Homomorphisms (Abstract Algebra)

[@Socratica]

Sign up to our email list to be notified when we release more Abstract Algebra content: snu.socratica.com/abstract-algebra

[@hatsuwr]

I rarely every subscribe to channels, but that ending earned it. Thanks for the great videos!

[@bmo_ri]

I"ll sub for that joke.

[@tekaaable]

same :D

[@NamTran-hb2te]

same here

[@gregoriousmaths266]

Same

[@raunitsingh676]

me too

[@n0nenone]

Seriously lol

[@MrMxy-ci5hx]

So far best abstract algebra tutor, in fact best tutor I've found on KZbin so far, thanks a lot

[@SyedAbbasAhmad]

The videos are so good that I have to come back to each video and check that I haven't forgot to like them. Really want to support you on patreon but am unable to. Only a Big Thaaaaank you for such quality content for free.

[@keepplayingnice]

This is beyond amazing. I read through notes and questions on these topics for 1 hour and couldn't connect the dots but this video and the related ones made it completely clear! I love how you simplify each thing you're explaining. I believe it's called the Feynman technique. Anyways, thanks loads!

[@alishacortes2398]

She makes abstract algebra easy to understand. I totally subscribed. Lol

[@Socratica]

Alisha Cortes Awesome! That's what we love to hear! Thanks so much for watching and subscribing!

[@notoriouswhitemoth]

+Socratica Don't apologize for your awesome math pun!

[@SzTz100]

This lady is an actress, not a trained mathematician. Kudos to her for teaching this subject.

[@devdeadcode11]

Your videos are really helpful to refresh the memory, by far the best that we have on Ab: Algebra in youtube. Would you mind doing videos on Isomorphism, cyclic groups, Galois theory, Sylow theory etc? They would really help lot of mathematics students. Again, really appreciate you work!

[@Socratica]

Thank you so much for your nice comment, devinda kariyawasam! We love to hear what other videos would be helpful! We are currently working on a video on isomorphisms - it will come out soon!

[@siddharthprakash8942]

You should try 3blue1brown's video series on linear algebra. They are also pretty good

[@MichaelGoldenberg]

The second example is a good one for seeing the difference between homomorphisms, which need not be injective, and isomorphisms & automorphisms, which by definition must be injective. I've been working my way through Benedict Gross' Harvard class on abstract algebra. The lectures are long and things can ramp up in abstraction and difficulty pretty quickly. Once in a while, I simply lose the thread completely. Having these far shorter and very well organized mini-lectures is proving really helpful thus far in clarifying and solidifying my understanding of key concepts. I don't imagine yours go nearly as far as the Harvard semester-long course does, but I'm happy for whatever I get from yours. Nicely done.

[@Socratica]

Thank you, Michael! We're starting off with a high level overview of Abstract Algebra, but in time we do hope to provide enough lessons to make an entire course. It will take some time to fill in all the gaps, though. :)

[@MichaelGoldenberg]

As I said, Ken R, he ramps up the abstraction and difficulty rather fast

[@RaminHonary]

This is a superb video. The concept of a homomorphism seems so simple when it is explained like this. If you haven't seen these videos before, just keep in mind that this one is one of a series, it is best to start from the beginning on the definition of an algebraic group.

[@vrendus522]

Like the ending, ha ha. One of the most precious math lecture endings I've ever seen.Thank you.

[@mohmd252]

ألف ألف ألف شكر لك، والله انك مبدعة ورائعة، الله يسعدك والله انت تنشرين العلم والمعرفة وتفيدينا نحن طلاب الجامعة وتسهلين علينا ، أنت إنسانة عظيمة جدااااااااااااا الف الف شكر الف الف الف شكر لك من المملكة العربية السعودية، إلهي يسعدك ويخليك ، ياناس هذه الإنسانة ماتوفيها كلمات الشكر وقليلة عليها ، بجد انا ممتن لك ، تقبلي تحياتي

[@HomerDinasty]

That pun earned my subscription, thank you ma'am.

