This is the most packed video of the series, content-wise. So many interesting and fascinating stuff mentioned, too quickly! It would be fantastic to have more videos on simple finite groups. You guys are the best!
I can't believe you guys have covered so much info in less than 10 minutes! That's really great, Socratica. Keep up the good work. (From Syria with love!)
Man, I have an exam tomorrow and I was looking for slow easy to understand videos with examples that drive the points home, but you're just as fast as my lecturer assuming that I already knew everything about math when I was born.
I am pretty new at group theory, so I did some 'research' (aka me typing it onto Google ) ... |A_13|= 13!/2= 3 113 510 400 Thats... a lot of subscribers you are asking for...
You could've simply taken the cardinality of S_13, 13!, and divided that by 2. The cardinality of even permutations in S_n is always the same as the number of odd (If n >= 2). You can prove it by defining a bijection between the two sets.
This is where my journey with your series ends, you have been a great help! This video in particular is very comprehensive! :D Thank you thank you thank you!
So, I am learning quantum mechanics and the abstract algebra is like a language you're using to talk about it. Although these lectures don't cover representations of groups and Lie's groups which are also needed for my quantum mechanics classes, I must say I'm in love! The concepts you're covering seem like they come in a natural way one after another and you want to know everything about every single concept. They don't seem just like a random topics you need to understand as fast as you can! The enthusiasm and the sort of an adventurous vibe I'm getting from the way you're talking is making me feel like I'm watching a movie! Thank you!!
First of all thanks for your all videos.. I don’t get time to comment, on every video, but let me tel u , Your explanation is just awesome ..👍👍keep it up 👍👍 Please every time keep trying to make abstract mathematics as a layman language subject as long as possible I know it’s hard to do every time , but that’s the only way we can convert maximum individuals to love higher mathematics ...❤️
I cannot believe I have followed from episode 1 to 22 and intend to keep going. You explain these abstract and difficult ideas in a much clear way than my any of my professor. Thank you so much! [I might find a small typo in episode 22 for Simple Groups at 03:04 in the second line (title not included) "Quotient groups are simple: (N_1/1), (N_1/N_2), (N_3/N_2)..." Is it intended to be (N_2/N_1)?]
We're so glad you are enjoying our videos! Your donation is SO appreciated. It will help us make more of these videos!! Thanks so much for your kind words and support.
Thank you for your kind comment! We're a small team of educators who make videos for UA-cam! You can read more about us here: www.patreon.com/socratica
Nice summary. I think it would be helpful to elaborate the correspondence between prime numbers and simple groups as follows: Every finite group (positive integer) can be expressed as a product of a unique set of simple groups (prime numbers) by the Jordan-Hoelder Theorem (Fundamental Theorem of Arithmetic). But a given set of simple groups can be multiplied in different ways to give different product groups (the extension problem you mentioned), whereas a given set of prime numbers can be multiplied in only one way to give a unique composite number. I guess the reason for this is that arithmetic multiplication is commutative but group multiplication is not.
Wonderful presentation!!! The videos are a great resource to understand the basics as well as some of the advanced concepts of Abstract Algebra neatly, quickly and efficiently... I'm a researcher in Applied Mathematics and the videos helped me a lot to revise my algebra concepts in a gist... Thanks a lot... Waiting for more topics on Advanced Mathematics to come...
I like your enthusiam ..... you sure have passion for the subject you are discussing ..... but I think that you need to add more examples .... and more important real life applications ..... the problem that makes alot of people hate math is that they feel it is irrelivant to thier every day life ..... one of the merets of educational videos on youtube is the appility to show people how science really affects thier life
To be honest, this video has a much higher requirements for the audience. Hence is not that consistent with the previous videos about abstract algebra. And the topics covered in this video is seldom used for a beginner of abstract algebra.
wow at least i came to this video.. finally i can understand why pol eq. of 5 or more grade have not a general formula as solution!!! thank you socratica team!
I like your videos on abstract algebra , but can you make videos on real sequences. Like bounced and unbound sequences , least upper bound greatest lower bound , infima , Suprema etc. if you do so then , It would be a great help .
