Isomorphisms (Abstract Algebra) 

On our website, we have an in-depth example of an isomorphism as a "Bonus Feature": www.socratica.com/subject/abstract-algebra

the quality of this video is incredible, the audio, the visuals, the pacing, the material, and the delivery

The best intuitive description of an "ismorphism" is to think in "analogies". Yep, an analogy itself is a good analogy for an isomorphism, you take some relationship and you change the context while maintaining that relationship in order to elucidate some property of the relationship. Of course, this is a very informal way to describe this. But it's a good intuitive insight.

Wow! I understand isomorphism now. This is the best explanation. Thank you :)

Our latest abstract algebra video is on *isomorphisms*! These are functions which tell you when two groups are identical. This is key, because the same group can appear in different places in wildly different guises. (You can also have isomorphisms between rings, fields, modules, etc. We'll cover those in separate videos.) #LearnMore

Thank you for this playlist... my friends and I are studying Abstract Algebra this summer before the class in the fall.

This is gold, I can't believe this series is free

I have no idea what Socratica is. I just stumble upon this wonderful video and I just want to say: thank you! This video is awesome! So well explained!

A lot of things clicked into place for me after watching this video. Thank you for so concisely expressing these concepts!

I’m too lazy to sit down and read a textbook sometimes. This engaging format also lends more memorability. I appreciate your demeanor! I’ve been looking for good abstract algebra resources for a while, and I think I’ve found what I needed.

I would like to really thank you for these videos. I am impressed by how well each concept is explained.

Just superb! Thank you so much.

Best explanation for isomorphism I ever heard. Thank you so much!

Wow, effective way to understanding. I appreciate you.

I'm really excited about this concept! Isomorphisms must be such a powerful tool to translate one type of group that can't be manipulated easily into a simpler one.

The beauty of mathematics is in simplicity of seemingly complex ideas .... thank you a lot !!! for unveiling this treasure💝💝💫

in my opinion , this is the best channel for everything mathematical .. Love you :)

Oh, the clarity!

These videos are just superb, thank you Socratica

thank you very much......helps to survive my semester...!!!

Best teaching style

I've been needing this exact video for a long time. Thank you!

Hi please do a video on cyclic groups... thanks

I believe that the correct description is that f NEED not be 1 to 1 (or onto). It CAN be, but doesn't HAVE TO be.

Please upload a video about Cayley &isomorphism theorem

I can understand better here than my professor lecture 🙂

Excellent! This helps me to understand isomorphism for the first time after school lecture! Thank you so much!

your channel and the presenter of these video series which is called "Abstract Algebra" are magnificent. I'm glad that I have you, guys. Also, I hope you'll continue your videos. Take care.....

Probably the best explanation of isomorphism in humankind. I think in less then 10 years youtube will replace all those sh*tty books we use in our classes.

you just save me from dying in my math class

wow! i understand isomorphism now.This is the best explanation

Thank you so much!!!

You should have used the definition of isomorphism as a morphism with a left and right inverse. Then give the intuition that a homomorphism maps group structure to an object and the inverse maps back from it, the existence of the two sided inverse would then necessitate the structure can be moved freely back and forth between the objects. This definition is not only equivalent in the case of groups, but it generalizes and unifies most mathematical objects. For example, you could draw the analogies with a familiar analogue: isomorphism of sets (ie: bijection), a visual/geometric analogue isomorphism of topologies (ie: homeomorphism) and then conclude by saying this concept (formed in this way) is the notion used in all of modern mathematics (ie: make a reference to category theory where the idea belongs). Personal Comment: - The set based definition you gave is a dated point of view which conceals elegant and intuitively simple mechanism by which the isomorphism preserves the structure of the group and is weighed down by set theoretic conceptual obstructions.

To be more precise: Isomorphisms are maps that preserve structures between objects (groups for instance) f s.t. you can find a different map g s.t. fg=Id, gf=Id. Since homomorphisms preserve structures between objects in groups, these are the type of maps we analyse to find isomorphisms. The only type of homomorphism with the property we look for are bijective homomorphism. This is the reason bijective homomorphisms are isometries in the category of groups. But an isomorphism is something more abstract. You might say that an isomorphism between two objects means that they have the same structure within the discussed category of objects. Isomorphic groups A,B for instance are essentially the same when discussing group theory, and this is why we really couldn't care less within group theory which of the two objects we discuss. However, if we look at our two groups A,B though the lens of a different theory, which cares for other properties they might hold, then they might not be isomorphic in that frame of discussion

Very clearly... CONGRATS

Very good and quality video ..thank you mam

Thanks mam

Lol, I totally loved your pun at last line, i thought it was another question but, isubscribed too😂😂🤸!!! Amazing background music, nice nice!!!

why the hell am I watching a random algebraic theory lesson at half past 1 on a Saturday night >.

I like your video! I really enjoyed watching it.

awesome channel, totally subscribed!

Absolutely excellent instruction!

