I hope you enjoyed this brief introduction to group theory and abstract algebra. If you'd like to learn more about undergraduate maths and physics make sure to subscribe!
Hey, thanks for the very positive feedback, that seriously means a lot! Those are channels that I really enjoy and think a lot of. There should be a new video up today as well!...
Damn, now I understood why 3Blue1Brown once said the matrices are transformations. By multiplying with the matrices, we indeed rotate / flip the image around!
Well made! You presented the ideas concretely unlike some other group theory video I came across, and your video definitely helped me understand some key things.
At 6:58, you say we conclude our set is closed by observing every element of our set appears exactly once in each row and column. Why do we need that? Don't we just need every element of the Cayley table to be an element of our set?
He's checking that it satisfies the Latin Square Property (that every element appears once in each column and row), which if true doesn't necessarily tell you it's a group, but if it's not true it tells you that it isn't a group.
Sometimes, a more rigorous definition of "binary operation" is used (S x S --> S) and it implies that groups must be closed. So sometimes you won't see the "closure" property as a group axiom, but rather just "a set with a binary operation, blah blah" with the rest of the axioms
@5:17 These aren't going to be a "y=x' or "y=-x" axes, they need to be at 30°, not 45°. "y=(1/√3)x" and the same with a minus sign would do the trick, if I'm correct. God bless from Poland!
In fact it seems your matrices represent the same transformations. An equilateral triangle at the origin with each side having the length of 1 is going to have its vertices at (0,0), (1,0) and (1/2, √3/2). Looks familiar, doesn't it?
in your definition set S doesn't contain the Identity element I, so how does it satisfy the identity axiom? Shouldn't the underlying set be S:{I,Q,R,Y,X1,X2}? Thanks
When I came to the part with matrices I tried to apply concepts from 3b1b series of linear transformation and kind of realized that each 2D array represent the exact same transformation that he described for the equilateral triangle
I don't quite understand what "mapping" from S to T entails. What kind of connections are valid? I'm a physics student trying to teach myself group theory, so forgive my ignorant question.
Ah maybe I should have been a bit more explicit! A map is basically just a rule that assigns elements of one set to elements of another. Any connections are valid for a general map, but you have to be a bit more particular when you're talking about functions. Hope that helped. P.S. I'm a physicist myself so expect to see some stuff on applications of group theory to physics in the future!
If you're referring to the symmetry group, I can see how you might be mistaken in that the operation is composition of the rotations, which is binary, not the rotations themselves.
The "dot" is supposed to be an arbitrary operation. It doesn't matter what symbol represents it. You could draw a dog's face each time if you wanted to. It's just that mathematicians like to draw an analogy between these arbitrary operations and the traditional multiplication in the real numbers, so they use a "dot" or other "times" symbol. This, however, does not actually mean that one should use multiplication. When he showed that example, the "dot" was standing in for/ substituting the operation of addition. He just wrote "." instead of "+" to be more general. He did this to show that most simple sets unified with simple binary operations, such as addition of real numbers, tend to go "out of bounds". That is, operations which combine two elements of a set must also produce an element of the original set or else the set under the operation cannot form a group. If he used the natural numbers instead of S as his base set, then obviously addition would never have caused the "out of bounds" error / violated the condition of closure.
This chap managed to cram 3 whole chapters of Further Mathematics in A Level in less than 11 minutes. This is what I call QUALITY video!
You managed to make me understand something in 10 min that I couldn't in 70 mins thank you so so much. You're a legend.
This video is amazing. What’s so surprising to me is how often the distributive property shows up in every area of mathematics
wow you've just summarized my professor's two weeks lectures, waiting for future videos on Group theory
You are making really amazing videos, kind of like a mash up of Khan Academy and 3Blue1Brown. Keep it up.
Hey, thanks for the very positive feedback, that seriously means a lot! Those are channels that I really enjoy and think a lot of. There should be a new video up today as well!...
So, who are you. A student, Professor... or Stanford Graduate like 3blue1brown.
Utsav Munendra I'm an undergrad from the UK. I'd kill to be a professor or a Stanford grad though!
@@scienceplease3364 I tried the binary operations of the Calley table @9:01. I didn't get any of those results.
Damn, now I understood why 3Blue1Brown once said the matrices are transformations. By multiplying with the matrices, we indeed rotate / flip the image around!
You are truly amazing.
You should make more videos.
You dont just have knowledge but you can also explain it to the others.
This was so good, it was an college-lecture like introduction with good sound quality and great visual examples! It was exactly what I was looking for
Well made! You presented the ideas concretely unlike some other group theory video I came across, and your video definitely helped me understand some key things.