Great video, But slightly misleading at 5:14 There is no general formula for degree 5 and higher *** IF *** you consider only using BASIC operations like +,-,*,/, roots, powers, exp(x), log(x), sin(x), cos(x) etc. Its a common misunderstanding that this hold for ALL types of multivalued functions you can consider. And in fact, there are GENERAL solutions for degree 5 and higher. Using elliptic functions, or jacobi theta functions, some others I can't even recall, hypergeometric etc.
Can you do a video on the monster group? John Conway thinks that he’s going to go to his grave without having learned why it’s there and that would be tragic
I think this channel is great. I love that it uses women to do scientific divulgation, I'm sure that this kind of projects helps to bring a little closer the gap that exists between men and woman in sciences and in mathematics. In particular I liked this episode a lot. Perhaps it appeals to a more specialized public than the others but It was useful for me to remember and understand some things. I have some comments in the content of the video: There is an important difference between the factorization of a positive integer as a product of prime numbers and the Jordan-Hölder composition factors from the composition series of a finite group. This difference is that you can't recover the original finite group from this composition factors alone, or in other words, there are different finite groups with the same composition factors. Nevertheless, there is a theory called "Group Cohomology" in the context of Homological algebra that is developed solely to understand this issue. Anyway, this analogy is useful to understand the result and in fact this two factorization results are related in other form that just the analogy. The Jordan-Hölder theorem implies the Fundamental theorem of arithmetic: Given a positive number n, one can prove that the factorizations of n as a product of primes are in bijection with the composition series of Z_n and under this bijection the compositions factors correspond to the primes in the factorization. Then, the uniqueness and existence of the composition series gives the uniqueness and existence of the factorization of n. This proof is the one that Bourbaki uses to prove the fundamental theorem of arithmetic (see Bourbaki's Algebra chapter 1).
Hi guys! Can anyone explain how the monster group can contain quotient groups? I thought that in order for a group to contain a quotient groups, it needs to contain normal subgroups. (simple groups do not contain normal subgroups, and the monster is a simple group) Thank you.
I looked it up. Apparently the happy family are all _subquotients_ of the Monster Group. Given a group G, a group K is a _subquotient_ of G if K is isomorphic to a quotient of a subgroup of G. In other words, K is a subquotient of G if there is some subgroup H ≤ G and some normal subgroup N ⊴ H so that K ≅ H/N.
1) Time will tell ?? (abstract) Perhaps there is a group between the "Happy Family" and the "Pariahs" 2) For patreon support do you accept also Bitcoins ?
1) People are researching ways to unify the sporadic groups. I'll need to check on the latest research to see what progress has been made. 2) We *do* accept bitcoins! :) You can find our address on our "About" page: ua-cam.com/users/SocraticaStudiosabout Thank you so much for considering supporting us!!
Stick to one of the names : like factor group or quotient group.. And similarly in other situations.. Maybe show an asterisk comment at the bottom of the video.. I'm a fan.. Trying to help..
Let G be a group and N be a subgroup of G. 1) gNg^-1 is a subset of N for all g in G 2) gNg^-1 = N for all g in G Statements 1 and 2 are equivalent. So either can be used as the definition.
@@rylieweaver1516 You can prove it! Statement 2 implies statement 1 without really any work. So most of the work goes into showing statement 1 implies statement 2. So suppose statement 1 is true: gNg^-1 is a subset of N for all g in G. Now fix an arbitrary g in G, and we will want to prove that gNg^-1 = N. To show two sets are equal, we can show they are subsets of each other. By statement 1 (which we are assuming), gNg^-1 is a subset of N. So all we have to do is show that N is a subset of gNg^-1. To do that, we should take an arbitrary element n of N. We want to find an element m of N so that n = gmg^-1. Can we do this? I think this is a good exercise for you to try on your own, but you are welcome to comment back again asking for the rest of the details, and I can provide them.
Is this connected to M theory in physics (because Monster group) and 26 dimensions that were needed before modern string theory allowed for 10 (before moving on to 11)..? P.S.: Very pop sciency.. I know..
