The Kernel of a Group Homomorphism - Abstract Algebra
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- เผยแพร่เมื่อ 14 ต.ค. 2024
- The kernel of a group homomorphism measures how far off it is from being one-to-one (an injection). Suppose you have a group homomorphism f:G → H. The kernel is the set of all elements in G which map to the identity element in H. It is a subgroup in G and it depends on f. Different homomorphisms between G and H can give different kernels.
If f is an isomorphism, then the kernel will simply be the identity element.
You can also define a kernel for a homomorphism between other objects in abstract algebra: rings, fields, vector spaces, modules. We will cover these in separate videos.
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We recommend the following textbooks:
Dummit & Foote, Abstract Algebra 3rd Edition
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Milne, Algebra Course Notes (available free online)
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Dear Socratica , I believe that your lecture series is just the most beautiful lecture series i have ever watched in abstract algebra. i am not afraid of abstract algebra any more thank you for such a beautiful series on math..... great work ....
true
Completely agree with that
Claim:
The kernel of G is a subgroup of G.
Proof:
We have established so far that the kernel is a non empty set containing elements of G, combined with the operation of G, *. We know that the identity 1G is always in the kernel by definition. Also, we know * is associative. Therefore we need to show that the kernel is closed under *, and that all elements of the kernel have unique inverses.
Consider two elements of the kernel of G, x and y. We know that f(x) = 1H and f(y) = 1H. Then f(x*y) = f(x) • f(y) = 1H • 1H = 1H. Thus x*y is in the kernel of G; the kernel is closed.
Now consider an element z of the kernel. Since homomorphisms map inverses to inverses, we know that f(z-1) = f(z)-1. But f(z) = 1H, and the identity is it's own inverse, so f(z-1) = 1H, and z-1 is in the kernel.
Thus the kernel of a group G with respect to a homomorphism f is a subgroup of G.
only explanation that i understood easily
Thanks for the solution, but, should the conclusion statement be that the kernel of a homomorphism f w.r.t. a group G is a subgroup of the group G? Since the "kernel" here isn't about the group but about the map f, stating the kernel of a group G might be somewhat misleading.
I notice everywhere you write kernel of G, but kernel is a property of homomorphism between two groups not the group, first correct your words, so I don't confuse.
You are proved kernel is a group but you missed to prove it is subgroup of G
To prove kernel is subgroup of G
We know that the definition of subgroup of a group
ab^-1 belongs to G since a,b belongs to H
We also know 1 is the only element present in kernel it is identity element also
Inverse of the Identity element is itself so we can claim kernel of a group G with respect to homomorphism f is subgroup of a group G
you are contributing to make a better world. thank you!
Oh my goodness, what a lovely thing for you to say. Our viewers are just the nicest. Thank you so much for watching!! :)
I wish you were my abstract algebra prof.
The best we can do is make more videos for you! Thanks for watching, and thank you for your kind comment! :)
^^^^
the grass is always greener elsewhere
Many years have passes since I learned this in the university.. It is a pleasure to recover that forgotten knowledge with such a wonderful teacher. Thank you!
Isn't it wonderful that you can pick up where you left off? Hooray for lifelong learning!!
Thank you for watching, and thank you for your kind comment! :)
University I spent 6 weeks to learn these = Here I use 20 min understand ... Thank You
I am sorry but you cannot really understand something without doing some exercise problems.
Man....I feel the same way!!!!!!!
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So, so true. People talk about math as if it’s a matter of understanding, which it is, but sometimes it’s neglected to mention that math is also a skill practiced by doing. You can understand the concepts, but applying them gives a more direct experiential familiarity with the processes actually going on, the repetition of which provides a greater and greater subconscious intuition that is absolutely invaluable and unlocks new maths frontiers for you.
Not doing practice problems is like watching archery on TH-cam and thinking you know enough to hit the bull’s-eye..
I like the way of teaching her. It's so lucid and made the content easy to understand. Thank you.
I like your challenge question at the end to show that the ker(f) is a subgroup of G.
For anyone who is a little stuck (this is a common feeling among mathematicians - it's OK to feel that way you're in good company!) just write down everything you know again on a sheet of paper.
So.... you have G,* and H,◊ and you have f: G -> H and you also know that f(x*y)=f(x)◊f(y). We also have our new definition for kernel which is ker(f) = { x in G | f(x)=1H}
All you need to do to show that this set, ker(f), is a subgroup of G is show that it's 1) closed under * 2) Has an identity 3) Each element in ker(f) also has it's inverse in ker(f) and finally 4) It's associative. Just like we did back in the fourth video "Group or not group"! That's it. It's fun and not too tough - hope that helps anyone who's stuck.
