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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[@Socratica]

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

[@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.

[@ProfessorDBehrman]

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

[@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.

[@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.

[@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!!

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I think subscribing can definitely STEM any tides of ignorance!

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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.

[@kaebynj.quintero2993]

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

[@MrGiuse72]

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

[@anujarora0]

Long time no see

[@Socratica]

Hello hello! Thanks for watching! 💜🦉

[@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.

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

[@distrologic2925]

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

[@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.

[@ViniciusSilva-vc4jt]

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

[@Djn77645]

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

[@Or0203]

Found my new favorite math channel. Thanks a lot

[@facepalm4e]

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

[@aaronnorman9755]

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

[@Icenri]

Groups! Again! YES!!!!!

[@Icenri]

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

[@Socratica]

@@Icenri WOW first viewer!! 💜🦉

[@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

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

[@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.

[@acquireknowledge8327]

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

[@Socratica]

Hello to our Socratica Friends in Pakistan! 💜🦉

[@markell1172]

Those deep concepts compacted in one video cool

[@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.

[@charlieng3347]

Your channel is just awesome

[@JohnDoe-oo9ll]

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

[@TheNetkrot]

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

[@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.

[@TheMachian]

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

[@boradmay]

I miss you

[@bazislapper6283]

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

[@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

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[@MateusSantos-zx2cz]

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.

[@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.

[@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.

[@cristiandavid706]

Thank you for videos like this!

[@ЙоаннаГичева]

Thank you very much, for the clear explanations

[@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???

[@luiskretzschmar7557]

wow this is very nice!!! keep it up

[@selvapeter2370]

Fantastic presentation.

[@isaacstamper7798]

This is so nice

[@GuadalupeAnimation]

how does she make the slides?

[@brianmiller1077]

5:07 At first I was confused but I see you switched them both on the columns and the rows.

[@suspendedsuplexchannel1000]

Please explain |x| = x when x>0 = -x when x

[@kelloginc1]

Yuuuuuuuuus

[@randomvideos3628]

It would be a little more helpful if you mention the corresponding group operation a bit earlier. Thanks for the nice video!

[@benterrell9139]

minor colour error at 6:10. -i is blue not purple. Excellent video though.

[@manodiprouth445]

Can you make a video on how to read a paper properly (effectively) ?

[@manodiprouth445]

*Scientific paper !

[@isabelamachado530]

Volta com o canal em português 💛

[@tee21.21]

Sooo good 🎉

[@nihalsuri4543]

Thank you so much!

[@naman.sharma1]

Please make more videos on algebra .

[@Lukas-br3zo]

Could someone explain to me in which sense the definition of a group homomorphism is connected to it 'preserving the group structure'?

[@yongjincho3965]

Thank you for the clear lecture. I have been studying abstract for some years for myself by myself, but what is the application? How to apply?

[@dialecticalmonist3405]

You have to find patterns in nature that are similar. Once you find those patterns, you can predict nature.

[@sarojpadhy5535]

So nice

[@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

[@c27Blay]

Lifesaver!

[@issacpetit8263]

thanks

[@thepianist7379]

Thanks!

[@Socratica]

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

[@tabasdezh]

This's great for matlab.... thanks.

[@ianrickey208]

Are more Python videos being produced and when will we be able to enjoy them?

[@prithak3263]

brilliant

[@drlangattx3dotnet]

thank you very much.

[@CreationTribe]

Magic! Funny, I was just talking to somebody about 'Donald in Mathemagic Land' earlier today. How can you not love math? It pretty much is magic.

[@somedude5782]

ok but how does this help me compute the determinant of a nXn matrix?

[@moab2192]

Make a Video about Infimum and Supremum please please please please please

[@Ad-wu8te]

Hi thank you very much! I just have one question. In some cases addition is used to check and in other cases multiplication is used to check for homomorphisms. How do you decide which operation to use? In the example you used addition but if you used multiplication it won’t show to be a homomorphism

[@MuffinsAPlenty]

You use whatever operation the group has. Remember: a group is a set _together with_ a binary operation (satisfying the properties that we state in the definition of a group). In Z/4Z, there are two natural operations you can talk about: addition and multiplication. But Z/4Z is _not_ a group under multiplication (you can show it doesn't satisfy the axioms), so when we talk about Z/4Z as a group, the operation we are referring to is definitely addition. Similarly, for complex numbers, you can talk about addition and multiplication, but the set {1, i, -1, -i} is _not_ a group under addition. But it is a group under multiplication. So when talking about {1, i, -1, -i} as a group, the operation we are referring to is definitely multiplication.