[@GPraimraj]

I have a coursework exam tomorrow based on Isomorphisms, Groups, and Rings. Your videos are helping me finally understand what my lecturer tried to teach. Thank you!

[@Socratica]

XioraA7X Awesome! That's great to hear. Good luck in your class!

[@Fishy100]

Did you pass afterall?

[@GPraimraj]

@@Fishy100 I did! I can’t remember what my grade was though

[@Fishy100]

HAHAHA I was not expecting a reply, I have an exam in a week on these topics haha, this channel really assisted me. I am glad you passed,@@GPraimraj

[@fredhasopinions]

i've got overlapping lectures and have never seen an algebra lecture so far this semester. you have no idea how much your video is saving me from eternal doom right now. jesus christ thank you

[@Socratica]

Eeeeeesh that's really hard. Good luck!! Keep us posted on how it goes!! 💜🦉

[@fredhasopinions]

@@Socratica thank you!! I’m sure it’ll be okay, thanks to the existence of people like you ☺️ makes catching up on content surprisingly doable and interesting, haha.

[@jonyb0b13]

Haha that joke at the end! HILARIOUS

[@Eyenn_n]

Please do more math videos, you are amazing and one of the best math teachers on youtube!

[@streampunksheep]

Thanks for making this video. I'm learning A.A. on my own, and this really helps me understand the minor details that I sometimes miss.

[@Sisa095]

I'm learning this as CS major at my uni, yet I feel like as if I wal learning it on my own bc the professors aren't really helpful lol

[@finnthompson3089]

Ffs I have a 50% exam on abstract algebra in a week and this channel is the only thing keeping me alive

[@jonathansalvacion7368]

I love that final joke. One of the reasons I keep coming back to these videos. The jokes aside, these videos help me a lot. I hope one day I will be able to help back in some way.

[@Socratica]

Thank you for your kind message!! One way you can help is by sharing our videos! We've found that we don't always show up in search results, so if you know anyone studying math/science/programming send them our way! 💜🦉

[@jonathansalvacion7368]

@@Socratica I will do exactly that.

[@samhenri7350]

These videos helped me pass my undergrad Group Theory viva! You guys rule!

[@cupajoesir]

very dense, well collected, good pace. I had to google many things but got the idea. great job and thanks!

[@shambo9807]

You had 8 years to edit out that pun. 8 years. You weren't sorry at all.

[@chrismanson3211]

A very good, brief overview of homomorphisms. I appreciate you taking the time do this video for us learners of abstract algebra. I took this course in 1998 for my BS in math and got a C, then again in grad school 2006 and dropped it because of life circumstances. Now, I'm taking it online and I am understanding so much more. This class and regression analysis were the toughest classes I have ever taken. I am good with algebra and calculus but this higher level modern math is intense. My favorite classes were diffy q's, vector calculus, graph theory, and linear algebra.

[@Socratica]

Thank you, Chris! Abstract Algebra is a pretty unusual course. It's often the first math class where you leave the realm of real and complex numbers and start exploring completely new structures. On top of that the focus is mainly on proving things. For me, I took abstract algebra after I had learned elementary number theory. As I learned group theory, I got quite excited at how easily I could now prove many of the theorems in number theory.

[@liamdacre1818]

You make abstract algebra so simple and easy to understand. I’ve subscribed

[@naixiancarucci242]

Thanks!

[@Socratica]

Thank you so much for your kind support!! 💜🦉

[@mohammadwaseembangash3612]

Initially, it was difficult to grasp. But watching repeatedly, it becomes easier and easier. Thanks for valuable video lectures of group theory.

[@Socratica]

This is one of our favourite things about KZbin - we can slow a video down, pause it, rewind, or even return and watch it later (and it usually makes more sense the second time after we've gone away and thought about it for a while). We're so glad you're watching!! 💜🦉

[@sanjursan]

Excellent! Great to see another super algebra video with Ms. Castro, packed with a lot of good information in four minutes. Maybe next show what happens to subgroups under a Homomorphism. Or maybe a little intro to Lie Groups.