Is it necessary to check all of those? I thought for subgroups less effort is required since we're talking about subsets of something which is a group. For example associativity is a given, I think
associative property need not be proved for subgroups.
These are helping me get a better overview of Abstract Algebra. Thank you!
Hope Socratica creates more Abstract Algebra videos as well as playlists on Topology and Analysis next.
I admire the presentation skill of the instructor. She presented it like a beautiful story.
Yay! I've really been enjoying the python/programming videos, but I'd honestly forgotten why I subscribed to this channel? This is why. Your abstract algebra videos are phenomenal. Keep them coming!
+GelidGanef not to mention she makes math seem so much interesting than it is in college
+William Huang True dat. If I had a maths teacher like her, I would be flying rockets to other planets, rather than my current part time job as a human resource. :-D
+GelidGanef Thank you for the helpful feedback! Many more abstract algebra and python videos are in the works.
True
This video stopped me from giving up in Abstract Algebra when I was on the edge of giving up. I'm deeply in your debt. As soon as I have a decent salary I will be contributing.
We're thrilled we could help, Russell. Your message really inspired us today - thank you so much for writing and letting us know. Thank you for watching! :D
It is 2020 and still watching this. Thank you, it really helped alot.
Lady Socratica; thank you so so so so so much. I have completely understood your video from the word Go to the word end.
What a blessing to have u on you tube. What a blessing, what a blessing from the LORD that you lady exist in Abstract Algebra. Thank you so much,really much and really much. An amazing video. U have humbled my minds down to learn.
This has helped me for one of my math modules. Explained succinctly and intuitively, can't ask for more! Thank you so much!
I learned more in this video than i have in the past 2 months of my abstract algebra class
Hi Socratica,
First of all, thank you so much for all these useful videos. Secondly, could you plz correct the negligible mistake at 3:35 f(x_2)=y -> f(x_2*x_2^(-1))=1_H
Nice video, but you should mention the cokernel and draw an analogy with "onto" maps. I find the dual construction very enlightening when trying to intuit kernels.
Beautifully presented! Thanks, Liliana and Socratica team!
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awesome video. ur organization is doing a great job.
your explanation is so clear.
please make more videos on concepts of abstract algebra.
Thanks! Many more Abstract Algebra videos are on the way. :)
Socratica we're waiting for it..
omg, this kernel is totally consistent with the kernel in linear algebra (as it should be). Gotta luv it when terminology and concepts are consistent! Question is, should you learn linear algebra first or abstract algebra first?
You can learn them in either order. However, if you learn Linear Algebra first, you'll be equipped with lots of examples for the ideas in abstract algebra. In fact, most abstract algebra textbooks assume you are familiar with matrices. So most people would probably find it easier to learn linear algebra first.
I AM SO LUCKY TO HAVE YOU MADAM SO THANKFUL TO YOU FOR HELPING ME OUT IN WHAT I THOUGHT IS IMPOSSIBLE TO ME AND MAKING IT POSSIBLE TO ME
The way that built up to ker(f) makes a lot more sense than the way i initially learned. Interesting mix of videos
I just want you to know I fell in love with your videos. although I am not a native English Speaker I completely got your explanation. Best Regards from Honduras!
Watching this in 2020 and it is so elegantly explained. Thank you so much.
I doubt there is a better video on this subject, but please prove me wrong with a reply! This whole series is fantastic.
I have watched many of these at this point. Besides being really a useful tool to learn a specific math topic which has a well deserved fame of being bit-cryptic and being able do it an efficient way, this innovative approach makes me think about how wrong the established approaches to transmit scientific knowledge is these days, being them on the 'math has to be dry and hard' or in 'math is fun' side. Learning should be a social experience, before becoming an individual one. IMHO this is the most important lesson I am taking from these classes.
My 2cents : ( correct me if wrong, as a progress of learning humbly ).
Definition of Subgroup S
I love that I was about to ask if the kernel is a subgroup of G, and then she said it was. I feel like I’m learning!
This is great! Have you considered making a video about orbits and stabilizers?
I first time in my life understand the meaning of kernel
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when I saw this video uploaded I got so excited!!! keep up the AMAZING work with abstract algebra, you guys are the best!
+Ilya Noskov Thank you! We're planning many more abstract algebra videos, and will be filming the next one this week!
+Socratica what are the topics you plan to cover? I think I am gonna be your patreon if there are more math videos!
+Ilya Noskov For abstract algebra we're going to cover the most important structures: groups, rings, fields, vector spaces and modules. We're also going to begin making number theory videos in the next few weeks!
