Group Homomorphisms - Abstract Algebra

[@Socratica]

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

[@yassineaghlaoui1906]

Absolutely the TEACHER OF THE YEAR. Far better than our teacher who just tells us to memorize the definition of an homomorphism without knowing the intuition behind it. Thank you so much. I really appreciate the great work you're doing. Keep doing so! Cheers from Morocco!

[@dialecticalmonist3405]

"Passing the test" is not isomorphic to understanding the material.

[@atrophysicist]

Moroccain here too, also came to learn about algebraic structures to pass the bac exam and i can tell you that our teacher didn't do a good job of explaning it either so im glad i found this series.

[@SaadTheGlad]

Hahahaah wash tanta mafhemti walu mn lcours ou jiti hna bash tfhem?

[@Cpt.Zenobia]

I found this channel while searching something in Number Theory for my cryptography class, but now I really got into pure mathematics and started studying it as a hobby & implementing what ever I can because of this channel. Keep up the good work.

[@yutopia7]

I always loved math and I still do. Everything upto calc 3 was easy for me. I even went to college as a math major. Unfortunately, once I took abstract algebra everything fell apart and my last 3 years in college turned into a living hell. I read the textbooks and nothing made sense to me. My professors didn’t do a good job explaining the concepts either. I really wish I had a teacher like you. Better late than never!

[@Dusk425]

We are in the same boat

[@hegelscat9423]

Some people cant do algebra but can do analysis, and some people can do analysis and not algebra, and then there is those fortunate enough that can do both. But teachers influence that a lot. I have been lucky enough to have a great algebra teacher. I've notice among the faculty at my school that their preference is influenced by undergraduate professors.

[@dialecticalmonist3405]

I never understood math. But when I started learning about Group Theory / Category Theory, it finally began to make sense to me. All that algebra nonsense started to actually mean something.

[@yutopia7]

@@dialecticalmonist3405 I genuinely envy you. I wish I had your brain.

[@yutopia7]

@@dialecticalmonist3405 I genuinely envy you. I wish I had your brain.

[@kemsekov6331]

This is the best of "Watch while eat. Think for day" thing ever! Your content give me joy incomparable to anything else!

[@Socratica]

We're so glad you're watching Socratica! You can subscribe for more great STEM videos here: bit.ly/SocraticaSubscribe Our Abstract Algebra playlist is here: bit.ly/AbstractAlgPlaylist

[@JoaoVictor-wx5lz]

Volta com o socrática português, o Brasil precisa de um incentivo para ter uma boa educação e seu conteúdo nos ajuda a ser estudantes melhores

[@dariuszg9991]

Your videos about SQL are super cool and funny!

[@kamranabdulkhaev1767]

Dear Socratica, Your videos are great, but I have difficulties in understanding the order of the playlist. Some videos starts with Abstarct Algebra in braces, some ends with this words and some even doesn't have them. Moreover the content (topic) sometimes jumps from one video to another. Is the playlist in a right order?

[@johnsun2580]

This video is much more understandable and beginner friendly than the Stanford lecture I've struggled to follow. Thank you!!

[@PunmasterSTP]

I think subscribing can definitely STEM any tides of ignorance!

[@MusicalInquisit]

Every time I watch these videos, I always feel that I am doing something epic (thanks to the music). Watching this is epic.

[@annushekhar6042]

I couldn't tolerate other channel's one hour video on group so I came here.....Thank you Socratica!

[@wellfederal]

Hi! I have a degree in physics, but unfortunately I don't work in the field. Your channel is inspiring. CONGRATULATIONS!! I am Brazilian and would love to see more videos on Socrática Português. In Brazil, whose STEM education is deficient, its channel is sorely missed. Congratulations again !!

[@MrGiuse72]

I praise you attitude of briefly clarifying the scopes, heuristically describing the motivations of definitions and theorems . THANK YOU

[@ProfessorDBehrman]

She speaks very clearly. Thanks for giving this introduction to the homomorphism.

