Hello everyone! We just released our latest *Abstract Algebra* video. In this lesson, we introduce the *Symmetric Group*. We *define* it, give some *examples*, and introduce a *compact notation for permutations*. Next up in the Abstract Algebra playlist: *homomorphisms*, *isomorphisms* and *kernels*. #LearnMore
Socratica Isn't it that every group is related to permutations, in a way? Because, you know, when you make a "multiplication table" for any group, then in each row or column, each element of a set can appear only once, no more no less. So each row/column is a permutation of the first row/column (for the identity element). If any element appeared more than once in a row/column, then there would be two ways to get there in a Cayley's diagram, and there would be an ambiguity of which of these operations we need to follow backwards to reverse it. And every operation in a group needs to be reversible, so such a "multiplication table" wouldn't be for a group.
My math examination is tomorrow and your videos will definitely help me pass this semester. I will recommend your videos to my friends as well. Although it's too late now.
My favorite way of representing permutations is to arrange all the elements of a set in a circle/ring, and for every element (an input) draw an arrow to the element it maps to (its output). This is the most visual way of doing it, and it often help in revealing some patterns in the permutation.
Another useful application of permutation groups is in cryptography, because cryptography is all about rearranging the sets of symbols (e.g. letters of the alphabet) to make a different alphabet.
Thank you very much for the well presented explanation. I really appreciate you for that. You give me confident to do my end-of-year semester exam for Abstract Algebra tomorrow. From a University in PNG.Once again thank you!.
I have not taken Abstract Algebra or specifically Group Theory before or did not know I was using the concepts in it in some of what I am currently doing. After watching the series of videos I felt like if it is fun to learn Abstract Algebra and thought of the so many things one could explore in and with it. A lot of Math professors, at least some of those I encountered, enjoyed a lot being Math-ish especially when presenting difficult concepts during their lectures. The tendency is for the students to feel a huge inferiority and thus hinder true learning and appreciation of the subject. I really appreciate how the topics are presented very well in this series of videos. Will be waiting for more.
Jeffrey Aborot Thank you for your thoughtful comment. It makes you wonder how much untapped potential is out there that is being turned off from certain subjects - like how SO many people we talk to say they are "bad at math" or "afraid of math." And it doesn't have to be that way! Thanks again for watching and we'd love to hear what other topics you would be interested in.
At 0:12, S sub n equals the group of permutations on the set { 1,2,3,...,n}. If A is a set of some other n elements, that symmetric groups is called S sub A.
May I request you videos on quotient group and normal subgroup. I search all the titles of your videos but can't find these two topics. If you discuss these topics in your videos, kindly give me the title of the video. Thanks for the works you did.
Great video, just heads up that the example for all the permutations for {1,2,3} is partially incorrect. the cycle (1 2 3) is the same cycle as (312) since 1 still maps to the 2, which maps to 3, which maps back to 1. is just that the cycle is written in different order. Therefore half of the permutations are just duplicates. The permutations you are missing out are: (1)(23) where 1 maps to 1, but 2,3 are cycles. and (2)(13) and (3)(12).
The example is not incorrect in the slightest. The issue is that there are multiple ways to represent permutations (elements of S_n). Here, you were expecting cycle notation, which is pretty much the only notation this video did not cover. Initially, one thinks of permutations as reordering the elements of a list/sequence. The sequence 1 2 3 can be reordered in the six ways shown on screen 0:41. Later, the video introduces the way to think about permutations as functions. The sequence 2 3 1, for example, is the same thing as having the function f(1) = 2, f(2) = 3, and f(3) = 1. This is then used to develop the "stacked" cycleish notation shown at the end of the video. The cycle notation you are describing in your comment here is the next step after this. You notice that you can make the notation _even more_ condensed by writing in cycles, rather than keeping track of where each element is sent.
Would it be a good guess that symmetry group S_n is also the group of symmetries of an (n-1)-dimensional simplex? For S_3 that would mean you get a 2d simplex or triangle, which you get to rotate around a point (0d) through 2d space or around an axis (1d) through 3d space; For S_4 you get a 3d simplex or tetrahedron, which can't really reach all 24 permutations of its vertices in 3d space, but I suspect it could if you let it flip along a plane (2d) through 4d space, and that an m-dimensional simplex could reach all permutations of it vertices if you generalize rotation/flip like this up to flip along (m-1)-dimensional space through (m+1)-dimensional space. Actually as I was writing this I had another thought, would it be right that if you can generalize rotation/flip like this, then flipping m-dimensional simplex along (m-1)-dimensional space could simply swap any two vertices and any sorting algorithm would then sufficiently prove that you can reach any permutation of vertices using just those swaps? I've got an intuitive feeling all of this works but don't know so I'd love any thoughts and insights my way!!