Thank you so much, I understood easily, I never forget about kernel.
I think the assumption x not equal 1 in 1:10 is not nececary. In fact, we can always choose x=1 and the proof still holds.
Clarity of your speech is helpful in seeing the terms and. relations apart .
way better than my prof explains it. Well-planned and executed video that makes algebra much easier to understand when ideas are explained fully since I don't remember them all yet
Great video! I watched a different one explaining isomorphisms/homomorphisms. So one way to prove a function is 1-1 is to say, Let f(x) = f(y)......x=y. Another way would be to say f(x)=identity iff x in Ker(f), or...?
I liked the way you teach with an authority. It makes the lecture more interesting!
This is the best explanation i have gone through till now. Thanks
You are doing a great job...finished all the abstract algebra vids in one sitting...Please upload more...thanks in advance :D
Thanks, Charlie! More Abstract Algebra videos are on the way! We filmed several more just last week.
Not a criticism, but around 3:35 some steps were skipped. Given x1 not equal to x2, we should show that x1 * x1~ and x2 * x1~ are distinct elements. As with previous uses of cancellation using inverse, it's not hard to do, but at the beginning level such details should be spelled out.
2021 !! And i found this videos what a great start of study with u..
Hi, This has been the most helpful thing during a pandemic when you can't go to uni! Thank you so much there is no way I could even attempt my coursework without you! :)
5 minute youtube video better for my understanding than 3 hrs of lectures. It's all good tho cuz my prof irl dumb handsome ;O
Solid Snake voice:
"Huh...
Kernel.
I'm trying to map to 1.
But I'm dummy thicc
And the elements of my group keep mapping to a non-identity"
underappreciated comment
This 4 min video takes my 1 hour to understand thoroughly not losing hope 😊
Super video. Short and *Concrete*
I love you. Thanks for these videos. they are very explanatory. Wish there were more math teachers in uni like you.
thank u madam for ur giving a good knowldege of mathematics ..i am very much impress to ur way teaching and understanding the concept of mathematics, i want to discuss the few general doubt of FUNDAMENTAL THEOREM OF HOMORPHISM OF GROUPS .i am grateful to u, if u prepare a video lecture on this topic plz maam...
I did my major in Physics.
I would never have come this far in abstract algebra series.
These lectures are tonic for my brain😅😅
We're so glad you're exploring with us!! 💜🦉
You are doing these videos quite interesting manner , We hope u will keep it up , I think u should cover whole content of this particular subject..
thank you so much for these - truly the simplest explanation of the subject, these videos have helped me so much !!
Great explanation! Have more confidence for the incoming midterm
You need to go over theorems in the Algebra playlist! Like Sylows theorem. thanks
When I watch this video it like , in French we say " une illumination" for me . Thank you very much
Dear socratica, your teaching method is too much impressive but your lectures ends before it started so please add few examples more plzzzzzzzzzzzzzz😘love 💕 from INDIA 🇮🇳🇮🇳
We're planning on adding more short example resources on our website! Thanks for the encouragement. 💜🦉
You can sign up for our email list so you'll get notified when new stuff arrives!
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Interesting : Being a Lecturer, it was really very fruitful lecture for me. Thank you
Sol of challenge:
(1) kernel is a homomorphism that contains all elements that map to identity of H, so it contains the identity of G
(2)if x in kernel then f(x)=identityH, if y also in kernel then f(y)=identityH, so f(xy)=f(x)f(y)=identityH*identityH=identityH
(x)if x in kernel then f(x)=identityH, so f(Identity G)=f(x&x^-1) = f(x)*f(x^-1)=Ih*f(x^-1)=f(x^-1) = Ih
Really your teaching style is so good ❤️❤️
That was very nicely put.very nice explanation.THANKYOU SO MUCH .it was realy useful.
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I really love the clear notation.
Awesome. This channel is exceptional!
The very first line blew me away.
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I took whole lot year while our lecture teaching. Only 5minn in socratica😎😍😘🥰
Are there any examples around which one might wrap his head? This is somewhat abstract.
Show that ker(f) is a subgroup of G:
It is already shown that ker(f) is a subset of G and that it contains the identity 1_G. ker(f) is also associative because its group operation is the same as of G.
To show ker(f) is closed, take any xa, xb ∈ ker(f).
xa * xb = x
f(xa * xb) = f(x)
f(xa) ♢ f(xb) = f(x)
1H ♢ 1H = f(x)
f(x) = 1H , therefore x ∈ ker(f).