[@TheTessatje123]

The notation on 7:43 clearifies a lot , because often the group operations of different groups have the same symbol.🙏

[@kaebynj.quintero2993]

I came to KZbin to look for philosophy answers and ended up here but this was pretty interesting to watch!

[@strncatt]

Exactly what I needed!!!

[@louis6331]

That was the most beautiful example of isomorphism. I didn't understand it until I heard it on TED Talk and I said what is isomorphic. That's when I found this channel. And it was so much easier to understand then any other thing that concerns any other thing that concerns algebra. I was never good at it. But this made me understand it in a whole new light. Thank you. Keep up the good work. This was the most mind-blowing and understandable way of explaining what I did not know. Thanks again.

[@Ray-jg5dj]

You talk really well. Perfect intonation and really satisfying. It makes the information easier to remember and prevent people from getting bored. Great !

[@John-lf3xf]

This brand you have built is very impressive. You’re brilliant!

[@katetranscribes]

The best Christmas present!

[@chasr1843]

This series is great

[@MadanKumar-xg6xy]

One of the best video explaining the motivation behind homomorphism in Group Theory. Thanks a lot.

[@edt8597]

Merry Christmas 🦌⛄🎄🎅 Thanks for all the useful stuff you posted this year👍

[@Socratica]

You are so welcome - thank you for watching!! 💜🦉

[@KazimirQ7G]

Thank you! I'm really glad you guys are back, despite the campaign to continue Python videos, it's good to see you're not restricted to that.

[@Aaravs21]

"Imagine We have 2 group which we will creatively call G1 and G2" Lmao

[@Djn77645]

I had so many aha moments watching your videos (10 days before my exam) cheers from France 🤙

[@ramumaha2779]

i love how u explain the each concept and include those awesome background music to emphasis the important points.

[@graphingwithgeogebra]

yes yes, the background music

[@facepalm4e]

So glad to see you guys are still making these videos! Such a good way to explain Homomorphisms!

[@sugongshow]

You're awesome! Thank you for allowing me as a part of your "group". :) Keep up the great work.

[@sugongshow]

By the way, in honor of Women's History Month, I am going to do a presentation on women in math. With that, I am going to try to give you a plug. Again, thank you for all that you do.

[@Socratica]

That's very kind of you. Good luck in your presentation!! 💜🦉

[@sugongshow]

@@Socratica , thank you very much. It went well even though I had a very short 10 minutes.

[@yousify]

Nice Nice Nice! I liked the sound engineering in the video and the way you speak!! I liked most the 9:40 "mag ... ic"!

[@RalphDratman]

This works perfectly for my speed of comprehension and level of knowledge about groups. Thank you!

[@DiegoMathemagician]

Best Christmas gift

[@stevejohnson1685]

Very nice video! A friend just wrote a paper on group theory, with mappings from R to R. I read through it, and had some thoughts about it (I'm not an abstract algabraicist), which were confirmed in this video. Happy! Another commenter asked for more on Haskell; I'd request the same, particularly with the connections to the mathematical underpinnings you're so good at explaining. Thanks again for your carefully scripted and clearly elucidated concepts!

[@desativa0000]

Parabéns pelo conteúdo, pela aula, só dê um pouco a mais de atenção ao Socrática Português por favor, continue postando vídeos com maior frequência! Abraço.

[@Gunth0r]

Oh boy, and I thought this was just a python channel. I know what I'm binge watching this week.

[@Vfdking]

You (people) make me gratuitous. Thnx for nice videos on my phone, helping with intuition.

[@distrologic2925]

Math is awesome. Especially algebra, which is like a language for mathematics. I wish more people could speak it.

[@Or0203]

Found my new favorite math channel. Thanks a lot

[@bradleymorgan8223]

you have compressed 3 lectures worth of monotonous droning into 10 minutes, very impressive!

[@aaronnorman9755]

I am amazed by your ability to move information through my thick skull, thank you very much!!!