Who wants to improve their study skills? We WROTE A BOOK FOR YOU!! How to Be a Great Student ebook: amzn.to/2Lh3XSP Paperback: amzn.to/3t5jeH3 or read for free when you sign up for Kindle Unlimited: amzn.to/3atr8TJ
Very effective and simple way . Thank you so much . I was totally confused in this topic but after watching this video everything is clear to me about the permutation of Sn .
+Abdulqadir Isaak (Jeilani) -- The right-hand function acts first, then the left-hand function operates on the output of the right-hand function. Hence: 1 --> 1 --> 4 ; 2 --> 3 --> 2 ; 3 --> 4 --> 1 ; 4 --> 2 --> 3.
So the operation of the group of permutations is not the mapping of the permutation but the composition of 2 or more permutations, represented by *. Right?
I have a question which hopefully someone at Socratica will answer. Let's look at an element of S_3: (123) (312) Now let's say I have three objects in a row A B C. My question is this: how does the group element operate on this set? It seems to me , there are 2 ways of interpreting this. Either (a) the 1st object is moved to the 3rd spot, th 2nd object is moved to the 1st position, etc or (b) it means the 3rd object is put first, the first object is moved to the second spot, etc. In case (a) ABC becomes BCA. In case (b) ABC becomes CAB Which interpretation is correct? I'm thinking in terms of these group elements acting as operators re-arranging objects. I've googled this but haven't found what I'm looking for. Can you help me understand this please? Maybe you could also do a video on this.
It depends on which group action of S_3 are you considering on the set ABC. Both your interpretation is correct up to isomorphism, as long as you're consistent with it.
Since, according to Cayley's theorem, every finite group is equivalent ("isomorphic") to some subgroup of a symmetric group, is it then true that every group is a permutation ? Or can some subgroups of a symmetric group Not be permutations ?
"Permutation group" is a name given to a larger class of groups than just symmetric groups. A permutation group is a group where every element is a permutation of a fixed set S, and the operation on the elements is composition. All symmetric groups are permutation groups, but not all permutation groups are symmetric groups. For example, any subgroup of a symmetric group is also a permutation group, even though not every subgroup of a symmetric group is itself a symmetric group. Cayley's Theorem reveals that every group is isomorphic to some permutation group, which is a very nice result.
I watch these just for my entertainment. I know this stuff already. I'm a PhD student studying algebraic geometry. EDIT: Im fascinated at how a woman with only a theater degree know algebra so well. Mad respect for her.
This video is great ... I learnt those permutations while doing bachelor's but now in my master's course there are symmetries of s3 and V4 and so on represented by geometrical shapes.... Im so confused ... I have my finals coming but have to do that too ... Someone help please.. :(
I've never had a mathematician give me a good answer to a simple question - OK, I have a set of objects {A,B,C} in that order. So I mix them up, let's say to BCA. How do I write this? Is it | 1 2 3 | | 2 3 1 | or is it | 1 2 3 | | 3 1 2 | ? Can anyone answer this? I would appreciate your help.
ABC maps to BCA, a permutation which sends A to B, B to C and C to A. There are many other permutations possible in this case. In this case: A maps to B, i.e first element maps to the 2nd, similarly second to third and third to first. So the correct representation would be: |1 2 3| |2 3 1| Which is your first option. The second option which you mentioned sends 1 to 3, i.e. first element to the third element. == A to C.. so ABC will go to CBA, which is not the case here.
There is a trap that Algebra Professors use to shave off points. Some Algebra books multiply from Left to Right. In the next Semester if you are NOT Careful, Multiplication can be switched to Right to Left. Make SURE You Discuss This Before You Get Burned By It!!!
Omg u r a queen i love u T_T thanks to this, i finally understand. I cant believe i couldnt solve such simple problems before dammit! I wish i found these earlier DX subscribing now
Sign up to our email list to be notified when we release more Abstract Algebra content: snu.socratica.com/abstract-algebra
Hello everyone! We just released our latest *Abstract Algebra* video. In this lesson, we introduce the *Symmetric Group*. We *define* it, give some *examples*, and introduce a *compact notation for permutations*.
Next up in the Abstract Algebra playlist: *homomorphisms*, *isomorphisms* and *kernels*.
#LearnMore
Socratica Isn't it that every group is related to permutations, in a way?
