# Homomorphisms (Abstract Algebra)

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…”}

### 100 thoughts on “Homomorphisms (Abstract Algebra)”

• October 27, 2017 at 8:28 pm

Why u said in last sorry .
And thank u
Because many things I learned.

• November 19, 2017 at 6:15 pm

I_subscribe totally got me….

• November 19, 2017 at 9:01 pm

I
subscribe

• November 21, 2017 at 9:00 am

Thanx mam…..

• November 21, 2017 at 12:36 pm

I love u

• November 21, 2017 at 12:37 pm

U r so bautiful

• November 28, 2017 at 2:54 am

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

• December 1, 2017 at 3:47 pm

• December 5, 2017 at 3:27 am

Do not be sorry. I subscribe.

• December 5, 2017 at 3:28 am

Excuse me. I am already a subscriber.

• December 24, 2017 at 10:37 am

• January 2, 2018 at 11:47 pm

HAHAHAHAHHAAH i-sub-scribe

• January 6, 2018 at 2:57 pm

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

• January 20, 2018 at 3:48 am

well expalaind

• January 27, 2018 at 11:45 am

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!

• February 17, 2018 at 8:24 pm

subbed for joke

• March 4, 2018 at 6:14 am

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

• March 6, 2018 at 12:03 pm

• March 7, 2018 at 4:47 am

Nice joke! haha

• March 7, 2018 at 4:48 am

And btw, you are so adorable!

• March 13, 2018 at 2:49 pm

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

• March 17, 2018 at 5:13 pm

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

• March 24, 2018 at 5:49 pm

hahahah nice joke at the end! I liked it

• April 9, 2018 at 6:26 am

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

• April 23, 2018 at 5:21 pm

i subscribe hahaahaaa…

• May 17, 2018 at 12:52 am

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

• May 17, 2018 at 7:16 am

Nice explanation but the video could have been less dramatic

• May 23, 2018 at 7:55 pm

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

• May 29, 2018 at 11:23 am

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• June 2, 2018 at 9:58 am

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

• June 9, 2018 at 1:12 am

She's smart and hot

• June 10, 2018 at 8:41 pm

• July 1, 2018 at 12:56 pm

i like you

• July 17, 2018 at 5:10 pm

So nice,,,,, new subscriber from India

• August 22, 2018 at 4:16 pm

typical math joke at the end

• August 24, 2018 at 2:51 am

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

• August 29, 2018 at 8:34 am

thanks

• September 21, 2018 at 2:42 am

what about strong homomorphism (vs isomorphism)?

• October 1, 2018 at 3:51 pm

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

• October 3, 2018 at 2:28 am

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)

• October 3, 2018 at 7:35 pm

I took this class last year and I subbed lol…

• October 14, 2018 at 4:47 am

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

• October 15, 2018 at 10:01 pm

1:24 This is important.

• October 25, 2018 at 3:02 pm

That joke was nice😊😊😊

• October 31, 2018 at 10:42 am

the sub joke is so original that i actually subbed

• November 1, 2018 at 5:47 pm

Can't get any easy understanding these little concepts 🤗

• November 3, 2018 at 2:51 pm

• November 6, 2018 at 1:40 am

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

• November 19, 2018 at 11:05 am

I need to find me a woman like this

• November 21, 2018 at 11:32 pm

subscribe for logical joke

• November 23, 2018 at 9:46 pm

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

• November 26, 2018 at 6:43 am

Wow 😮

• November 28, 2018 at 5:09 am

Was ex2 isomorphic caused she didn't state it?

• December 7, 2018 at 5:16 am

I love this channel, thank u!

• December 11, 2018 at 2:30 am

Beauty and Brains

• December 11, 2018 at 2:57 am

ugh, I had to sub after that one.

• December 24, 2018 at 1:46 pm

Thanks 😀

• December 26, 2018 at 10:45 pm

😀 OK, subscribed

• January 24, 2019 at 4:32 am

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

• January 25, 2019 at 4:30 pm

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!

• February 20, 2019 at 9:46 pm

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

• March 2, 2019 at 9:10 pm

I don't think which would be failed if you are his/her teacher…
I am yr friend ..thanku

• March 3, 2019 at 8:35 pm

You are so beautiful. I fell in love with you

• March 4, 2019 at 6:13 pm

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

• March 19, 2019 at 12:48 pm

great video

• March 26, 2019 at 2:34 pm

Good work…👌

• March 27, 2019 at 4:53 pm

She is so sweet..I already subscribed

• March 31, 2019 at 10:23 am

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

• April 7, 2019 at 4:38 pm

one good lecture after many days

• April 24, 2019 at 4:45 pm

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

• May 1, 2019 at 5:06 am

heyy you are so beautiful

love yaa

• May 6, 2019 at 2:17 pm

when she says sorry :p

• May 16, 2019 at 11:02 pm

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.

• May 17, 2019 at 2:45 pm

dat joke :))

• May 19, 2019 at 3:11 pm

Best videos on Group Theory by Socratica!

• May 23, 2019 at 4:16 pm

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

• May 27, 2019 at 4:16 pm

Can you make a video on the fundamental theorems of isomorphism

• July 13, 2019 at 6:24 pm

pllllz..make vdo on Topology😆😆👌

• August 16, 2019 at 6:14 pm

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

• August 24, 2019 at 5:31 pm

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

• August 28, 2019 at 7:54 pm

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.

• September 2, 2019 at 12:33 pm

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

• September 6, 2019 at 8:33 pm

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

• September 15, 2019 at 9:16 am

Nice smile

• October 6, 2019 at 7:18 am

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

• October 8, 2019 at 7:25 pm

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• October 14, 2019 at 11:47 am

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

• October 16, 2019 at 4:38 pm

that joke cracked me up, u r awesome

• October 23, 2019 at 2:39 am

what an excellent and beautiful teacher!

• October 23, 2019 at 2:39 am

how could we not subscribe ?

• October 25, 2019 at 10:57 am

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

• November 3, 2019 at 7:54 am

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

• November 7, 2019 at 12:55 pm

omG U re soo pretty!!!!!

• November 11, 2019 at 12:41 pm

• November 17, 2019 at 4:53 pm

• November 19, 2019 at 1:12 pm

Thanks for creating this video i understand a lot.

• November 22, 2019 at 8:00 pm

Is she trying Italian 🍝

• November 25, 2019 at 5:18 am

you explained it so well :))