[@tareksaad1393]

came across this video a year ago, then landed back here again... am in a Bitcoin rabbit hole, and am looking for something just dont know what it is... but its somewhere here ... just wanted u to know i love ur videos ... u made me love math again ... appareciate math... and be like a kid trying to figure how the rabbit came out of the hat! peace!

[@shubh1606]

I dont have any exam related to group theory in future and its 1 january 2:30 am and i am still watching this video because i love group theory nice explanation,Happy new year

[@bryanchambers1964]

Thanks Patreon. It has been a dream of mine to teach myself Galois theory. Your quick vids give me some momentum for that.

[@Junnabeaverchewy]

😓 where was this video last semester?! I'm done with Abstract Algebra but i still love watching these videos!

[@okoyoso]

One is not simply 'done' with abstract algebra.

[@chaitanyatuckley4666]

Finished the entire playlist. It was wonderful.

[@Socratica]

Thank you for sticking with us!! 💜🦉

[@mohanbhattrai2752]

Your all videos are excellent . you are the export on abstract algebra .I salute you on your knowledge.

[@Socratica]

Did you know we have an Abstract Algebra newsletter? You can sign up here! snu.socratica.com/abstract-algebra

[@MadanKumar-xg6xy]

Wonderful

[@hammademuhammad3266]

Thanks Socratica you are very helpful and assisting teacher of mathematics you have made my many question simple for me

[@anujarora0]

Long time no see

[@Socratica]

Hello hello! Thanks for watching! 💜🦉

[@wjshood]

Its weird but I've forgotten all my analysis, but I remember this stuff perfectly. I love algebra everything just makes so much sense

[@elvinsamedov547]

*English is not my native language. But I admire your English accents. I hope one day I will have perfect English like you. I envy you...:) for your Perfect English and Mathematics.. Best Regards from Azerbaijan. I just discovered your channel and subscribed.*

[@thepianist7379]

Thanks!

[@Socratica]

Thank you so much for your kind contribution! Much appreciated!! 💜🦉

[@SubhomoyHaldar]

Even though my semester on Modern Algebra is over, I'm still watching this. Thanks for helping out with the A!

[@rambo3rd471]

Same!

[@Icenri]

Groups! Again! YES!!!!!

[@Icenri]

Also, first time I catch a video with 0 views 🤭

[@Socratica]

@@Icenri WOW first viewer!! 💜🦉

[@MBAVD5]

wow this is so useful to understand mapping between groups...thank you for these videos...happy holidays

[@qadeer.ahmad123]

what a awesome teaching method i ever got.

[@chimachan6652]

i can just say : you r the best keep going

[@ViniciusSilva-vc4jt]

Uma sugestão. Poderiam fazer vídeos sobre como alguns grandes matemáticos pensaram para chegar a essas conclusões(nesse caso específico, teoria dos grupos).

[@acquireknowledge8327]

You are Owsome brilliant mind blowing teacher... watching from Pakistan 🇵🇰

[@Socratica]

Hello to our Socratica Friends in Pakistan! 💜🦉

[@bazislapper6283]

Thanks from MSU . Greate material, it was very important for me to know this now.

[@MrGiuse72]

you ARE a goddess of didactic !! REALLY. Never heard a mathematician going to the "meaning" or the "origin" of a concept. That is what te student is looking to create a context for the topics. Usually articles, text and scientists act coldly in generality and abstractism. PRAISE TO YOU

[@uniquechannel3086]

The table explanation gave me more reality perspective and helped to understand the formula

[@cameronspalding9792]

@2:56 an easy way to verify that is to see that the integers is an infinite set and {0,1} has size 2

[@JohnDoe-oo9ll]

Man this is fire

[@romywilliamson4981]

Please do more advanced topics with modules! And also PIDs, EDs, UFDs etc. You're so good at explaining this kind of thing but it doesn't go far enough for my Rings and Modules course and I'm finding it really hard to get an intuition for a lot of the concepts involved.

[@ANURAGSINGH-pn2uu]

Hello madam. Today I was having my C.S practicals of class 12. Ulka Madam's videos of python helped me a lot kindly convey my thanks to her.

