Symmetric Groups (Abstract Algebra)

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  • Опубліковано 24 гру 2024

КОМЕНТАРІ • 155

  • @Socratica
    @Socratica  2 роки тому +4

    Sign up to our email list to be notified when we release more Abstract Algebra content: snu.socratica.com/abstract-algebra

  • @Socratica
    @Socratica  10 років тому +61

    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

    • @twilightsparkle6756
      @twilightsparkle6756 9 років тому

      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.

    • @davidpal1378
      @davidpal1378 7 років тому

      Socratica please solve my query . I have already mentioned my query in the comments

    • @davidpal1378
      @davidpal1378 7 років тому

      Socratica please make a video on transposition too .
      And relationship between transposition and permutation

    • @sreejaps2428
      @sreejaps2428 6 років тому

      Mam pls do a video on permutation group

    • @Grassmpl
      @Grassmpl 3 роки тому

      I should find an open cover for this notation and ask you to give me a finite subcover.

  • @anshul9856
    @anshul9856 5 років тому +108

    She sounds like the woman on a screen from a video game where you're stranded on an island in 2050

  • @AbhijitDas-wg8mb
    @AbhijitDas-wg8mb 6 років тому +102

    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.

    • @Socratica
      @Socratica  6 років тому +13

      Ahhhh good luck!!!! :D

  • @ifathameedshora7063
    @ifathameedshora7063 5 років тому +17

    U cant imagine how ur videos are helping me in understanding this abstract algebra which was like ghost stuff to me

  • @nothingisimpossible.1357
    @nothingisimpossible.1357 4 роки тому +7

    Clarity not confusion to key of success
    Your lec is just amazing for me

  • @twilightsparkle6756
    @twilightsparkle6756 9 років тому +20

    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.

    • @Grassmpl
      @Grassmpl 3 роки тому

      There needs to be a video on representation theory

  • @bonbonpony
    @bonbonpony 9 років тому +39

    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.

  • @TobbyKEM
    @TobbyKEM Рік тому +4

    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!.

    • @Socratica
      @Socratica  Рік тому +1

      We're rooting for you!! 💜🦉

  • @rishabhnegi1937
    @rishabhnegi1937 3 роки тому

    the best simplified explanation we can get on internet

  • @JeffreyAborot
    @JeffreyAborot 9 років тому +25

    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.

    • @Socratica
      @Socratica  9 років тому +7

      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.

    • @bonbonpony
      @bonbonpony 9 років тому +1

      Jeffrey Aborot It's because mathematicians seem to dislike thought-revealing names ;J

    • @10MANOEL
      @10MANOEL 5 років тому

      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

  • @markmathman
    @markmathman 6 років тому

    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.

  • @devins9402
    @devins9402 4 роки тому

    I read several websites to find the explanation and this one is the best.

  • @ponho9715
    @ponho9715 3 роки тому +1

    This teacher is so great, and clear.

  • @bhupendrosalam6037
    @bhupendrosalam6037 6 років тому +5

    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.

  • @pittdancer85
    @pittdancer85 Рік тому

    This makes so much sense connecting S3 back to the triangles!

  • @nujranujranujra
    @nujranujranujra 4 роки тому +3

    Test tomorrow. Binging your videos.

    • @Socratica
      @Socratica  4 роки тому +1

      Good luck!! 💜🦉

  • @avs4838
    @avs4838 2 роки тому +1

    good explanation ...keep going 💕💕

  • @PunmasterSTP
    @PunmasterSTP 3 роки тому +1

    I have been binge-watching these amazing and illuminating videos since early this morning. The only thing I can say is: “Factori-aw yeah!”

  • @ZeeshanShah-ur7fd
    @ZeeshanShah-ur7fd 3 місяці тому

    I really like your videos. It clears the complete concept.

  • @WahidBakhshSumalani
    @WahidBakhshSumalani 3 роки тому

    Wow! What a lucid beautiful illustration of terminologies...

