it distinguishes from school algebra so much!! All content on my language
(Russian) is so complex and formal, that it is almost impossible to comprehend how algebra works. But this videos saved me! Thanks for all your work, please, continue to make videos!
This is an amazing introduction to the subject of groups. I hate how they were taught but this lady explains so well with the illustrations. Keep up the great work!
Ok... I usually get weird recommended from TH-cam, but this just takes it to a whole other level.
This is called 'recommendation system', what it really means is that folks who you hang with all the time betrayed you and went to study math behind your back.
I feel so smart after watching this, thank you for the self-esteem boost ❤️
How you use animation in your lecture
I have gone through most of other's videos but didn't understood... And finally found your video. You made it so easy to understand with examples. Thankyou 👍👍👍
I'm watching this now prior to my comprehensive examination in my Master's degree. Cheers and thank you for your hard work in producing videos that are understandable to an extent. :)
We're so glad you're watching Socratica! We have a whole series of videos about Abstract Algebra: bit.ly/AbstractAlgebraSocratica
Subscribe here to see our latest videos: bit.ly/SocraticaSubscribe
bye socratica. i am your fan and admirer.when i watch your videos i fell as if godess saraswati is dancing on your tongue
thank you so very much for creating this video. it was so clear and easy to follow. I've been wanting to learn abstract algebra but didn't know where to start. I'm gonna go watch all your other videos on abstract algebra.
do you have any recommendations for textbooks for self study on the subject? i think it would help me supplement your lectures to get an even deeper understanding. thanks
Thank you very much ma'am
A day will come when i will be a good teacher like her
Her teaching method is amazing
@@s.k.potdarpotdar8377 4
I don't know how much time taken by mathematicians to define a group, but I only knew that it took me only 8 min to understand.
Purely Simplified ❤
Great concise presentation/visualization and a clear voice. Many thanks.
This is an amazing video! Thanks a lot!
I just started learning abstract algebra but had no idea what a symmetry group is. The video kind of explains all my confusion.
Algebra is the abstraction of arithmetic. Thus, abstract algebra is the abstraction of the abstraction of arithmetic. 😁
What about the abstraction of the abstraction of the abstraction of the abstraction of the abstraction of the abstraction of the abstraction of the abstraction of abstract algebra? ;-)
I guess that would be category theory.
en.wikipedia.org/wiki/Category_theory
@@benhur2806 Abstractception Algebra.
That'll keep me awake, thanks man
@@benhur2806 This is precisely what you don't want to do in mathematics. The "duality" of something does come up but very rarely.
Great production quality. Hiring an actress, writing a script, and producing the visuals....that costs time and money. I hope you guys can keep it up!
wow thank you ....a whole week of attending lectures covered simply in an 11 min video....amazing 🔥🔥🔥🔥
I'm taking a mathematics for machine learning course. I understood everything until they brought up groups. They provided a very academic source explaining groups which I barely understood. This video made it so easy to understand. THANK YOU!
This is awesome. Math taught the way it's meant to be.
Sign up to our email list to be notified when we release more Abstract Algebra content: snu.socratica.com/abstract-algebra
Thank you very much. You are my angel. You made my day!!!
In the triangle example why is r.f taken as an operation but f.r is not considered as one of them(since its non-commutative group its a separate operation right)
I'm really speechless of how good of a teacher you're thank you for doing this
Without any doubt, your videos are extremely easy to understand and very well organized.
I would like to see a bearing on this sort of material being introduced to students earlier on than post-high-school. At the moment, the mathematical curricula that schools push out is protracted for far too long. This is what mathematics is really about.
The education system needs to catch up
I think they underestimate what children are capable of learning
@@studiohq they always do
@@studiohq you say they need to catch up, but a surprising number of children and adults can't do basic numeracy. If anything they need to teach them the basics for longer.
@@FairyNuffMuffin2 interesting!
We either need to teach them the fundamentals for longer or, we need to stop letting kids advance in their education without learning these fundamentals…
Great video series! Can you please make a video on Lie Group and Lie Algebra?
Actually a nifty exercise to appreciate how crucial associativity is in the business of groups is the fact that courtesy associativity the left and right identity element of group are the one and the same, and the left and right inverses of any element of the group are the one and the same. Good work, keep it up :)
Her voice is so soothing, but not in a way that puts me to sleep. Its clear and well supported, what a beautiful sound. I'm currently an undergrad studying biology, I've only taken beginner calculus and statistics, but I am considering a minor in mathematics. These videos are definitely a big inspiration, as well as having a great teacher.
