A homomorphism is a way to compare two groups

for structural similarities. It’s a function between two groups which preserves the group

structure in each group. Let’s be more specific. Suppose we have two groups G and H. Remember,

G and H are different groups, and they may have entirely different group operations.

To help us keep G and H separate in our minds, let’s use an asterisk for the group operation

in G and a diamond for the group operation in H. We’ll still use the word “times,”

but we’ll write the correct symbol for the operation. Now pick any two elements X and Y in G. Let’s

say that X times Y equals Z. Next, suppose we have a function F that maps G to H. The

three elements X, Y and Z are mapped to elements in H; X is mapped to F-of-X, Y is mapped to

F-of-Y, and Z is mapped to F-of-Z. The whole point of a homomorphism is to find a structural

similarity between two groups. So if X times Y equals Z in G, then we’d like F-of-X times

F-of-Y to be equal to F-of-Z in the group H. So (F-of-X) times (F-of-Y) equals (F-of-Z).

But Z equals X times Y. Substituting this gives us (F-of-X) times (F-of-Y)=F-of-(X-times-Y).

This is the definition of a homomorphism. It’s a simple definition, but it captures

a big idea – a way to compare two groups. Let’s see an example. Let G be the group

of real numbers under addition, and H be the positive real numbers under multiplication.

G is an abelian group with identity element 0, and H is an abelian group with identity

element 1. Here’s the homomorphism. Define the function F that maps X to (e-to-the-X).

To make sure this is a homomorphism, we need to verify that F-of-(X-plus-Y)=(F-of-X)

times (F-of-Y). Don’t forget that the group operation for G is addition, while the group

operation for H is multiplication. By the definition of F, this says that e to the (x

plus y) equals (e-to-the-x) times (e-to-the-y). And this is true! This is one of the rules

of exponents. So F is a homomorphism. Here’s another example of a homomorphism.

G will be the real numbers under addition, and H will be the complex numbers with absolute

value 1. These complex numbers form a group under multiplication. Showing this is a good

exercise, so why don’t you pause the video and show that H is a group. Here’s a hint

– every complex number with absolute value 1 can be written as e to the i-THETA for some

angle theta. Define the function F as F-of-X equals e-to-the-i-x.

For F to be a homomorphism, we must have F-of-(X-plus-Y)=F-of-X times F-of-Y . This is true only

if e-to-the- i-times-(x plus y) equals (e-to-the-i-x) times (e-to-the-i-y). We can see this is true

by expanding the left hand side… then using the rules of exponents. It’s true; F is

a homomorphism. By the way, I’d like to point out that F-of-zero equals F-of-(2 pi).

Both are equal to 1. In fact F of ANY multiple of 2 pi is 1. So this is NOT a one-to-one

function. Homomorphisms do NOT have to be one-to-one. Take a closer look at the word “homomorphism.”

It’s a combination of “homo” which means “same” and “morph” which means shape.

But don’t be misled by this name. If two groups are homomorphic, it doesn’t mean

they’re the same. It just means they’re SIMILAR. Also, not every pair of groups will

have a non-trivial homomorphism between them. Some groups are so different from one another there are

no structural similarities at all. One final note. How would you simplify this

expression? Well, the square root of negative one is i, so this leaves us with “i subscribe”.

{mouths the phrase:: “I’m so sorry…”}

Why u said in last sorry .

I already subscribed.

And thank u

Because many things I learned.

I_subscribe totally got me….

I

subscribe

Thanx mam…..

I love u

U r so bautiful

Abstract algebra blows, no disrespect to the video…just the subject

Can she please be my teacher? PLEASE?

Do not be sorry. I subscribe.

Excuse me. I am already a subscriber.