At 7:27 the instructor mentions that monster group contains 20/26 sporadic groups as either subgroups or quotient groups. But as monster group is a simple group, then it shouldn't have any normal subgroups right? And hence we shouldn't be able to form any quotient groups? Can someone please comment on what I'm missing here.
I looked it up. Apparently the happy family are all _subquotients_ of the Monster Group. Given a group G, a group K is a _subquotient_ of G if K is isomorphic to a quotient of a subgroup of G. In other words, K is a subquotient of G if there is some subgroup H ≤ G and some normal subgroup N ⊴ H so that K ≅ H/N. This does not contradict the simplicity of the monster group because H will not be normal in G there. This is not a failure on your part, though, because the video didn't say this.
7:33 How is it that the monster group contains any of the other members of the happy family as quotient groups? I'm under the impression that simple groups can't have quotient groups.
I looked it up. Apparently the happy family are all _subquotients_ of the Monster Group. Given a group G, a group K is a _subquotient_ of G if K is isomorphic to a quotient of a subgroup of G. In other words, K is a subquotient of G if there is some subgroup H ≤ G and some normal subgroup N ⊴ H so that K ≅ H/N. This does not contradict the simplicity of the monster group because H will not be normal in G there.
Thank you for making UA-cam a better place to pass time.
literally have an abstract algebra exam tomorrow. videos are undoubtedly a great help!
Thank you to the entire team who worked hard to produce this great video series.
This is the most packed video of the series, content-wise. So many interesting and fascinating stuff mentioned, too quickly! It would be fantastic to have more videos on simple finite groups. You guys are the best!
I can't believe you guys have covered so much info in less than 10 minutes! That's really great, Socratica. Keep up the good work. (From Syria with love!)
That zooming-in and Monster group music! So suddenly intense, just like the rate of my learning after finding Socratica several days ago!
I can only imagine how much time and effort and knowledge is required to put out a video like this.
It's so sad that they have stopped making videos, now who will teach me more of such awesome things
"releace a video every hour" Nooo, I will witerally spand my life watching your videos...
Man, I have an exam tomorrow and I was looking for slow easy to understand videos with examples that drive the points home, but you're just as fast as my lecturer assuming that I already knew everything about math when I was born.
I am pretty new at group theory, so I did some 'research' (aka me typing it onto Google ) ... |A_13|= 13!/2= 3 113 510 400
Thats... a lot of subscribers you are asking for...
T series has surpassed both pewds and Music to become no 1. But even they've got apprx 107M subs.
You could've simply taken the cardinality of S_13, 13!, and divided that by 2. The cardinality of even permutations in S_n is always the same as the number of odd (If n >= 2). You can prove it by defining a bijection between the two sets.
Thank you so much Socratica! You make mathematics very intriguing!
This is where my journey with your series ends, you have been a great help! This video in particular is very comprehensive! :D Thank you thank you thank you!
We're so glad you found our videos helpful! Thank you so much for watching. Please share with anyone you think we could help! 💜🦉
You resumed 300 years of mathematics in just 8:52 minutes. Thank you.
I am afraid to going to sleep today and have bad dreams because of this monster group. Thanks making us understand these concepts.
This is a nice introduction to finite simple groups! Thank you, Socratica!
So, I am learning quantum mechanics and the abstract algebra is like a language you're using to talk about it. Although these lectures don't cover representations of groups and Lie's groups which are also needed for my quantum mechanics classes, I must say I'm in love! The concepts you're covering seem like they come in a natural way one after another and you want to know everything about every single concept. They don't seem just like a random topics you need to understand as fast as you can! The enthusiasm and the sort of an adventurous vibe I'm getting from the way you're talking is making me feel like I'm watching a movie! Thank you!!
well this 8 mins video is the best investment of my life till now....thank you madam.....
That is so nice of you to say, thank you! We're so glad we could help. 💜🦉
First of all thanks for your all videos..
