Group Homomorphisms - Abstract Algebra

A group homomorphism is a function between two groups that identifies similarities between them. This essential tool in abstract algebra lets you find two groups which are identical (but may not appear to be), only similar, or completely different from one another.
Homomorphisms will be used through abstract algebra. You will study homomorphisms between groups, rings, fields, modules, vector spaces, and more.
Correction note: at 6:10 -i in the second table should be outlined in purple. Thank you to viewer Power Max for catching this!
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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)
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Teaching​ ​Assistant:​ ​​ ​Liliana​ ​de​ ​Castro
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[@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.

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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!

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We are in the same boat

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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.

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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.

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[@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

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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?

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

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I think subscribing can definitely STEM any tides of ignorance!

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

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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!

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

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The best Christmas present!

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

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Best Christmas gift

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

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[@Socratica]

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

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Wonderful

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

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

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Finished the entire playlist. It was wonderful.

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

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Those deep concepts compacted in one video cool

[@bradleymorgan8223]

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

[@facepalm4e]

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

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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.

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

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what an awesome introduction into abstract algebra

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Kind of mind blowing seeing this for the 1st time

[@TheMachian]

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

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

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

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One is not simply 'done' with abstract algebra.

[@MBAVD5]

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

[@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

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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!

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Thank you for videos like this!

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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

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Thanks!

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[@Socratica]

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Fantastic presentation.

[@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.

[@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.

[@ViniciusSilva-vc4jt]

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

[@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....

[@ЙоаннаГичева]

Thank you very much, for the clear explanations

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

[@ajinruben4184]

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

[@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.

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

[@FelineBlender]

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

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These are very helpful!!

[@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.

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Liliana castro ensinando matemática incrível ❤❤❤

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what excellent way you have

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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!

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

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Official Math Rockstar

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Great videos! Maybe some graph theory?

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Mindblowing👍

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I know this.....but I remember myself seeing her videos when I was reading this.....it helped a lot then...

[@nihalsuri4543]

Thank you so much!

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That was very good.

[@aliadams516]

114 ÷ 2 = 57 median(1..57) = 29 57 × 29 = 1653 16th prime = 53 There are 16 primes with prime digit sums up to 114. There are 53 composites with composite digit sums up to 114. 506 ÷ 2 = 253 median(1..253) = 127 253 × 127 = 32131 32nd prime = 131 There are 42 primes with non-prime digit sums up to 506. There are 131 composites with non-composite digit sums up to 506. 114th prime = 619 506th composite = 621 where 114 + 506 = 620 114th composite = 150 114th composite with composite digit sum = 220 114th composite with non-composite digit sum = 436 where 506 = 436 + 220 - 150 Full details are at heliwave.com/506.pdf 1 IS NOT PRIME: Division is supposed to divide a number into smaller parts but division by 1 leaves it undivided. 1 is indivisible. It is the Unit for building all numbers. [think PROTON] Prime number is divisible by itself only. [think ATOM] Composite number is divisible by itself and others. [think MOLECULE] Ali Adams God > infinity

[@ravishanker785]

great work

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I miss you

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This is so nice