This was quite helpful...

Home girl is so funny. I love the way she talks. I feel like I'm watching a crime show with the eerie music in the backround. lol.

Wish you did videos on cyclic groups and quotient groups!

good presentation

Awesome explanation

Plz give more lectures on group theory

Superb explanation mam ,thank you

Thank you so much.

you are savior of students who suffer from bad teachers

Well done

Can you please upload the lecture about caley's theorem?

Smart teaching thanks

Me to the Iconfuseda. This is one of the times when I actually would like links. Links to the videos that need to be understood BEFORE this.

WHERE WERE YOU BACK THEN ?!!!???

really nice even thought i didnt understand a thing. but i would like to say keep going your amazing.

thank you madam...........

At 4:15, it is stated that Cx is not isomorphic to S1. However, in the chapter on isomorphisms in Gallian, in the section on Cayley's theorem (last paragraph) it says "... the group of nonzero complex numbers under multiplication is isomorphic to the group of complex numbers with absolute value of 1 under multiplication." And there is a reference to a paper with a complicated proof I couldn't understand 😅 So, I'm confused, is Gallian talking about something diffferent or is Cx isomorphic to S1? The paper referred in Gallian is: "The punctured plane is isomorphic to the unit circle" by James R Clay

At last I'm subscribing

Honestly I was looking this up cuz I saw a keyboard that was isomorphic and idk what it meant. Now I know so much idek what to do with this info

thank you ❤️

excellent video !! thanks

prove that g= a+b√2 a b€a and b are not both zero is a subgroup of r under the group operation. Can you please answer these.

Done!

Tnku.... Nd plz say about cyclic group........

Thanks, good stuff, keep it up

wowwww you explain sooo good maam

You are a GREAT Algebraist. I love math more each time I watch your video. Can you be my personal teacher? I want to specialize in Abstract Algebra.

wow!! thank you so much.

Thank you soo much!!

I don't understand why at 1:44 you show that f(x*y) = f(x) + f(y). I thought the condition for homomorphism was that f(x)*f(x) = f(x+y) ?

thanks tom

So in what way can you prove that it is an isomorphism given the imaginary entry to be 0

@socratica am I the only one confused here, the range is not all real numbers, log(x) is no defined at x=0

Its not an isomorphism because it was not one one, as the graph of f'(x) >0 and f'(x)

Mam it was amazing class.. but can you help me how to find out one such function exist between two functions? Thanks in advance ❤️

good job

*jerk jerk jerk* hooo, hufff, hooo, haff.... Phew! Man, this video was so helpful to me... much more so than my dry, stale textbook really was. I feel like I have really... "internalized" these concepts now.

I think the illustration at @0:54 for surjection is reversed, the function should map to all H and not _necessirely_ from all G.

Couldn’t you map all points on the unit circle to a unique point on the real line using stereographic projection? If anything it’s perfect because you’re losing 0 in the domain and you have to lose either 0 or infinity in the range

How's example 1 an isomorphism when G is defined in R+, while H is R? Isn't R+ half the size, how can it be an onto? Or is this another one of those classic math things where if two groups are infinite, we're gonna look at them like they're the same size (even though ones obviously bigger) ?

4:50 “Isomorphism” is actually not a great name, because it can be misleading. “Equal shape” sounds practical for ‘Top’ or ‘Ten’. Isomorphisms are invertible, which is what makes them more interesting than homomorphisms. The name doesn’t imply invertibility. But, it’s not a bad name either; especially because it’s so widely used.

2:11 You got confused here between range and image. The image of a function is the set of all outputs of the function, but a range is any set that contains the image. So even if a function has a range of all real numbers, it doesn't mean that function will be onto in this example. For example sin(x) has a range of all real numbers, even though it's image (if the domain is the real numbers) is [-1,1].

how ring monomorphism and epimorphism can be characterized by using kernel and image

what is meant at 1:30 when she says all real number under addition? and all positive real numbers under multiplication?

How prove this? Let S^1={z ϵ complex numbers: |z|=1}, and let H be the additive group of real numbers. Use the first isomorphy theorem to show that H/ is isomorphic to S^1. Please help :(

how can u call something which is not a bijection as a function(i.e. in case of homomorphism)???

nice voice

Sometimes we say that two groups are isomorphic and we dont specify the function , is that corect ??

You are a smart women. And I can't help but love a smart wowen like you. Hail maths!

At 2:00 we denote by definition to the logarithm base 10 by "log" and logarithm bas e by "ln" so to get x we must rise 10 to (log(x)) and not e .... is this true ?

Doesn't 1:54 only show that the logarithmic function itself is "1-1" instead of the mapping of the domain G on the codomain H?

Nice

So an isomorphism of a set is basically a relabeling of the set.

Do homomorphisms have to be surjective

do you have vedio on application of field:?

THIS LADY INTENSE AS FUCK....SHET

0:56

Nothing about Automorphisms?