To show every element in ker(f) has an inverse, choose x1, x2 ∈ G such that
x1, x2 → y
as shown at 3:35 this yields:
f(x1 * x2^-1) = 1H and by the same reasoning
f(x2 * x1^-1) = 1H
Call these:
x1 * x2^-1 = xr ∈ ker(f)
x2 * x1^-1 = xs ∈ ker(f)
We can invert one of these step by step:
x1 * x2^-1 = xr
x1 * x2^-1 * xr^-1 = xr * xr^-1
x1 * x2^-1 * xr^-1 = 1G
x1^-1 * x1 * x2^-1 * xr^-1 = x1^-1 * 1G
x2^-1 * xr^-1 = x1^-1
xr^-1 = x2 * x1^-1 = xs
This shows that xr is the inverse of xs.
so sweet ,keep going, give ur self some times to learn c programing ,it will be amazing with algebra , belive me ,and it will not take long ,u can learn fundmentals in a week.algebra more difficult and complex than programming in beginner level
Lets consider a case let {Xp |p=1,2...m }and{ Xq |q=1,2...n}
Maps to y1 and y2 and the rest is 1-1 map ,how would u define a kernal in such case ? Do we require 2 kernals in that case ?
Is the phrase "sends inverses to inverses" equivalent to "preserves inverses?" The "sends/to" phrasing is used throughout the video, but I didn't hear anything about preserving. Should I not use the latter phrasing?
Yes. Mathematicians do say that homomorphisms "preserve inverses" and that's what the speaker means by "sends inverses to inverses."
Thanks, Leonard!
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Thank you for the clear explanation!
thank you very much this was very helpful
Thank for those free I was searching for Abstract Algebra professor And finally I got it 😊.Yes, I have solved the challengeThank You
do you have a video about ring homomorphisms and the kernel of those?
Beautiful explanation!!
Excellent classes
I think the proof that group homomorphisms send identities to identities isn't quite right. It doesn't work if G = {e}, where e is the identity of G. In other words, G is the trivial group. The argument doesn't work because there is no x in G distinct from e. However, the assumption that x ≠ e is irrelevant to the argument. In particular, since the only element that is guaranteed to be in G is e, we might as well use that.
In more detail, let f: G -> H be a group homomorphism. Then e=ee, so f(e) = f(ee) = f(e)f(e). Multiplying both sides by the inverse of f(e), which exists because H is a group, implies that f(e) is the identity in H.
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This series is great but it needs to be completed by covering more topics.
i love the creators of this channel
love your videos. you make my interest in algebra. and im very thankful.
more videos please...
Thank you so much for watching! We're so glad you are finding our videos helpful. More on the way soon! :)
great videos the socratica team
Please make a video on Reproducing kernel operator with cyclic vector
Fantastic work!
Mam, you make abstract algebra simple...thanks a lot
i am inspired from her way of expressing.. math as poems of Shakespeare
Wow, what a great tutoring!
Is it f(x2)*f(x2^-1) in the definition of kernel. In the video it is given f(x2)*f(x1^-1)
Let f be a non trivial homomorphism from ℤ10 to ℤ15. Then which of the following
holds?
A) Im f is of order 10. B) Ker ݂f is of order 5.
C) Ker f is of order 2. D)f is a one to one map.
What is the answer for this question?
Note that im f is a subgroup of Z15, so |im f| divides |Z15| by Lagrange's Theorem. Additionally, by the First Isomorphism Theorem, im f ≅ Z10/(ker f). By Lagrange's Theorem, |Z10/(ker f)| = |Z10|/|ker f|.
This gives that |im f| divides |Z15| and |Z10|. But since f is non-trivial, |im f| is NOT 1. Can you figure it out from there?
About the proof, is it not false if "f(1G)=/=1G", "1G" is not the identity in the group of "f(x)" and/or "1G=/=1H"?
And is the Kernel not only a subgroup of "G->H"?
Please tell me the name of background music you play for this video.
Makes group theory easiest to understand 😍😍😍
According to definition of homomorphism:
F(1g*1g)=f(1g)$f(1g)
Now, f(1g) is an idempotent element of {H, $}
But the only idempotent element elements of a group is its identity, thus, f(1g) = 1h
How does this sound?
I think you're showing that ker(f) is a subset of G, but the challenge asks showing that it is a /subgroup/ of G; that is, a set equipped with an operation * (the same as G) which fulfills all four requirements of being classified as a group
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Very good content, thanks!
So that's why in Linear Algebra they have two similar terms - Null Space and Kernel. One of them is native to LA, the other comes from Abstract Algebra?