I don’t get time to comment, on every video, but let me tel u ,
Your explanation is just awesome ..👍👍keep it up 👍👍
Please every time keep trying to make abstract mathematics as a layman language subject as long as possible
I know it’s hard to do every time , but that’s the only way we can convert maximum individuals to love higher mathematics ...❤️
Would love for this series to eventually get to an explanation of what E8 is and why it is considered such a beautiful mathematical object.
This lady knows so much. How about a video on cohomology groups?
I've heard that there is only ONE mathematician alive now who understands the whole 10000 pages of simple groups. S a d.
And his name is?
I cannot believe I have followed from episode 1 to 22 and intend to keep going. You explain these abstract and difficult ideas in a much clear way than my any of my professor.
Thank you so much!
[I might find a small typo in episode 22 for Simple Groups at 03:04 in the second line (title not included) "Quotient groups are simple: (N_1/1), (N_1/N_2), (N_3/N_2)..."
Is it intended to be (N_2/N_1)?]
I wondered about that (N_1/N_2) as well. Glad it's not just me.
I love seeing you go a little deeper into abstract algebra, nice job you earned a donation!
We're so glad you are enjoying our videos! Your donation is SO appreciated. It will help us make more of these videos!! Thanks so much for your kind words and support.
I got goosebumps when I saw Monster group :O And the music was whaaat
I study business but I really like these videos
This is a wonderful presentation -- thank you! What exactly is Socratica?
Thank you for your kind comment! We're a small team of educators who make videos for UA-cam! You can read more about us here: www.patreon.com/socratica
Thanks for making me understand a bit in the ocean. I am struggling very hard to get the essence of it.
Nice summary. I think it would be helpful to elaborate the correspondence between prime numbers and simple groups as follows: Every finite group (positive integer) can be expressed as a product of a unique set of simple groups (prime numbers) by the Jordan-Hoelder Theorem (Fundamental Theorem of Arithmetic). But a given set of simple groups can be multiplied in different ways to give different product groups (the extension problem you mentioned), whereas a given set of prime numbers can be multiplied in only one way to give a unique composite number. I guess the reason for this is that arithmetic multiplication is commutative but group multiplication is not.
Wonderful presentation!!! The videos are a great resource to understand the basics as well as some of the advanced concepts of Abstract Algebra neatly, quickly and efficiently... I'm a researcher in Applied Mathematics and the videos helped me a lot to revise my algebra concepts in a gist... Thanks a lot... Waiting for more topics on Advanced Mathematics to come...
Thank you in these 9 min video u explained a lot and in a simple way
Slide at 6:27, “intervertible” is written, “invertible” was spoken
Thanks for the video series, although I don't speal English, there are so useful for me. My best greetings from Arequipa - Peru
There is a typo at 3:10: (N1/N2) should probably be (N2/N1).
Congratulations and many thanks for the excellent video!
I like your enthusiam ..... you sure have passion for the subject you are discussing ..... but I think that you need to add more examples .... and more important real life applications ..... the problem that makes alot of people hate math is that they feel it is irrelivant to thier every day life ..... one of the merets of educational videos on youtube is the appility to show people how science really affects thier life
Your way of explanation is wonderful
To be honest, this video has a much higher requirements for the audience. Hence is not that consistent with the previous videos about abstract algebra. And the topics covered in this video is seldom used for a beginner of abstract algebra.
Thank you alot for these series
|A13| is approximately 3 billion. Way to go, Socratica!😃
wow at least i came to this video.. finally i can understand why pol eq. of 5 or more grade have not a general formula as solution!!! thank you socratica team!
Sign up to our email list to be notified when we release more Abstract Algebra content: snu.socratica.com/abstract-algebra
@8:23 The number she’s aiming for is half of 13 factorial which is 3.1 *10^9
The gist of Galois theory in under 10 min! The groups might be simple, but this video is certainly not. Outstanding work, as usual Socratica...
Love your indetails information on group.❤️❤️❤️
I love this channel. Thanks for the great video.
I like your videos on abstract algebra , but can you make videos on real sequences.
Like bounced and unbound sequences , least upper bound greatest lower bound , infima , Suprema etc. if you do so then ,
It would be a great help .