[@Socratica]

Thank you so much for sharing and commenting, sanjursan! We love it when our viewers let us know what other videos would be helpful - more videos coming soon!

[@aabhasvijz]

Well my algebra exam is one week from now. Counting on your videos, and totally subscribing. Thanks!

[@pianokatti9766]

I have to say that this is the first math-channel on KZbin I've come across that is actually worth watching. The others that I've seen cover mathematics as calculus and over simplify concepts to the extent that they're not correct. The structure of mathematics is never revealed - the words definition and proof are either not used or used incorrectly. For example, Numberphile's "proof" that 1+2+3+4... = -1/12 is painful to watch. A physicist reasoning that the result must be true since there are applications for it in some theory in physics... Anyway, I immensly value the hard work you've put into these videos and the fact that you actually know what you're talking about!

[@nadimuddin362]

I just liked the video because the way it ended.....it worth our subscription

[@SolomonLug]

To show the group {z \in C : |z|=1} under multiplication is a group, is it as follows...: Identity elt : \theta=0 since e^(0)=1 and e^(i\theta) * 1 = e^(i\theta) for all members of the set. Inverse elt exists since e^(i\theta)*e^(-i\theta)=e^0=1 for all -2pi

[@fernandascovino7805]

I'm having this subjects in my graduation course now, loved the video! Simple and clear explanation! (and u got me on the end, subscribing right now)

[@vincentorute564]

I LIKE SO MUCH REVIEWING LECTURES IN CLASS WITH THOSE KZbin ESPECIALLY BY SOCRATES

[@vitalchance5768]

The best video plus great English and artistic voice and pleasant looks. Its like the best explanation from the princess of math. Thanks

[@lookthesky9632]

Hope this will save my exam lol! It looks very simple to understand, and yet abstract algebra is one of the most difficult math course in my major. Indeed, this course is like if I choose wrong assumption for "G", I will get roast for understanding the material. Thank you lecturer!

[@guitarttimman]

I earned an A in linear algebra, but that's been a long time ago. Your videos are awesome, and this brings back so many memories. Do you have one on eigenvectors and eigenvalues?

[@Vannishn]

Around 1:35 isn't it R+* under multiplication that is abelian ? Coz 0 has no multiplicative inverse to be in that group, right ?

[@Socratica]

Socratica Friends, are you trying to improve as a student? We wrote a book for you! How to Be a Great Student ebook: amzn.to/2Lh3XSP Paperback: amzn.to/3t5jeH3 or read for free when you sign up for Kindle Unlimited: amzn.to/3atr8TJ

[@rdxskgaming8785]

Explained really well!! Hats-off

[@deepaksinghrawat7571]

i kinda like your videos bc they are short and full of information but here a little suggestion being a student is that "include a bit more content in videos. so that it could be more useful for us." Thanks :)

[@HDQuote]

I love your videos so much! The quality is top notch and your are very sympathetic

[@brandonallen2301]

In that final example mapping Rationals,+ to Complex Roots,*, could you also introduce an intermediate homomorphism and say, Rationals,+ -> Rationals%(2pi),+ | f(x) = x%2pi Rationals%(2pi),+ -> Complex Roots,* | f(x) = e^(ix) To obtain the same homomorphism?

[@OpPhilo03]

Thank you mam . Your English language is very amazing and Voice is very sweet. I don't know much English but I can understand you. 😊

[@nukalasaikirankashyap3376]

one good lecture after many days

[@arzoil96]

omg you saved me :D this seemed so hard when i was reading the book and i watched this video and i cant beleave this is so easy :D

[@muazuyakububalarabe5309]

the videos is so help in this area of mathematics

[@opeyemioyewumi5960]

@Socratica, Please what's the difference between homeomorphism and homomorphism in the abstract algebra. You can as well use the idea of groups to explain the difference please!