Your look is killing me

HAHAHAHAHHAAH i-sub-scribe

great video but maybe it should be noted that homomorphisms doesn't have to be between groups only.

if anyone else was slightly confused like me, see this:

https://math.stackexchange.com/questions/29944/difference-between-linear-map-and-homomorphism

well expalaind

I have to say that this is the first math-channel on Youtube 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!

subbed for joke

does she really have a command on these topics or is she just reading out what's displayed on the screen? coz she looks like a model yo

very helpful thanks

Nice joke! haha

And btw, you are so adorable!

i know what homomorphism is. Now can we skip to the part where you marry me?

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

hahahah nice joke at the end! I liked it

Ffs I have a 50% exam on abstract algebra in a week and this channel is the only thing keeping me alive <3

i subscribe hahaahaaa…

i was disqualified 3 times from The Voice, only ´cos im a refugee!

Nice explanation but the video could have been less dramatic

Hey! Can anyone help me in this that (Q,+) and (Q+,*) are isomorphic or not?

I LOVE HER!!!!!!!😍😍😍😍😍😍😍

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

She's smart and hot

the joke made me subscribe!!!

i like you

So nice,,,,, new subscriber from India

typical math joke at the end

I subscribed to the channel from 2 accounts (1 as thanks and other as a bonus for her skills and as well for hotness).

thanks

what about strong homomorphism (vs isomorphism)?

I am a Mathematician and I know all of these things, but I'm here just because she's so beautiful.

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)

I took this class last year and I subbed lol…

Great explanation, her looks are second only to her voice like a symphony of symmetries.

1:24 This is important.

That joke was nice😊😊😊

the sub joke is so original that i actually subbed

Can't get any easy understanding these little concepts 🤗

thanks madam

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

I need to find me a woman like this

subscribe for logical joke

Oof, I'm in 10th grade geometry, and this kind of math seems so much more interesting, but understood absolutely nothing.😂😅

Wow 😮

Was ex2 isomorphic caused she didn't state it?

I love this channel, thank u!

Beauty and Brains

ugh, I had to sub after that one.

Thanks 😀

😀 OK, subscribed

I was going to give a like….but when she said that joke i subscribed instantaneously jajajajajaja she is so mm sweety or sexy i dont know but i like her jaajaja

you look like the girl in the tv show blindspot! BTW, I love these videos. They are incredibly helpful! I wish you made more math videos. Like Real Analysis or Differential Geometry!

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

I don't think which would be failed if you are his/her teacher…

I am yr friend ..thanku

You are so beautiful. I fell in love with you

I had 3 books of abstract algebra.. yet i was horrified.. now.. after watching this videos i think i am an genius 😂😂😂

great video

Good work…👌

She is so sweet..I already subscribed

I have been watching this playlist for a month but didn't subscribed but I just for this joke .

You should try stand up comedy also I'll definitely subscribe that channel too

one good lecture after many days

Prof. McCarthy at University of Illinois at Urbana-Champaign, thank you, for those brilliant hours of logic lecture.

heyy you are so beautiful

love yaa

when she says sorry :p

She is very attractive however that’s not why she’s doing this. She’s trying to help people gain a better understanding of mathematics. I’ve no doubt she would much prefer to know she’s doing a fantastic job of that. Which she is. Thanks.

dat joke :))

Best videos on Group Theory by Socratica!

hope these videos save my semester….awesome explanation..btw i am an old subscriber

Can you make a video on the fundamental theorems of isomorphism

pllllz..make vdo on Topology😆😆👌

Joke at they end was a shocker! 😲… Still kind of funny though🤭

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

I think she looks better with short hair.

Also, this is a great channel. If I can get all the way through, I'll donate.

I already subscribed, if I do it again I become a -1 !!!

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

Nice smile

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

YES GIB ME DATS NICE JOKE +sUB

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

that joke cracked me up, u r awesome

what an excellent and beautiful teacher!

how could we not subscribe ?

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

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

omG U re soo pretty!!!!!

God bless your soul lady

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

Thanks for creating this video i understand a lot.

Is she trying Italian 🍝

you explained it so well :))

I pass this subject thanks to her, thanks alot😊, by the way: i loved the way she talks, it is just cute like she is…great content😊

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