[@himasishghosal7818]

And with respect to multiplication Z is not forms a group also

[@steven1andsteven229]

Have a Q maybe irrelevant, how do students do their homework these days? I remember it was quite some deep thinking and joy when I solved a homework problem: for non-trivial p group G, prove G' is proper. Decades later today I just found out the answer is a few clicks...

[@gunwookim4047]

nice video

[@realzhella6817]

What's a mod?

[@nataliagomez8275]

Great tutorials!! Oh Yeah

[@duality4y]

1:20 looks like a truth table

[@euler73]

The Cayley table isn't quite a truth table - it is more like a multiplication table ("times table") that shows the output of the operation given any two inputs.

[@duality4y]

@@euler73 I just want to say that a Logic operation given two inputs also produces a output. I also said it looks like a truth table, and found it a curious observation :)

[@b43xoit]

So think about the truth table for a specific logical operation, out of the sixteen possible: XOR. {true, false} under XOR is a group.

[@brianmiller1077]

Reminded me of Karnaugh maps from digital classes.

[@unotwotriquatre]

cayley tables and truth tables are precisely the same thing when there are exactly two objects. other than that, the difference becomes clear

[@lylechen8881]

I like this 小姐姐 (Xiao Jiejie)'s class. ^_^

[@G.Faustini]

Os videos do canal br é tão antigo que o cabelo dela ainda tá curto.

[@bandosantos]

💕

[@lathavedachalam89]

Thank u mam💖❤

[@khoavo5758]

What is the link between group homomorphism and comparsion? Isomorphism I can understand. But what does homomorphism mean?

[@metakatana]

An isomorphism is just a bijective homomorphism. A homomorphism preserves the structure of the object.

[@khoavo5758]

@@metakatana I understand what isomorphism is, but what is homomorphism? Why does it "preserve the structure of the object"?

[@metakatana]

@@khoavo5758 For a group homomorphism h:(G,*)->(H,#), the preserved structure are the operations *, # of G and H, respectively. Under h, applying * to any two elements g1,g2 of G yields the application of # to the images h(g1),h(g2) of H. That is, under h, g1 * g2 maps to h(g1) # h(g2). That is what is meant by preserving the structure. If you're asking why the definition is what it is, one reason might be that homomorphisms help us compare groups, vector spaces, etc and identify if they're alike in how they preserve certain structures but not necessarily isomorphic.

[@unotwotriquatre]

VDK think of a function that maps the integer numbers to people. you know that 1+2=3 in the integers. in order to that function be called an homomorphism, there must be, in particular, a binary operation on the people such that, when you operate the person assigned the number 1 with the one assigned 2, the result is the person assigned the number 3.

[@khoavo5758]

@@unotwotriquatre And why would you want that? Why is it called "homomorphism"?

[@marcc1179]

KZbin University is better than mine.

[@ابنرشد-ش6ض]

Why and why z and z/2z are not isomorphic

[@MuffinsAPlenty]

For starters, in order to be isomorphic, they have to have the same cardinality (number of elements). This follows from the fact that an isomorphism is a bijection. Essentially, we want to relabel the groups into each other. It's not possible to do this if they have a different number of elements. Z has infinitely many elements and Z/2Z has 2 elements. So they can't be isomorphic. There are other justifications you can give, too.

[@MatheusDeLimaDilima]

Keep it up the good work, and try to tune the Socratica in portuguese too. Videos about Abstract Algebra are rarely good like these. Cheers!

[@Delgen1951]

What about A+B=c-d ??!! or d+c=A+b-1 ??!! how does this help me to find anything? now if i define a as 1 and b as 3 and d as -1 and c as a+b-d then i get 3 as c and am graded as F what have i learned ? Math sucks, by the by I am dyslexic and it all reads the same to me both ways. this is what "new math" of 1963 was about and was tried to be tough to seventh graders, Hated Math ever scents and I am now 68.

[@brucewayne-mh7ul]

Smart lady

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saudades dos videos em português-brasil

[@davis7652]

Spain

[@EduardoFox]

Traz de volta o Socrática PORTUGUÊS :(

[@joaotv7387]

Volta com o socrática br

[@DavidVasquez91]

O canal agora é em inglês ?! E aquele esforço por ensinar em nossa língua ?!

[@TheTessatje123]

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

[@kyle-silver]

The best Christmas present!

[@Socratica]

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

[@MadanKumar-xg6xy]

Wonderful

[@edt8597]

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

[@Socratica]

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

[@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.

[@DiegoMathemagician]

Best Christmas gift

[@LuizFernando-ux1cz]

Não vou recriminar pois canal em inglês é bem mais rentável no KZbin,mas que faz falta o conteúdo em português faz...