  • @SphereofTime
    @SphereofTime 4 місяці тому +1

    5:20 every finite g is

  • @avinashbhatia3303
    @avinashbhatia3303 4 роки тому

    i would like to thank you as i needed only the basic info but most channels were discussing about useless things.thx a lot

  • @harrywang6792
    @harrywang6792 2 роки тому +1

    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).

    • @MuffinsAPlenty
      @MuffinsAPlenty 2 роки тому +1

      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.

  • @descuddlebat
    @descuddlebat 4 роки тому +1

    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!!

  • @Socratica
    @Socratica  3 роки тому

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  • @rajdeepsaha439
    @rajdeepsaha439 6 років тому +1

    hey can you please make a video on permutation group

  • @tosinpedro16
    @tosinpedro16 10 місяців тому

    Pls how did we get the second one of 4,2,1,3 when it is multiplied?

  • @GunanesanSingarajah-jj1jb
    @GunanesanSingarajah-jj1jb Рік тому

    Could you upload video in abstract algebra group action please

  • @sammie1824
    @sammie1824 Рік тому

    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!!!

  • @StyleReDesign
    @StyleReDesign 5 років тому +8

    God😍😍😍😍mam, your lecture is ....SUPERRRRRRRRR AWESOMEEEEEEEEE.... THANK YOUUUUUUUUU😍😍😍😍😍😍

  • @mc4444
    @mc4444 10 років тому +1

    You should add links from the video in the description for people on mobile devices. Great video btw.

    • @Socratica
      @Socratica  10 років тому

      mcmcmcmc4444 Thanks for this really helpful suggestion. We're on it! :)

    • @Socratica
      @Socratica  10 років тому

      Done! Thanks for catching that. :)

    • @mc4444
      @mc4444 10 років тому +1

      I caught something else: The video from the description looks familiar..., too familiar. :)

    • @Socratica
      @Socratica  10 років тому

      mcmcmcmc4444 YIKES! Clearly more coffee needed here.

  • @davidpal1378
    @davidpal1378 7 років тому +1

    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 .

    • @davidpal1378
      @davidpal1378 7 років тому

      When I start multiplying the permutations ,
      (14)(12)(13)
      I am get the answer , (1342)
      But it should be (1423) .
      Please solve this .

    • @Grassmpl
      @Grassmpl 3 роки тому

      @@davidpal1378 yeah not everything is clear to you about Sn.
      Both your answers are wrong. It's (1234).

  • @sherazahmed1677
    @sherazahmed1677 Рік тому

    Love your way, beautifully spoken words, voice amuses my mathematical brain

  • @Jannahlifeskills
    @Jannahlifeskills 9 років тому +1

    Can someone explain me to how they got 4 2 13 at 3:56

    • @kevinbyrne4538
      @kevinbyrne4538 9 років тому +1

      +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.

    • @MoyoSore
      @MoyoSore 8 років тому

      she multiplied the permutations from right to left like she did with the previous one. God bless.

    • @gibrilladumbuya847
      @gibrilladumbuya847 3 роки тому +1

      @@kevinbyrne4538 Thank you.

  • @dpheneghan2
    @dpheneghan2 6 років тому

    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?

  • @rohanv609
    @rohanv609 6 років тому

    How is the order mentioned in the video different from the order of elements? I'm confused.

  • @germ1saba
    @germ1saba Рік тому

    Which subject , can be applied?, for example in physics... etc...

  • @scientiaetveritas40
    @scientiaetveritas40 6 років тому

    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.

    • @Grassmpl
      @Grassmpl 3 роки тому

      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.

  • @arpitnama420
    @arpitnama420 6 років тому +2

    You are doing a great job! Love your videos.

  • @ugaleadinath
    @ugaleadinath 3 роки тому +1

    It's really really helpful
    You are amazing teacher.
    God bless you.