Abstract algebra♡♡ keep on post videos i love it
Support on patreon..!!
extra ordinary videos very helpful
God, why isnt this the first course TH-cam recommends on group theory, it's leagues clearer than the others. You've earned a sub
I am just as a obsessed with the absolute beauty of mathematics as you may be. I love this!
Great video !
I would have liked to have you as my teacher back in school !
Keep going on this amazing series of Math videos ! ❤️ from France !
Finally I got to know about Groups, so good explanation,
I'm happy today that I found you!! You're an awesome Teacher 😊💯 I re-built my concept just by your help 🌸 Continue your efforts for students like us!!
Thank you so much for explaining this so good never had a teacher who explained it so good
please upload videos on MULTIVARIABLE CALCULUS and VECTOR CALCULUS. ..... THANKS.....
waiting for you to upload these material to enable us easy learning
Please we want a lessons about "Nilpotent Group or ideal" and and about "sylow " 😭💔🙏🙏
Yup. I'mma gonna hafta rewatch that a few times more to git all them concepts in solidified in that grey matter.
Very well done video! This is the first video I found that summarizes the basic concepts of advanced algebra in a way that is relatable. You're a smart cookie. 👍
Can't believe I've been struggling so much when this page actually existed. Sincerely from the bottom of my heart THANK YOU!
We're SO glad when we hear our videos are helping people. It really inspires us to keep making more videos! 💜🦉
Q: What's purple and commutes?
A: An Abelian grape.
🍇
XD
No one: What‘s algebra
Me: when you draw your root out of an unknown in the morning
This was awesome! I thought I would need to re-watch it a bunch of times before I understood, but this was so clear that once was enough. :)
You gotta be kidding me. I have an exam about this in two days! Your timing couldn't have been any more perfect. Thanks!
In the end, in the definition you gave the identity should be mentioned prior to the inverse. Without the existence of identity there is no reason of speaking about inverse
Amazing teaching abilities. It's incredible how one-hour spending reading a famous math textbook did not teach me the same amount of information as you did in this video. Would you know why textbooks are not done in this way? Since authors are so smart, why can't they realize books should be written in this way? It's really an honest question if you don't mind answering.
It's a mystery that the best mathematical minds are unable to penetrate...
I agree! It’s almost like mathematicians take pride in making their textbooks as dense and opaque as possible. Math is hard enough! I don’t know why math books can’t break the concepts down in both an informal way and then with more rigor *after* the concept has first been simply explained. These videos are absolutely outstanding!
This is the most awesome channel I've seen so far, since I took number theory course on college that I've wanted to learn this subject, and this is a very graphic and fun way to do it. Thanks for the dedication!
Thank you so much for your kind words! We're so glad you've found us! :D
Great lecture. It's very helpful for deep understanding of group definition. Thank you so much!
Add some videos on Real analysis and calculus.
More math videos coming soon!
Doesn't mod stand for modulo (or sometimes modulus)?
Pierre Soubourou yes
modulus is (mathematics) the base with respect to which a congruence is computed while modulo is (computing) the operation or function that returns the remainder of one number divided by another. I also discovered there is a difference between mod and remainder when you divide negative numbers! 🙂
Pierre Soubourou it is called "module" actuallu
Yes, it's "modulo" -- the Latin ablative case form of "modulus". The meaning of the ablative case form is something like "using the modulus". So in 7 ≡ 2 (mod 5), pronounced "seven is congruent to two mod five", "mod 5" means "using the modulus 5", or "five being the modulus".
This is definitely a (minor) error in the script of the video.
just thinking how much effort you have put into the video.Hats off guys...
For some reason, my spectroscopy professor could'nt explain group theory in such a clear fashion. What a great channel!
You are perfect for the next COSMOS! Carl Sagan, Niel deGrasse Tyson and you!
Yep - and her Python robot counterpart.
1) "We call this transformation R-squared as we applied it twice." Why R-squared and not 2*R.
Is it because "x" is a dimensional operator and not scaler as "•" the dot operator? In other words, if the triangle was moved to the right or left, 'translated', would that be -2X (left) and 2X (right)? And if the triangle was moved to the up or down would that be translation onto the -i (imaginary vs. real number) axis? -2IX and 2IX?
2) If one can rotate it 180° about a vertical axis why not 'swing' 360° it about a horizontal axis positioned at the apex?
It's called R^2 because you are applying the R operation twice: RR, or R times R, or R squared.
