Isomorphisms (Abstract Algebra)

[@Socratica]

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

[@huttarl]

I went there and found the PDF you're talking about under "Isomorphisms for Groups." But when I clicked on the "BUY" button, nothing happened.

[@azuboof]

@@huttarl xd cmon

[@woahitsben]

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

[@Kaje_]

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.

[@sr-kt9ml]

Reading GEB right now, this helps

[@alejrandom6592]

You might be stepping intl category theory

[@Socratica]

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

[@alishacortes2398]

Socratica Will you be adding a video on automorphisms?

[@SilverArro]

+Alisha Cortes Automorphisms are just special cases of isomorphisms where the function maps a group to itself.

[@martijn130370]

Fantastic videos esp because of the clear concrete examples!

[@shafiullah627]

@@SilverArro Plz explain why we can do this mapping in group itself ?

[@---gi9kf]

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

[@PunmasterSTP]

Iso understand it now as well!

[@lynettemojica6503]

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

[@Socratica]

That's fantastic! We're so glad we're part of your independent summer school! 💜🦉

[@toasteduranium]

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.

[@alejrandom6592]

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

[@mountain3301]

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

[@Socratica]

That's so amazing to hear. Thank you for letting us know our videos are helping! 💜🦉

[@rubempacelli6815]

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!

[@kemaltezerdilsiz4126]

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

[@jasonbourne9798]

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

[@MuffinsAPlenty]

There's a very subtle detail here to be careful about! In the video, it was stated that f is not an isomorphism. This _does not_ mean that C^x and S^1 are not isomorphic. It just means that this _particular function_ is not an isomorphism. Other functions could be isomorphisms between C^x and S^1. The isomorphism between C^x and S^1 is much more complicated than the function shown in the video.

[@jasonbourne9798]

​@@MuffinsAPlentyAh yes! Feels so obvious now that it's been pointed out, but couldn't sort it out myself. Thanks for replying!

[@chasr1843]

I can't believe how simply she puts in in such a simple explanation

[@MattRichards711]

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.

[@Rishabh_Joshi_]

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

[@rapturian8228]

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.....

[@anikanowshin459]

Even though i am not good at abstract algebra but i have to say as ur explanation are just so simple and fun, it makes me love math again ( currently i hate it ) Thank you

[@raymangoel9327]

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

[@AM-rb4ps]

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

[@陈十七-z9u]

you just save me from dying in my math class

[@sanjursan]

Just superb! Thank you so much.

[@Socratica]

Thank you for watching, sanjursan!

[@sirelegant2002]

These videos are just superb, thank you Socratica

[@1995amittai1]

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

[@jadekan72]

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

[@lugia8888]

Black people lol

[@abrahamsweetvoice7687]

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.

[@riturajsingh6938]

Wow, effective way to understanding. I appreciate you.

[@AnastasisKr]

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.

[@bcthoburn]

Anastasis K So true, even though I’m still just learning about this

[@mownistark5770]

I can understand better here than my professor lecture 🙂

[@navjotsingh2251]

Because here she is teaching us and building our intuition, something professors seem to fail in doing

[@mehmetedex]

you are savior of students who suffer from bad teachers

[@narendrakhadka9598]

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

[@tomau3946]

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.

[@tanjinaaktar1146]

Best teaching style

[@Shaan_Suri]

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) ?

[@nicodemusmd]

Oh, the clarity!

[@arghyagemini]

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

[@Socratica]

Arghya Haldar We are so glad you are finding our videos helpful! Thanks so much for watching.

[@avinaysingh3904]

Can I ask you a serious question, what's the purpose of this math? How to apply it?

[@nephildevil]

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

[@xXx-un3ie]

bruh....same here wtf what are the odds?

[@LastvanLichtenGlorie]

It just means you have good tastes.

[@timmy18135]

To see Aleph null

[@DragonKidPlaysMC]

It’s 2 am currently lol idk too HAHAHA

[@anilmethipara]

@@xXx-un3ie What this is a such a coincidence lollll

[@tahaanouar2453]

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 ?

[@HassanJMandour]

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

[@MuffinsAPlenty]

No, the diagram represents exactly what they want it to represent. A homomorphism does _not_ need to be a surjection, so it doesn't have to map onto all of H. That's why they show it only mapping to part of H. For the record, by the definition of a function, since the domain is G, _all_ of G has to be mapped somewhere.

[@HassanJMandour]

@@MuffinsAPlenty Yup, thank you, for some reason, I thought they were trying to illustrate surjectivity (to say it's not the case that), but your point makes more sense. And for the domain part, I was just being dump for some reason :"D

[@AbhishekBhal]

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

[@zracklfr1334]

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

[@MuffinsAPlenty]

A group is a set together with a binary operation. You need both elements and an operation. The "real numbers under addition" means that the set of elements you have consists of _all_ real numbers (positive, 0, negative), where the operation is addition. The "positive real numbers under multiplication" means that the set of elements you have consists only of positive real numbers (no negative, no 0, but everything positive is there), where the operation is multiplication.

[@khansaheb7991]

Please upload a video about Cayley &isomorphism theorem

[@malikahsan4535]

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

[@tylerbakeman]

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.

[@MuffinsAPlenty]

How is it misleading?

[@rajdeepsarkar5721]

Very good and quality video ..thank you mam

[@simplesalmon1604]

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?