    • @Socratica
      @Socratica  3 роки тому

      Thank you for your kind comment. Be well, Socratica Friend! 💜🦉

  • @benishmushtaq820
    @benishmushtaq820 4 роки тому

    upper and lower central series in symmetric groups and dihedral groups please mam explain this theorem

  • @Helena-fl9ff
    @Helena-fl9ff 3 роки тому

    I love the dramatic music in the background x>

  • @stephaniesheehan1026
    @stephaniesheehan1026 4 роки тому

    Is a permutation an isomorphism?

  • @143mathematics
    @143mathematics 7 років тому

    great approach to algebra

  • @oyaoya2468
    @oyaoya2468 2 роки тому

    Thanks a lot for such nice explanations. The video never get old

  • @radicalfox4416
    @radicalfox4416 Рік тому

    Thanks for this video, your explanations are perfect !

  • @deenice
    @deenice 7 років тому +1

    YOU SAVED MY LIFE

  • @kevinbyrne4538
    @kevinbyrne4538 9 років тому

    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 ?

  • @sunitasingh-rj6jq
    @sunitasingh-rj6jq 6 років тому

    So if we say Sn denotes the group of permutation on G then that n should be order of Group G ?

    • @ttttt_
      @ttttt_ 6 років тому

      Narendra Salvi n is the amount of elements,but there are n! Permutations, hence |Sn|=n!

  • @deepakmirchandani1348
    @deepakmirchandani1348 4 роки тому

    thanks madam for such informative video. great job. further pl show how to find centre of Sn. thanks

  • @niteshmishra5416
    @niteshmishra5416 3 роки тому

    Nice content, you make a vedio like calculus , analytical geometry

  • @charlesbrunelle
    @charlesbrunelle 4 роки тому

    You would not believe how many times your series on groups saved my ass.
    Thank you so fucking much.

    • @Grassmpl
      @Grassmpl 3 роки тому

      Composition series of finite length.

  • @miriamborja263
    @miriamborja263 4 роки тому +1

    Well explained

  • @vivekvaghela2274
    @vivekvaghela2274 2 роки тому

    After clicking this video, after 3 minutes I realised that I was here for group theory

  • @pashaATX
    @pashaATX 8 років тому +1

    very clear very understandable and very attractive :) thank you for the video

  • @chynxyune3546
    @chynxyune3546 4 роки тому

    I do recommand all the students studying philosopher learn the algebra

  • @rakhisiwach1611
    @rakhisiwach1611 3 роки тому

    Thanks
    Very nice explanation 👍👍

  • @d-shiri
    @d-shiri 6 років тому +2

    well-explained, thanks

  • @vijayshankar102
    @vijayshankar102 3 роки тому

    Thank you for making this video

  • @karimkhan1312
    @karimkhan1312 7 років тому

    please tell me about cayley digraph= thank u madam

  • @ghulamali925
    @ghulamali925 4 роки тому

    Lecture about permutabl in group

  • @sergebyusajabo2138
    @sergebyusajabo2138 8 років тому +2

    This is so helpful.
    Thanks

    • @Socratica
      @Socratica  8 років тому

      We're so glad to hear our videos are helping, Serge! Thanks for watching! :)

  • @veixquadron
    @veixquadron Рік тому

    symmetry as syntax, symmetry cardinality

  • @eleazaralmazan4089
    @eleazaralmazan4089 6 років тому +1

    Thank you for making this less complicated than it needs to be!

  • @learningsuper6785
    @learningsuper6785 7 років тому

    Why is it not called the permutation group?

    • @MuffinsAPlenty
      @MuffinsAPlenty 7 років тому

      "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.

  • @SameekshaBodh27
    @SameekshaBodh27 8 років тому

    Please Make a video on Cyclic Group too.
    thanks.

    • @Socratica
      @Socratica  8 років тому +4

      We just finished filming a video on cyclic groups! It will be ready ... soonish? :)

    • @SameekshaBodh27
      @SameekshaBodh27 8 років тому

      Thank you!! :)

  • @Grassmpl
    @Grassmpl 3 роки тому +3

    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.