A13 is a huge group !
At 3:05 there is a mistake. I think that is not N1/N2 but N2/N1
Great lecture
Group is very interesting chapter in abstract algebra
Bravo, I love this video, it’s so fascinating and helpful
amazing work!! keep up
3:04 can we take N1/N2, where N2 is bigger than N1? I think not, since N2 has to be a normal subgroup of N1 to be able to take a quotient group.
seems like a mistake?
Great video,
But slightly misleading at 5:14
There is no general formula for degree 5 and higher *** IF *** you consider only using BASIC operations like +,-,*,/, roots, powers, exp(x), log(x), sin(x), cos(x) etc.
Its a common misunderstanding that this hold for ALL types of multivalued functions you can consider.
And in fact, there are GENERAL solutions for degree 5 and higher. Using elliptic functions, or jacobi theta functions, some others I can't even recall, hypergeometric etc.
Thank you teacher 😊
Can you do a video on the monster group? John Conway thinks that he’s going to go to his grave without having learned why it’s there and that would be tragic
Would love that. Want to learn more about it!
Interesting that two of the concepts in the video are named after 19th century Norwegian mathematicians, Abel and Lie
These are actually really nice videos, I'm impressed!
This is really awesome! Great work. 😍🤗🤗
Hola
Muy buenos videos, excelente calidad
Me gustaría que volvieran en español
Great Socratica❤❤❤
hello! why do we get R^(n^2) ? why is n^2 a dimension?
I hate Maths but John Conway video got me here 😂😂😂😂😂😂😂😂
Great...Please upload more videos on abstract Algebra...also in linear algebra and real Analysis
I think this channel is great. I love that it uses women to do scientific divulgation, I'm sure that this kind of projects helps to bring a little closer the gap that exists between men and woman in sciences and in mathematics.
In particular I liked this episode a lot. Perhaps it appeals to a more specialized public than the others but It was useful for me to remember and understand some things.
I have some comments in the content of the video: There is an important difference between the factorization of a positive integer as a product of prime numbers and the Jordan-Hölder composition factors from the composition series of a finite group. This difference is that you can't recover the original finite group from this composition factors alone, or in other words, there are different finite groups with the same composition factors. Nevertheless, there is a theory called "Group Cohomology" in the context of Homological algebra that is developed solely to understand this issue.
Anyway, this analogy is useful to understand the result and in fact this two factorization results are related in other form that just the analogy. The Jordan-Hölder theorem implies the Fundamental theorem of arithmetic: Given a positive number n, one can prove that the factorizations of n as a product of primes are in bijection with the composition series of Z_n and under this bijection the compositions factors correspond to the primes in the factorization. Then, the uniqueness and existence of the composition series gives the uniqueness and existence of the factorization of n. This proof is the one that Bourbaki uses to prove the fundamental theorem of arithmetic (see Bourbaki's Algebra chapter 1).
Plz give video's on some examples on abstract algebra like inverse , order of an element..etc.
Hi guys!
Can anyone explain how the monster group can contain quotient groups?
I thought that in order for a group to contain a quotient groups, it needs to contain normal subgroups.
(simple groups do not contain normal subgroups, and the monster is a simple group)
Thank you.
I looked it up. Apparently the happy family are all _subquotients_ of the Monster Group. Given a group G, a group K is a _subquotient_ of G if K is isomorphic to a quotient of a subgroup of G. In other words, K is a subquotient of G if there is some subgroup H ≤ G and some normal subgroup N ⊴ H so that K ≅ H/N.
I am just amazed! I only can thank you..
1) Time will tell ?? (abstract) Perhaps there is a group between the "Happy Family" and the "Pariahs"
2) For patreon support do you accept also Bitcoins ?
1) People are researching ways to unify the sporadic groups. I'll need to check on the latest research to see what progress has been made.
2) We *do* accept bitcoins! :) You can find our address on our "About" page: ua-cam.com/users/SocraticaStudiosabout
Thank you so much for considering supporting us!!
A masterclass presentation.