Because, you know, when you make a "multiplication table" for any group, then in each row or column, each element of a set can appear only once, no more no less. So each row/column is a permutation of the first row/column (for the identity element). If any element appeared more than once in a row/column, then there would be two ways to get there in a Cayley's diagram, and there would be an ambiguity of which of these operations we need to follow backwards to reverse it. And every operation in a group needs to be reversible, so such a "multiplication table" wouldn't be for a group.
Socratica please solve my query . I have already mentioned my query in the comments
Socratica please make a video on transposition too .
And relationship between transposition and permutation
Mam pls do a video on permutation group
I should find an open cover for this notation and ask you to give me a finite subcover.
She sounds like the woman on a screen from a video game where you're stranded on an island in 2050
Yeah, or Skynet.
My math examination is tomorrow and your videos will definitely help me pass this semester. I will recommend your videos to my friends as well. Although it's too late now.
Ahhhh good luck!!!! :D
U cant imagine how ur videos are helping me in understanding this abstract algebra which was like ghost stuff to me
Clarity not confusion to key of success
Your lec is just amazing for me
My favorite way of representing permutations is to arrange all the elements of a set in a circle/ring, and for every element (an input) draw an arrow to the element it maps to (its output). This is the most visual way of doing it, and it often help in revealing some patterns in the permutation.
There needs to be a video on representation theory
Another useful application of permutation groups is in cryptography, because cryptography is all about rearranging the sets of symbols (e.g. letters of the alphabet) to make a different alphabet.
Thank you very much for the well presented explanation. I really appreciate you for that. You give me confident to do my end-of-year semester exam for Abstract Algebra tomorrow. From a University in PNG.Once again thank you!.
We're rooting for you!! 💜🦉
the best simplified explanation we can get on internet
I have not taken Abstract Algebra or specifically Group Theory before or did not know I was using the concepts in it in some of what I am currently doing. After watching the series of videos I felt like if it is fun to learn Abstract Algebra and thought of the so many things one could explore in and with it.
A lot of Math professors, at least some of those I encountered, enjoyed a lot being Math-ish especially when presenting difficult concepts during their lectures. The tendency is for the students to feel a huge inferiority and thus hinder true learning and appreciation of the subject.
I really appreciate how the topics are presented very well in this series of videos. Will be waiting for more.
Jeffrey Aborot Thank you for your thoughtful comment. It makes you wonder how much untapped potential is out there that is being turned off from certain subjects - like how SO many people we talk to say they are "bad at math" or "afraid of math." And it doesn't have to be that way!
Thanks again for watching and we'd love to hear what other topics you would be interested in.
Jeffrey Aborot It's because mathematicians seem to dislike thought-revealing names ;J
Same here :( . I have tried this course 3 times and I feel bad for don't get finish. It's like there's a abyss between the professors and their alumns
At 0:12, S sub n equals the group of permutations on the set { 1,2,3,...,n}. If A is a set of some other n elements, that symmetric groups is called S sub A.
I read several websites to find the explanation and this one is the best.
This teacher is so great, and clear.
May I request you videos on quotient group and normal subgroup. I search all the titles of your videos but can't find these two topics. If you discuss these topics in your videos, kindly give me the title of the video. Thanks for the works you did.
This makes so much sense connecting S3 back to the triangles!
Test tomorrow. Binging your videos.
Good luck!! 💜🦉
good explanation ...keep going 💕💕
I have been binge-watching these amazing and illuminating videos since early this morning. The only thing I can say is: “Factori-aw yeah!”
I really like your videos. It clears the complete concept.
Wow! What a lucid beautiful illustration of terminologies...
5:20 every finite g is
i would like to thank you as i needed only the basic info but most channels were discussing about useless things.thx a lot
Great video, just heads up that the example for all the permutations for {1,2,3} is partially incorrect. the cycle (1 2 3) is the same cycle as (312) since 1 still maps to the 2, which maps to 3, which maps back to 1. is just that the cycle is written in different order. Therefore half of the permutations are just duplicates. The permutations you are missing out are: (1)(23) where 1 maps to 1, but 2,3 are cycles. and (2)(13) and (3)(12).
The example is not incorrect in the slightest. The issue is that there are multiple ways to represent permutations (elements of S_n). Here, you were expecting cycle notation, which is pretty much the only notation this video did not cover.
Initially, one thinks of permutations as reordering the elements of a list/sequence.
The sequence 1 2 3 can be reordered in the six ways shown on screen 0:41.
Later, the video introduces the way to think about permutations as functions. The sequence 2 3 1, for example, is the same thing as having the function f(1) = 2, f(2) = 3, and f(3) = 1. This is then used to develop the "stacked" cycleish notation shown at the end of the video.