  • @markmathman
    @markmathman 6 років тому

    Good Lecture!

  • @sichenwang9994
    @sichenwang9994 6 років тому

    Thank you so much! you saved my life...

  • @moularaoul643
    @moularaoul643 2 роки тому +1

    thanks

  • @mirat9155
    @mirat9155 9 років тому

    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.. :(

  • @sasi0505
    @sasi0505 5 років тому

    We need more videos

  • @Grassmpl
    @Grassmpl 3 роки тому

    Symmetric group doesn't have to be finite. Cayley's theorem applies to all groups.

  • @fatmaboyal9789
    @fatmaboyal9789 3 роки тому

    Bunun ispatı nedir

  • @scientiaetveritas40
    @scientiaetveritas40 6 років тому

    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.

    • @TrueZenquiorra
      @TrueZenquiorra 5 років тому

      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.

  • @lovishjain5002
    @lovishjain5002 7 років тому

    What is the name of background music at the start of video ? Great videos btw love the presentations

  • @awesomewinter3103
    @awesomewinter3103 8 років тому

    This was very helpful, thank you!

  • @Anujkumar-fy8bo
    @Anujkumar-fy8bo 6 років тому +1

    Your method is very helpful, thanks

  • @MrZeeg617
    @MrZeeg617 9 років тому

    Thanks, clear explanation.

  • @mijaelrodriguezsaavedra2615
    @mijaelrodriguezsaavedra2615 9 років тому

    This was awesome!!! Thank you :)
    /A person with learning disability

  • @niallransford7647
    @niallransford7647 3 роки тому

    4:57 groups are helpful for studying groups

  • @RalphDratman
    @RalphDratman 5 років тому

    That is excellent! Thank you.

  • @claytonbenignus4688
    @claytonbenignus4688 10 місяців тому

    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!!!

  • @mathwithaliraza4465
    @mathwithaliraza4465 4 роки тому

    Very helpful

  • @saurabhsingh-ow7ue
    @saurabhsingh-ow7ue 4 роки тому

    thank you madam.......

  • @imppie3754
    @imppie3754 4 роки тому

    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

  • @jack002tuber
    @jack002tuber 8 років тому

    The notation you use looks like matrix notation and I think you can even use matrix math to do what you're doing too

    • @Grassmpl
      @Grassmpl 3 роки тому +1

      Of course. They are permutation matrices.
      You may not know this but you're literally crying out representation theory.

  • @bikingaround7429
    @bikingaround7429 3 роки тому

    This is great!!

  • @ritikanegi1604
    @ritikanegi1604 7 років тому

    Thank you ma'am

  • @magnushelliesen
    @magnushelliesen 5 років тому +1

    The music gives me Twin Peaks-vibes.

    • @Socratica
      @Socratica  3 роки тому

      Our favourite!! 💜🦉

  • @kunslipper
    @kunslipper 7 років тому

    Thank you so much.

  • @smithsmitherson9449
    @smithsmitherson9449 8 років тому +1

    Thank you.

  • @notoriouswhitemoth
    @notoriouswhitemoth 9 років тому

    I can think of another reason to study symmetry groups: it's fun.

    • @Grassmpl
      @Grassmpl 3 роки тому

      It is indeed. It can be coded easily.

  • @StyleReDesign
    @StyleReDesign 5 років тому

    Awesome 😍😍😍😍😍

  • @143mathematics
    @143mathematics 7 років тому

    nice video

  • @adarshdwivedishailendra770
    @adarshdwivedishailendra770 5 років тому

    Thank you 🙏💕

  • @katya08
    @katya08 5 років тому +1

    Спасибо большое!!!

  • @EclipZeMuzik
    @EclipZeMuzik 6 років тому +1

    this is freaking awesome dude!

  • @ssvemuri
    @ssvemuri 2 роки тому +1

    Is this a real person or an AI generated bot?