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.
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 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.
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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 .
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 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.
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!!
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 ?
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
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.
"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.
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!!!
+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.