Thanks! I'm almost certainly going to make some more group theory videos in the future! Really glad the video helped.
At 6:58, you say we conclude our set is closed by observing every element of our set appears exactly once in each row and column. Why do we need that? Don't we just need every element of the Cayley table to be an element of our set?
He's checking that it satisfies the Latin Square Property (that every element appears once in each column and row), which if true doesn't necessarily tell you it's a group, but if it's not true it tells you that it isn't a group.
When you defined X1 and X2, I think you meant to use it as a rotation along axis tilted 30deg from the horizontal and not 45deg...
Sometimes, a more rigorous definition of "binary operation" is used (S x S --> S) and it implies that groups must be closed. So sometimes you won't see the "closure" property as a group axiom, but rather just "a set with a binary operation, blah blah" with the rest of the axioms
@8:32 one can prove that the set of symmetries on an equilateral triangle is associative by using the matrix representations
Welcome back
Please talk about ai somnium
@5:17 These aren't going to be a "y=x' or "y=-x" axes, they need to be at 30°, not 45°. "y=(1/√3)x" and the same with a minus sign would do the trick, if I'm correct.
God bless from Poland!
In fact it seems your matrices represent the same transformations. An equilateral triangle at the origin with each side having the length of 1 is going to have its vertices at (0,0), (1,0) and (1/2, √3/2). Looks familiar, doesn't it?
What a wonderful video! Thank u very much!
Extremely good video
in your definition set S doesn't contain the Identity element I, so how does it satisfy the identity axiom? Shouldn't the underlying set be S:{I,Q,R,Y,X1,X2}? Thanks
Yeah that's correct. Just a typo that hasn't been pointed out thus far. Although I'm pretty sure that con be infered from the context...
Some of the drawings are hard to see, but other than that it's a great video. Thanks for making these.
Thanks for letting me know! I'll try to improve the render quality next time around :)
Science Please, you are using dark colours that blend into black background. Some of your viewers are colour blind, whereas I am night blind.
When I came to the part with matrices I tried to apply concepts from 3b1b series of linear transformation and kind of realized that each 2D array represent the exact same transformation that he described for the equilateral triangle
That was quite obvious, by glancing at those matrices you could tell
Which app/software are you using to make the videos?
Excellent! Where are the next videos?
sounds like data types in programming... this is cool
I think the first example for R had an error; you said R would rotate by 120 but then the diagram actually did so by 240.
He specifies 120° anticlockwise
Should I take real analysis before group theory?
Very good video lecture
I don't quite understand what "mapping" from S to T entails. What kind of connections are valid?
I'm a physics student trying to teach myself group theory, so forgive my ignorant question.
Ah maybe I should have been a bit more explicit! A map is basically just a rule that assigns elements of one set to elements of another. Any connections are valid for a general map, but you have to be a bit more particular when you're talking about functions.
Hope that helped.
P.S. I'm a physicist myself so expect to see some stuff on applications of group theory to physics in the future!
Oops, isn't S at 8:05 missing the identity element?
Thank You.
Sorry for my silly question .you said group need binary operations but transformation you have used are unary operation.please explain.
If you're referring to the symmetry group, I can see how you might be mistaken in that the operation is composition of the rotations, which is binary, not the rotations themselves.
thanks you helped me.
The "dot" is supposed to be an arbitrary operation. It doesn't matter what symbol represents it. You could draw a dog's face each time if you wanted to. It's just that mathematicians like to draw an analogy between these arbitrary operations and the traditional multiplication in the real numbers, so they use a "dot" or other "times" symbol. This, however, does not actually mean that one should use multiplication. When he showed that example, the "dot" was standing in for/ substituting the operation of addition. He just wrote "." instead of "+" to be more general. He did this to show that most simple sets unified with simple binary operations, such as addition of real numbers, tend to go "out of bounds". That is, operations which combine two elements of a set must also produce an element of the original set or else the set under the operation cannot form a group. If he used the natural numbers instead of S as his base set, then obviously addition would never have caused the "out of bounds" error / violated the condition of closure.
@6:28 your missing the identity element
Hey why is that matrix table messed up in the diagonal? Just asking
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Thanks! Took others 30+ minutes to explain same.
You know that there are some _vowels_ you can use sometime? :q
wow that really helped
Thanks :) I've got a lot more group theory videos planned, I'll probably go over the basics in a better way as well.
ভাল
so this channel is dead?
I couldn't watch more than two minutes of this I'm afraid, the speaker was talking far too fast.
Try to speak more clearly.
your introduction is too too too too too difficult.TT
maybe because I am just yr1 in undergraduate