Michael Sommers
Why does applying the rotation twice have to mean R*R? Why can’t we understand stand it as R+R (i.e. 2*R).
@@Hutt-px1jx its because we call the operation on that group "multiplication" so applying x and then y is written x times y so r times r is r^2.
Its just convention. If we called it "plus" like we did for clock arithmetic it would be 2*r.
@@danielmartchenkov7437 If the triangle was moved to the right or left, 'linearly translated', would that be -2X (left) and 2X (right)? And if the triangle was moved to the up or down would that be translation onto the -i (imaginary vs. real number) axis? -2IX and 2IX?
If it's just convention then call it cosine. Clock arithmetic would seem to be "plus" because it's a scaler and rotational translation has to be multi-dimensional so it's multi-plied ... no? ... 4x5=20 (sq. units) vs. 4•5=20 (scaler).
@@randyzeitman1354 You can also see it's not normal addition nor normal multiplication since in the symmetries of a triangle example it's not even commutative.
Also the complex extension of multiplication doesn't apply since calling it multiplication is just a convention. Like the examples with dot products, cross products you bring up-you wouldn't extend those to the complex plane, even though they're still called "multiplication".
And about calling it cosine.. the reason it's given the specific name of addition or multiplication is because of how it was discovered: the operation on the modular group on the integers very closely resembles addition, and a common convention is to associate composition of "actions" (in this case the action of doing something to the triangle) to multiplication. I'm not entirely sure where that started, but you can see it in linear algebra when you consider matrix "multiplication" (even though it is technically the composition of the linear functions that each matrix represents, which are transformations (action) of the vector space. And again! not commutative).
In fact, similarly to the matrices, the elements of a symmetric group (which in this case were the rotations and flips) are often denoted by functions. So the rotation clockwise would be the function that takes 1->2, 2->3, 3->1, if you imagine how the triangle with the numbered corners looks after that transformation. Then, doing an reflection, r, followed by a flip, f, is the function f(r), which takes the top-most point to f(r(1)), etc. This means that, like with the matrices in LA, the operation is in fact *function composition*, but when we intuitively consider it to be doing certain actions one after the other, we just write it as multiplication.
This video was really very helpful for any undergraduate as myself. Thank u so much for providing us with such a well defined and well organised explaination.
This was so helpful!! I've been struggling to grasp this concept and this really cleared it all up! Thank you!!
Cortana became human?
why we have no vertical rotation in triangles?
Because that can be made of the operations she did include?
You are MORE than amazing! The right person is in the right place. Thank you!!
You are awesome. I’m taking abstract algebra and discrete math (combinatorics and graph theory) this semester. It’s videos like these that convinced me to switch my major to mathmematics a year ago. I will be sure to make a contribution.
-Marty (University of Victoria, Canada)
I love your job! I will do everything to follow you! But I have my limits and I miss the Portuguese Socratica. Please look for sponsorship (example: DUOLINGO) and re-invest in the language content of our good old Brazilian literary (João Guimarães Rosa, Machado de Assis, Cecília Meireles). I and other MILLIONS of Brazilians, sometimes limited by our labor and financial conditions, want many stools without power to follow the EDUCATION you sponsored. Help us to help you come to our sad reality of not often recerber a BILINGUAL education. I and other MILLIONS of internauts are grateful, may God be with you in the struggle for a quality education in this NATION! (Brazil)
I have been searching for something like this for quite some time. Thank you for putting up such a tutorial
Although I already watched many other videos, I still have not understood the Group Definition until I watch this video, Thanks a lot, you are a good teacher.
I'm so glad to have found this video. I'm teaching a survey/liberal arts class in mathematics, and students are required to do math history presentations. One student is presenting on Galois but needs help in understand what Group Theory is (at the most basic level). This is a FANTASTIC introduction - short and accessible and accurate. I will be sharing this video with him. Thank you!
Thank you so much for sharing with your student!! We're rooting for his success! Be well, Socratica Friend! 💜🦉
Thank you so much for making these videos.
It made me understand how this works quickly.
Thanks again.
This is the best explanation of group, very clear presentation, am grateful.
Well done for this video. My lecturer and textbook fail to lay this out so well. They just scramble a bunch of these points together with no explanation making something so simple feel impossible. So thanks for the clarity.
Thank you sooooo much, don't know how to thank you 😭😭💜💜
I was in huge confusion about the real use of group theory and the real concept behind it, but you made it so simple to understand, THANK YOU ❤️