[@Silentgluttony]

what an awesome introduction into abstract algebra

[@TheMachian]

6:12 without using the table, the isomorphy is evident from e^(i k pi/2), k=0...3.

[@craigruchman7007]

Kind of mind blowing seeing this for the 1st time

[@JohnDoe-oo9ll]

I love that two times the set of all integers includes only all even integers

[@markell1172]

Those deep concepts compacted in one video cool

[@charlieng3347]

Your channel is just awesome

[@pedroendo4617]

Congratulations. You make the abstract Math more tactile. Maths always will be part of our mankind, our civilization, but I have firm belief that Math language of the future civilazation is not invented yet. Current Math simbols, specially as those invented in 20 Century, are designed for Mathematicians. ...People that are not mathematicians are more prone to understand abstract things visually by graphs, arrows, lines, ....in summary...the geometry seems to be the future! I cannot belief the future of Math be presented as complicated math simbols with lots of hypotesis and demonstrations without using any figures ou examples...But, you are changing the Math using the thing most powerful in the youtube which is:The visualization technics to teach abstract maths!...I think the abstract things should be explained visually....I think Euclids starts the "formal " Math as it should be by using Geometry for his desmostrations and hypotesis....

[@ayushdwivedi3519]

I'm fond of your way of teaching. It's wonderful and it has let me to love abstract algebra. Can you please make video on automorphism and inner automorphism.

[@evansjahja711]

Thank you for making me interested in the weird sides of math

[@fredericoapuleio9583]

Eu adorei o vídeo. Gostaria de ver mais. Parabéns pelo excelente trabalho. Eu não sabia que vcs prosseguiam fazendo vídeos. E o fato de ser de matemática é melhor ainda. Sugiro uma série em problem solving. Grande abraço!

[@sebastianparamera2424]

Very well explained.

[@jeremylaughery2555]

It's mathemagical! Great work!

[@othmanabdalamohamedadanabd1978]

I proud to watch your lecture🙏🏼🙏🏼❤❤

[@FelineBlender]

I love it when the No! flies in forcefully after she poses the question 😂

[@dialecticalmonist3405]

I don't see why the complex plain can't be represented by theta and -theta instead of 1 and -1. Just use theta and then + -theta rather than multiply. The addition would essentially be represented as an "inverse" within the complex plane. Seeing it as an inverse makes the correpsondence to the number line far more clear. The number line serving as an "extension" while the complex plane serving as the "inverse". This would also explain the square root of -1 and why in the complex plane, it isn't a number but rather an "inverse".

[@cristiandavid706]

Thank you for videos like this!

[@HelloWorlds__JTS]

I question the validity of the first example: Why can you say Z --> Zmod2 by first breaking Z into Z_even + Z_odd, but you don't use the same trick in the opposite direction to show Zmod2 --> Z? It seems to me this example is inconsistent, and it really should be a three-step mapping Z --> Z_even + Z_odd --> Zmod2. Then the reverse mapping Zmod2 --> Z_even + Z_odd --> Z also works. BTW, your abstract algebra videos are amazing! Why aren't you making more???

[@danilodossantoscamargo8949]

Liliana castro ensinando matemática incrível ❤❤❤

[@Jabber_Wock]

Great 👍 thank you for an excellent update. And a very happy Holiday Season to you, friends and family.

[@jaylearn2671]

Surely the integers mod 2 is 2Z and Z/2Z is the group of cosets that partition Z (ie. the sets of numbers with same remainders when divided by 2). This isn't a homomorphism or even an isomorphism, it is the same set described by different terminology. Evens means the congruence class that is the identity coset a Odds means the other congruence class.