[@opeyemioyewumi5960]

@@hyperduality2838 you have confused me more

[@mramirkhan5982]

please tell us how to count number of homomorphism from any group to any group

[@usman5954]

Best videos on Group Theory by Socratica!

[@second_second_]

how come i didnt realize this video exists for such a long time.. i was stuck on this topic

[@Socratica]

We're so glad you've found us!! 💜🦉

[@ameliech5134]

Amazing video! So clear and concise.

[@demidevil666]

This is the best explanation for homomorphisms I have heard, by far! Hands down. Thank you so much.

[@lokeshvarkumar2705]

Your way to explain is really good Mam.

[@zeeshanahmed4866]

oh my god 😂 That ending was just LOVE ❤ I had Already Subscribed but I wish there would be another button to subscribe so I would have Subscribed Million times just because of that ending💞

[@millicentayippey8009]

Can't see some of the writing because of the subtitles, but it was very helpful.

[@harshlodha2056]

In example 2 : H = { Z ε C, abs(z) =1} How can it be group , each element in H has more than one inverse θ2 = θ1 +2κπ, whereas inverse should always be unique???

[@chanyui8838]

I love this series. Hope making more videos about maths

[@arijitmishra2435]

Can you please come up with a video on cyclic groups?

[@xthene]

So 2 groups are structural similar if there exits a homomorphism f between these 2 groups.

[@minhajfahim8306]

Can you make a video on the fundamental theorems of isomorphism

[@crypticcrazy3672]

There's something amazing about Liliana​ ​de​ ​Castro. I thought for a moment she was AI-generated but I see she's real with an off the charts IQ combined with beauty and poise. Scary. Looks like I'll be learning more group theory.

[@othmaniarij7983]

i study basically every subject including algebra in french but this came in handy more than why my teacher said

[@eldeee143]

My apologies for my lame question but why are you trying to use e^x as a way of mapping G to H?

[@gregoriousmaths266]

2:17 do you mean magnitude?

[@rob6129]

Your videos are really helpful, great as an addition to my class ressources

[@lelelanaa2500]

Thank God for this lady! I finally understand and I have a test...thanks to her I'll get a better score

[@pawelsogra623]

please tell us about the homomorphisms of cyclic groups

[@janglaser6130]

the sub joke is so original that i actually subbed

[@Hythloday71]

What sort of structure does it preserve ? Why are we interested / is there a physical motivation ? I'm wondering / thinking: is loss of structure = loss of information ? Information in the purest sense of distinctions ? Any other thoughts or ideas ?

[@sss-ol3dl]

Look up Cayley tables, it gives a nice visualisation, but I struggle to think of a way to explain it other than what the formula already tells you. Its a way to link 2 seperate binary operations. The motivation for it is that if 2 groups are homomorphic they usually share a lot of properties, if they're isomorphic they both share a LOT of properties, like being Albian or infinite.

[@kristinatodorovic6573]

There is something in the way she talks about algebra that makes me want to know more about it..

[@jtjuan7]

I am currently taking abstract, and these videos are very helpful, thank you!

[@Socratica]

We are so glad to hear you are finding our videos helpful, Juan Tafolla! We'd love to hear what other videos you would like to see.

[@abhishekkumar-yd3os]

love ur way to express every thing and beautiful way of asking to subscribe

[@MichaelGoldenberg]

The choice of e as the base in the first example is arbitrary, isn't it? That homomorphism exists using the exponential function for any positive base, right?

[@KaelynWillingham]

Yes. Because for any real number c, c^(a+b) = c^a × c^b. I'm guessing they chose e because that can also show that the same homomorphism extends to the complex numbers.

[@cameronspalding9792]

One thing that bothers me is that in topology the word homomorphism is used for continuous bijections between two spaces but in group theory it doesn’t have to be bijective or surjective

[@MuffinsAPlenty]

No, no. The word in topology is not "homomorphism". It is "homeomorphism". That little letter "e" makes a big difference. Also, be careful! A homeomorphism is not merely a continuous bijection between two spaces. It is a continuous function _with continuous inverse._ A continuous bijection will have an inverse, but that inverse need not be continuous. Therefore, a continuous bijection need not be a homeomorphism.