The cycle notation you are describing in your comment here is the next step after this. You notice that you can make the notation _even more_ condensed by writing in cycles, rather than keeping track of where each element is sent.
Would it be a good guess that symmetry group S_n is also the group of symmetries of an (n-1)-dimensional simplex?
For S_3 that would mean you get a 2d simplex or triangle, which you get to rotate around a point (0d) through 2d space or around an axis (1d) through 3d space; For S_4 you get a 3d simplex or tetrahedron, which can't really reach all 24 permutations of its vertices in 3d space, but I suspect it could if you let it flip along a plane (2d) through 4d space, and that an m-dimensional simplex could reach all permutations of it vertices if you generalize rotation/flip like this up to flip along (m-1)-dimensional space through (m+1)-dimensional space.
Actually as I was writing this I had another thought, would it be right that if you can generalize rotation/flip like this, then flipping m-dimensional simplex along (m-1)-dimensional space could simply swap any two vertices and any sorting algorithm would then sufficiently prove that you can reach any permutation of vertices using just those swaps?
I've got an intuitive feeling all of this works but don't know so I'd love any thoughts and insights my way!!
Who wants to improve their study skills? We WROTE A BOOK FOR YOU!!
How to Be a Great Student ebook: amzn.to/2Lh3XSP Paperback: amzn.to/3t5jeH3
or read for free when you sign up for Kindle Unlimited: amzn.to/3atr8TJ
hey can you please make a video on permutation group
Pls how did we get the second one of 4,2,1,3 when it is multiplied?
Could you upload video in abstract algebra group action please
your channel is amazing I'm so glad I found it I've been very lost in my course. these are such great clear summaries!!!
God😍😍😍😍mam, your lecture is ....SUPERRRRRRRRR AWESOMEEEEEEEEE.... THANK YOUUUUUUUUU😍😍😍😍😍😍
You should add links from the video in the description for people on mobile devices. Great video btw.
mcmcmcmc4444 Thanks for this really helpful suggestion. We're on it! :)
Done! Thanks for catching that. :)
I caught something else: The video from the description looks familiar..., too familiar. :)
mcmcmcmc4444 YIKES! Clearly more coffee needed here.
Very effective and simple way . Thank you so much . I was totally confused in this topic but after watching this video everything is clear to me about the permutation of Sn .
When I start multiplying the permutations ,
(14)(12)(13)
I am get the answer , (1342)
But it should be (1423) .
Please solve this .
@@davidpal1378 yeah not everything is clear to you about Sn.
Both your answers are wrong. It's (1234).
Love your way, beautifully spoken words, voice amuses my mathematical brain
Can someone explain me to how they got 4 2 13 at 3:56
+Abdulqadir Isaak (Jeilani) -- The right-hand function acts first, then the left-hand function operates on the output of the right-hand function. Hence: 1 --> 1 --> 4 ; 2 --> 3 --> 2 ; 3 --> 4 --> 1 ; 4 --> 2 --> 3.
she multiplied the permutations from right to left like she did with the previous one. God bless.
@@kevinbyrne4538 Thank you.
So the operation of the group of permutations is not the mapping of the permutation but the composition of 2 or more permutations, represented by *. Right?
That is correct!
How is the order mentioned in the video different from the order of elements? I'm confused.
Which subject , can be applied?, for example in physics... etc...
I have a question which hopefully someone at Socratica will answer. Let's look at an element of S_3:
(123)
(312)
Now let's say I have three objects in a row A B C.
My question is this: how does the group element operate on this set?
It seems to me , there are 2 ways of interpreting this. Either (a) the 1st object is moved to the 3rd spot, th 2nd object is moved to the 1st position, etc or (b) it means the 3rd object is put first, the first object is moved to the second spot, etc.
In case (a) ABC becomes BCA.
In case (b) ABC becomes CAB
Which interpretation is correct? I'm thinking in terms of these group elements acting as operators re-arranging objects. I've googled this but haven't found what I'm looking for. Can you help me understand this please? Maybe you could also do a video on this.
It depends on which group action of S_3 are you considering on the set ABC. Both your interpretation is correct up to isomorphism, as long as you're consistent with it.
You are doing a great job! Love your videos.
It's really really helpful
You are amazing teacher.
God bless you.
Thank you for your kind comment. Be well, Socratica Friend! 💜🦉
upper and lower central series in symmetric groups and dihedral groups please mam explain this theorem
I love the dramatic music in the background x>
Is a permutation an isomorphism?
great approach to algebra
Thanks a lot for such nice explanations. The video never get old
Thanks for this video, your explanations are perfect !