[@o.s.h.4613]

If you got confused at all, try not to overthink of homomorphisms in terms of multiplication tables. Homomorphisms are not the result of two multiplication tables being identical; you can have a homomorphism from a commutative to a non-commutative group and vice versa, i.e. even if their tables are completely different (commutativity is shown by symmetry about the main diagonal). The real point is that it doesn’t matter whether you do the operation in G or in H, the result is the same. The “operation” is different, but the “denotation” is the same. However, when you have a homomorphism φ from an abelian group G (domain) to a non-abelian group H (codomain), only the IMAGE of φ in the codomain is abelian, only a subset of the codomain respects that structure; in other words, a homomorphism between an abelian and non-abelian group cannot be surjective. Likewise, a homomorphism ψ in the other direction from a NON-abelian group H to an abelian group G cannot be injective, for the same reason. For example, the determinant map from GLn(R) to R*; it’s a homomorphism from a non-abelian group to an abelian one, and is not at all injective; an infinite amount of matrices can get squished into a single real number via the determinant mapping-(verify that f(xy)=f(x)f(y) => Det(AB)=Det(A)Det(B)). If either or both of these are true, then these homomorphisms φ and ψ cannot be bijective-i.e. cannot be isomorphisms. Keep in mind that square matrix multiplication is not commutative, AB does not necessarily equal BA. While it’s true that the determinant outputs a scalar value, Det(A)Det(B) is not necessarily equal to Det(B)Det(A) for all cases, this only holds for square matrices (which won’t affect us here since A,B are in GLn(R))-this is because the determinant is a polynomial, and commutativity is important because it allows us to rearrange the terms in a polynomial multiplication without changing the result; The determinant of a non-square matrix is not well-defined, so we can’t extend the rule Det(AB)=Det(A)Det(B) for non-square matrices. However, since we are indeed operating in GLn(R), the image of the determinant map in the codomain (which is the entire codomain) is abelian.

[@ajinruben4184]

thank you so much.It helped me a lot. And saved my time.

[@merdenbryant8872]

Great presentation. Thank you!

[@sherifffruitfly]

The isomorphism-via-color-coded-multiplation-table discussion seems flawed: one might have listed the row/column headers in an "unfortunate" order, resulting in the color pattern NOT looking the same. Fix seems to be adding in the condition "if there exists a permutation of the headers such that the multiplication table is identically color-coded..."

[@dscmtr686]

Great videos! Maybe some graph theory?

[@TheNetkrot]

than you... thank you... thank you..... dear Socratia I am getting a bit further with your help.

[@selvapeter2370]

Fantastic presentation.

[@victorrodrigues1407]

Olá, tudo bem? Eu ainda não visitei toda a biblioteca do canal, mas já gostaria de fazer uma sugestão: um vídeo sobre idiomas e oratória. Que tal? Tenho certeza de que seria um tema muito bem vindo, pois é só olhar os top comments do socrática em qualquer uma das três línguas nas quais o conteúdo é disponibilizado para perceber que a maneira de falar da Liliane é muito admirada pelos inscritos, mais até, feliz ou infelizmente, que o conteúdo de alguns vídeos em si. Agradeço pelo ótimo trabalho que compartilham e peço desculpas pelo comentário em português.

[@mateus-santos-npm]

Olá, moça do Socratica. Vocês não irão mais postar vídeos no Socratica português? Por favor, não nos deixem, adoramos as suas aulas e você; esse seu jeito incrível de falar e com ótimos conteúdos.

[@ЙоаннаГичева]

Thank you very much, for the clear explanations

[@luiskretzschmar7557]

wow this is very nice!!! keep it up

[@imppie3754]

PLLLEEEAAASEEE MAKE A VIDEO SERIES ON LINEAR ALGEBRA PLLLLEEEAAASSSEE I BEG UUU

[@lucaolmastroni6270]

Thanks, very nice videos. Do you plan to create some videos to give an overview of 'Representation theory' ? Same question for 'group action on a set' ?

[@IamInloveWithEther]

These are very helpful!!

[@power-max]

6:10 color of `-i` in 2nd table should be purple :)

[@Socratica]

Thank you, good catch! We wish KZbin let us add an annotation like they used to. Will add to description. 💜🦉

[@philosophyversuslogic]

Wait a second, when you compare both groups in 1:25, then there are major difference between them: there are 'equality' (=) and 'equivalence' (≡) which are different things. So, I think there must have been put more logic into it.