[@andinomie8988]

That is precisely what I thought. I believe it ought to be R+ under + as the second point.

[@SphereofTime]

0:56

[@SphereofTime]

2:00 check log is 101 and onto,

[@SphereofTime]

3:00 Every point is z=r&*e**itheta

[@ninosawbrzostowiecki1892]

awesome channel, totally subscribed!

[@rayharmuth8587]

I like your video! I really enjoyed watching it.

[@CrimsonKnightmare1]

That was really cool!!! Thanks for sharing :D

[@nafrost2787]

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].

[@MuffinsAPlenty]

"a range is any set that contains the image" I have never encountered any text using such a definition of "range". Every text I have encountered which uses "range" uses it synonymously with "image".

[@nafrost2787]

Ok I searched, sometimes range is defined like the image, and sometimes it's defined like I described.

[@iwantaoctosteponmyneckbut3545]

My uni's abstract algebra textbook, Dummit and Foote, uses "range" and "image" synonymously (which matches how "range" was defined in my high school math classes). The set which a function maps onto, one which includes the image, is instead called the "codomain"

[@MuffinsAPlenty]

@@nafrost2787 Can you provide me with a name of a textbook which makes the distinction between image and range like you said in your post?

[@bottleimp007]

Absolutely excellent instruction!

[@signature.smile.4]

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

[@masterstghm]

Wish you did videos on cyclic groups and quotient groups!

[@savitasondhi7690]

Awesome explanation

[@isaacahiazu3695]

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.

[@Grassmpl]

I'll teach you

[@phyziks878]

Superb explanation mam ,thank you

[@aishwaryadash4166]

This was quite helpful...

[@jairobonilla7980]

Very clearly... CONGRATS

[@chowhan13]

At last I'm subscribing

[@Socratica]

HOORAY!!! :D

[@mohit0901]

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

[@mubahaliqbal5063]

Plz give more lectures on group theory

[@moularaoul643]

Thank you so much!!!

[@NeerajSingh-kl1dl]

good presentation

[@reymarkpaquiao8964]

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.

[@Mycrosss]

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) ?

[@enterthepleasuredome8602]

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.

[@keylee85]

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.

[@MaxxTosh]

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

[@jairoselin5119]

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

[@Klebtomaniac]

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

[@cruizergaming1738]

Isomorphism term in this video is maths😅

[@stormzykirey6552]

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

[@naziabno]

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

[@Gipsy4u]

Thanks, good stuff, keep it up

[@SHASHANKRUSTAGII]

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

[@liketsontobo8463]

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

[@rajarshichattopadhyay8407]

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

[@yahya5308]

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

[@nuradinamin1628]

do you have vedio on application of field:?

[@coolquitepowerful]

Smart teaching thanks

[@alancristopher3539]

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 :(

[@lugia8888]

Find a map between the sets (has something to do with Euler Formula) and prove it is a bijection and homomorphism

[@alessiodenny6123]

excellent video !! thanks

[@larbibenghrieb]

thank you ❤️

[@kunslipper]

Thank you so much.

[@saranegi8316]

wowwww you explain sooo good maam

[@akankshamadhuriraj1565]

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

[@tuikolovatufalemaka2096]

If two groups are abelian with the same order, do they automatically isomorphic?

[@Socratica]

Unfortunately, no. For example, the groups Z/2Z x Z/2Z and Z/4Z are both of order 4, but are non-isomorphic. (Here, Z/nZ are the integers mod n.) There is a very nice theorem that describes all finite abelian groups which we'll talk about in an upcoming video.

[@Grassmpl]

If you replace abelian with cyclic then you are right. In general, the number of nonisomophic abelian groups for a particular finite order is the product of some integer partition numbers.

[@Mathgodpi]

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

[@Socratica]

It's a relabeling that also preserves the group operation. For example, suppose in group G you have a*b = c. And with an isomorphism from G to H you relabel a, b, c as x, y, z, then because a*b = c in G, you want to have x*y = z in H. It's possible to have relabelings that do not preserve the group operation. These would simply be 1-1 mappings, and not isomorphisms.

[@adventhouse6506]

Well done

[@cameronspalding9792]

Do homomorphisms have to be surjective

[@MuffinsAPlenty]

Homomorphisms are not required to be surjective. They are also not required to be injective. On the other hand, isomoprhisms are required to be both surjective and injective.

[@user-lg7mf8sx4w]

Thank you soo much!!

[@codethegamer]

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

[@thanushathisanthan5290]

wow!! thank you so much.

[@VicksGamingWorld]

Thanks mam

[@lemyul]

thanks tom

[@sananseyidbeyli3073]

good job

[@saurabhsingh-ow7ue]

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

[@PunmasterSTP]

Did you know that if you put this on 0.75x, you get a...slomomorphism?

[@joansola02]

hahaha

[@helloitsme7553]

If you have two groups with equal order, can you always find an isomorphism

[@MuffinsAPlenty]

No, you cannot. The smallest example is order 4. Z/4Z and Z/2Z x Z/2Z are both groups of order 4, but they are not isomorphic. Z/4Z has an element of order 4, but Z/2Z x Z/2Z has no elements of order 4.

[@straighttothepointcoding4067]

Nothing about Automorphisms?

[@TheHdz2011]

THIS LADY INTENSE AS FUCK....SHET