YOU SAVED MY LIFE
Since, according to Cayley's theorem, every finite group is equivalent ("isomorphic") to some subgroup of a symmetric group, is it then true that every group is a permutation ? Or can some subgroups of a symmetric group Not be permutations ?
So if we say Sn denotes the group of permutation on G then that n should be order of Group G ?
Narendra Salvi n is the amount of elements,but there are n! Permutations, hence |Sn|=n!
thanks madam for such informative video. great job. further pl show how to find centre of Sn. thanks
Nice content, you make a vedio like calculus , analytical geometry
You would not believe how many times your series on groups saved my ass.
Thank you so fucking much.
Composition series of finite length.
Well explained
After clicking this video, after 3 minutes I realised that I was here for group theory
very clear very understandable and very attractive :) thank you for the video
I do recommand all the students studying philosopher learn the algebra
Thanks
Very nice explanation 👍👍
well-explained, thanks
Thank you for making this video
please tell me about cayley digraph= thank u madam
Lecture about permutabl in group
This is so helpful.
Thanks
We're so glad to hear our videos are helping, Serge! Thanks for watching! :)
symmetry as syntax, symmetry cardinality
Thank you for making this less complicated than it needs to be!
Why is it not called the permutation group?
"Permutation group" is a name given to a larger class of groups than just symmetric groups. A permutation group is a group where every element is a permutation of a fixed set S, and the operation on the elements is composition. All symmetric groups are permutation groups, but not all permutation groups are symmetric groups. For example, any subgroup of a symmetric group is also a permutation group, even though not every subgroup of a symmetric group is itself a symmetric group.
Cayley's Theorem reveals that every group is isomorphic to some permutation group, which is a very nice result.
Please Make a video on Cyclic Group too.
thanks.
We just finished filming a video on cyclic groups! It will be ready ... soonish? :)
Thank you!! :)
I watch these just for my entertainment. I know this stuff already. I'm a PhD student studying algebraic geometry.
EDIT: Im fascinated at how a woman with only a theater degree know algebra so well. Mad respect for her.
Good Lecture!
Thank you so much! you saved my life...
thanks
This video is great ... I learnt those permutations while doing bachelor's but now in my master's course there are symmetries of s3 and V4 and so on represented by geometrical shapes.... Im so confused ... I have my finals coming but have to do that too ... Someone help please.. :(
We need more videos
Symmetric group doesn't have to be finite. Cayley's theorem applies to all groups.
Bunun ispatı nedir
I've never had a mathematician give me a good answer to a simple question -
OK, I have a set of objects {A,B,C} in that order. So I mix them up, let's say to BCA.
How do I write this? Is it
| 1 2 3 |
| 2 3 1 |
or is it
| 1 2 3 |
| 3 1 2 | ?
Can anyone answer this? I would appreciate your help.
ABC maps to BCA, a permutation which sends A to B, B to C and C to A. There are many other permutations possible in this case.
In this case: A maps to B, i.e first element maps to the 2nd, similarly second to third and third to first. So the correct representation would be:
|1 2 3|
|2 3 1|
Which is your first option. The second option which you mentioned sends 1 to 3, i.e. first element to the third element. == A to C.. so ABC will go to CBA, which is not the case here.
What is the name of background music at the start of video ? Great videos btw love the presentations
This was very helpful, thank you!
Your method is very helpful, thanks
Thanks, clear explanation.
This was awesome!!! Thank you :)
/A person with learning disability
4:57 groups are helpful for studying groups
That is excellent! Thank you.
There is a trap that Algebra Professors use to shave off points. Some Algebra books multiply from Left to Right. In the next Semester if you are NOT Careful, Multiplication can be switched to Right to Left. Make SURE You Discuss This Before You Get Burned By It!!!
Very helpful
thank you madam.......
Omg u r a queen i love u T_T thanks to this, i finally understand. I cant believe i couldnt solve such simple problems before dammit! I wish i found these earlier DX subscribing now
The notation you use looks like matrix notation and I think you can even use matrix math to do what you're doing too
Of course. They are permutation matrices.
You may not know this but you're literally crying out representation theory.
This is great!!
Thank you ma'am
The music gives me Twin Peaks-vibes.
Our favourite!! 💜🦉
Thank you so much.
Thank you.
I can think of another reason to study symmetry groups: it's fun.
It is indeed. It can be coded easily.
Awesome 😍😍😍😍😍
nice video
Thank you 🙏💕
Спасибо большое!!!
this is freaking awesome dude!
Is this a real person or an AI generated bot?
liliana de castro