[@hengsokgich8353]

You help me with my today homeworks, professor. Thank you so much

[@jbjbjbjbjbjbjbjbjbjbjbjbjbjb]

Love the ending. I needed the laugh.

[@OmarAhmed-ic4fw]

But what is the motivation for such a concept "Homomorphism"?

[@santhoshwagle9857]

those groups are homomorphic for that particular mapping function??? if mapping function is different then those groups are not homomorphic???

[@rob6129]

Yes it all depends on the function f()

[@brandomiranda6703]

what about strong homomorphism (vs isomorphism)?

[@natekidwell9099]

Amazing series. Quick question here regarding example 2 If |z| = 1 and can be written as e^i*theta for SOME theta, I'm not sure how we can deduce here that ALL theta make |z| equal to1. But isn't that necessary for f to ensure that its range is within H. Because otherwise how can we be sure that e^ix (to which f maps x) has an absolute value of 1?

[@MuffinsAPlenty]

It's an if and only if statement. If z is a complex number, |z| = 1 if and only if z = e^(iθ) for some real number θ. If you're concerned about going backwards, here's an argument. We need two facts. Euler's formula: e^(iθ) = cos(θ) + i·sin(θ) and |a+bi| = sqrt(a^2 + b^2). Putting these two facts, together: |e^(iθ)| = |cos(θ) + i·sin(θ)| = sqrt(cos^2(θ) + sin^2(θ)) = sqrt(1) = 1, regardless of the value of θ.

[@goodjunksam8149]

Please, how do I find page 1. where I jumped in has been sequential, which is nice. Blessings for bringing the squiggles of the greats of math to an understanding. Huge concepts in single letters, who'd a thought.

[@alvinlepik5265]

that iscribe thing made me smile :D a very simple explanation and easily understandable examples. you got yourself a scriber :)

[@thrineshs3829]

Mam your teachings helped me a lot ,thanks a lot.........

[@annafeting9741]

Can anybody please explain why z=e^itheta if |z|=1 and z belongs to complex numbers??

[@kamikamen_official]

If I wasn't already subscribed, I would have subscribed for that "i_{scribe}" joke

[@sreejaps2428]

Thanks mam.ur videos are realy helpful.pls upload a video on permutation groups.it is my humble request

[@jenniferimogie1676]

It's always the last segment for me I subscribed 😂

[@tokajileo5928]

actually the square root is defined that it is a valid function only for nonnegative numbers so square root of -1 is not mathematically rigorous or correct notaion because of the definition of the sqrt fucntion. i is defined in a way that i^2 = -1

[@MuffinsAPlenty]

One could very easily define sqrt(z) for any z in the complex numbers. This is often done by defining a principal argument of a complex number, and using that principal argument to define a principal square root. Typically, the principal argument is defined as being in the inveral (−π,π], since this interval produces the nicest properties. (For example, "conj(z)^conj(w) = conj(z^w) when z is not a negative real number" is a true statement precisely when we take the argument to be the principal argument). I know many analysts say that there are other problems we care about where we do not wish to define a principal square root, but there's nothing wrong with it.

[@yakopro49]

Was ex2 isomorphic caused she didn't state it?

[@PrasannjeetSingh]

the joke made me subscribe!!!

[@Jaya-yt4nx]

Your videos r so helpful👍👍 would u mind doing videos on real analysis, metric spaces?

[@giuliogemino6407]

I would like to know the formulas about how to calculate geodesic domes components... struts leght, how many hubs, frequency...

[@moularaoul643]

Thank you so much!!

[@AzatDzhanybekov]

Very good examples to consider. Best wishes!

[@suppandi1000]

Loved it, thanks! A big help for 6th form kids interested in maths trying to do some self study

[@chantalgray8627]

The joke at the end got me. Smile

[@nadaadel878]

Your videos are very helpful 👍👍

[@PeterPhan]

I subscribed for that closing joke.