Stick to one of the names : like factor group or quotient group.. And similarly in other situations.. Maybe show an asterisk comment at the bottom of the video.. I'm a fan.. Trying to help..
unique presentation information ... thanks a lot ..
Briliant work!
Nice explanation
Order of A13 is 3,113,510,400
Very clear explain...
Thanks.
Nice
At 1:01, it should say that N is normal if gNg^-1 is a subset of N, not equal
Let G be a group and N be a subgroup of G.
1) gNg^-1 is a subset of N for all g in G
2) gNg^-1 = N for all g in G
Statements 1 and 2 are equivalent. So either can be used as the definition.
@@MuffinsAPlenty How do you know that the statements are equal?
@@rylieweaver1516 You can prove it!
Statement 2 implies statement 1 without really any work. So most of the work goes into showing statement 1 implies statement 2.
So suppose statement 1 is true: gNg^-1 is a subset of N for all g in G.
Now fix an arbitrary g in G, and we will want to prove that gNg^-1 = N. To show two sets are equal, we can show they are subsets of each other. By statement 1 (which we are assuming), gNg^-1 is a subset of N. So all we have to do is show that N is a subset of gNg^-1.
To do that, we should take an arbitrary element n of N. We want to find an element m of N so that n = gmg^-1. Can we do this?
I think this is a good exercise for you to try on your own, but you are welcome to comment back again asking for the rest of the details, and I can provide them.
You guys want 3+ billion subscribers?
Please do a video on the Monster!
|A₁₃| = 3 113 510 400 = 2⁹×3⁵×5²×7×11×13
A big number indeed.
typo: 1. no *normal* subgroup H (missing normal) 2. quotient groups are simple: N1/1, * N2/N1 *, N3/N2, ...
Is this connected to M theory in physics (because Monster group) and 26 dimensions that were needed before modern string theory allowed for 10 (before moving on to 11)..?
P.S.: Very pop sciency.. I know..
Thanks
3:05 There is a typo/error. N2/N1, not N1/N2. The latter doesn't make sense.
At 7:27 the instructor mentions that monster group contains 20/26 sporadic groups as either subgroups or quotient groups.
But as monster group is a simple group, then it shouldn't have any normal subgroups right? And hence we shouldn't be able to form any quotient groups?
Can someone please comment on what I'm missing here.
I looked it up. Apparently the happy family are all _subquotients_ of the Monster Group. Given a group G, a group K is a _subquotient_ of G if K is isomorphic to a quotient of a subgroup of G. In other words, K is a subquotient of G if there is some subgroup H ≤ G and some normal subgroup N ⊴ H so that K ≅ H/N.
This does not contradict the simplicity of the monster group because H will not be normal in G there.
This is not a failure on your part, though, because the video didn't say this.
Ma'am plz make one vedio on P group, sylow p-subgroups and related theorem
It's well undertaken video
视频讲的很清楚，获益匪浅！
S_n is not simple for any n>1 because A_n is a normal subgroup with index 2 of it!
But, A_n is simple for n>4.
Hausdorff Space and T2 space is also T3 space ?
is it Right?
I need your lectures about real analysis and topology
A factor group is also known as a quotient group
Este canal es increíblemente bueno
Vf
thank you so much mam
(from India)
3:04 Looks like ( N1 / N2 ) should have been ( N2 / N1)
Proof of 10 thousands of pages. I wonder how many people have actually read it and verified that there is no error or miss in the proof.
you say "remember manifolds?" but i am not able to find a video of yours covering manifolds. which one is it? thx
Unique!...🙏
7:33 How is it that the monster group contains any of the other members of the happy family as quotient groups? I'm under the impression that simple groups can't have quotient groups.
I looked it up. Apparently the happy family are all _subquotients_ of the Monster Group. Given a group G, a group K is a _subquotient_ of G if K is isomorphic to a quotient of a subgroup of G. In other words, K is a subquotient of G if there is some subgroup H ≤ G and some normal subgroup N ⊴ H so that K ≅ H/N.
This does not contradict the simplicity of the monster group because H will not be normal in G there.
thank you