From my analytics, 86.2% of viewers watching are not subscribed, so if you did enjoy the video, do consider subscribing! I might not upload as frequently, but hopefully every upload is a good one! Do consider logging your math levels here: forms.gle/QJ29hocF9uQAyZyH6, because it helps me cater for your math levels when making these videos!
This is very nice and clear presentation In this presentation, you are able to address like 60% of group theory topics. This is very nice applied presentation.
Very intuitive. Very brilliant. I just couldn't stop watching. It would be nice to see what you could say about sylow theorems, and about real world applications of group theory in other fields of study, or every day life problems (engineering, cryptography, rubiks cube, etc).
Thanks for the compliment! I generally want to divert from a more traditional approach, and since I can't think of a way to intuitively understand Sylow theorems just yet (not in a way significantly different from the traditional proofs), but when I could find a way, I might make a video about it. The other suggestions are noted as well.
That was actually a quite nice way to Illustrate this fact. To be Honert ist was nothing new, since its not to hard to prove the simplicity of A_5 algebraicaly. But its always nice to see your results properly illustrated.
Well, it isn't actually that easy to prove that A5 is isomorphic to the group of rotational symmetry of icosahedron in the first place. So if we adopt the algebraic definition of A5, then the algebraic way of proving simplicity is a simpler route; but of course, this is a nice demonstration of simplicity of A5, if you accept that the two groups are isomorphic.
@@mathemaniac Thats true. The only way to show the isomorphie with which I can come up quickly (havent thought longer than a few seconds but still) is the fact that each simple group with 60 Elements is isomorphic to A_5 :D . But I supose you see the problem with that :P
@@johanneskunz9096 The much more direct way of seeing isomorphism between the two groups is letting A5 act on five regular tetrahedra with vertices being the centres of the faces of the icosahedron. But of course, it is still not very easy to see the connection there.
A 72-degree rotation can't magically become a 144 degree rotation by rotating your perspective (which is what we are doing in considering conjugacy classes), but for the purpose of proving simplicity, sure!
Excellent video. A doubt remains: are the symmetry groups of the faces, the edges and the vertices related somehow to the symmetry group of the icosahedron? I've learned with this video that they are not subgroups, but their symmetries are all among the symmetries of the icosahedron. If so, how to call this relationship? Or is this a property not of the groups themselves, but of the geometrical realization of it? Thanks
I am not sure what you mean by symmetry groups of faces. Do you mean the stabilisers of faces, i.e. symmetries fixing a particular face. Those are subgroups because composing any two symmetries that fix face A still fixes face A, the inverses of the symmetries fix A, and the identity of course fix A. I guess what you meant is the conjugacy classes which are coincidentally *collections of stabilisers of faces/edges/vertices*. Once you consider a collection of subgroups, it might not be a subgroup anymore.
From my analytics, 86.2% of viewers watching are not subscribed, so if you did enjoy the video, do consider subscribing! I might not upload as frequently, but hopefully every upload is a good one!
Do consider logging your math levels here: forms.gle/QJ29hocF9uQAyZyH6, because it helps me cater for your math levels when making these videos!
This is very nice and clear presentation
In this presentation, you are able to address like 60% of group theory topics. This is very nice applied presentation.
Thanks for the appreciation!
Very intuitive. Very brilliant. I just couldn't stop watching. It would be nice to see what you could say about sylow theorems, and about real world applications of group theory in other fields of study, or every day life problems (engineering, cryptography, rubiks cube, etc).
Thanks for the compliment! I generally want to divert from a more traditional approach, and since I can't think of a way to intuitively understand Sylow theorems just yet (not in a way significantly different from the traditional proofs), but when I could find a way, I might make a video about it. The other suggestions are noted as well.
@@mathemaniac thanks for your efforts to make it more intuitive :))
That was actually a quite nice way to Illustrate this fact. To be Honert ist was nothing new, since its not to hard to prove the simplicity of A_5 algebraicaly. But its always nice to see your results properly illustrated.
Well, it isn't actually that easy to prove that A5 is isomorphic to the group of rotational symmetry of icosahedron in the first place. So if we adopt the algebraic definition of A5, then the algebraic way of proving simplicity is a simpler route; but of course, this is a nice demonstration of simplicity of A5, if you accept that the two groups are isomorphic.
@@mathemaniac Thats true. The only way to show the isomorphie with which I can come up quickly (havent thought longer than a few seconds but still) is the fact that each simple group with 60 Elements is isomorphic to A_5 :D . But I supose you see the problem with that :P
@@johanneskunz9096 The much more direct way of seeing isomorphism between the two groups is letting A5 act on five regular tetrahedra with vertices being the centres of the faces of the icosahedron. But of course, it is still not very easy to see the connection there.
Rotational symmetry of Isosahedron category 5:38
Thank you so much for covering these things in an excellent way
You're welcome! Hope that it helps!
Thanks so much for your sharing. Love your channel very much.
Thanks so much for the kind words!
This comment is for the algorithm to boost this channel up!
Thanks so much for the appreciation!
I watch some video from your channel time to time just to reach inner balance and sense that everything is just perfect :)
You are very right. The icosahedral group is indeed perfect. Henri Poincaré missed this fact by the way.
Wonderful Series
I love it ❤️
Thanks!
Thank you!! I needed that video for my math class
Glad that it helps!
I have a hw problem of representation of A5 and it was hard for me to visualize icosahedron. Thank you.
Excellent video!
Glad you liked it!
8:10 you can also merge the 72- and 144-degree rotations.
A 72-degree rotation can't magically become a 144 degree rotation by rotating your perspective (which is what we are doing in considering conjugacy classes), but for the purpose of proving simplicity, sure!
Very Excellent.
Thanks!
It would be fun to ask students to find classes and symmetries and then tell them that all the other answers they wrote are actually correct.
Fantastic
Excellent video.
A doubt remains: are the symmetry groups of the faces, the edges and the vertices related somehow to the symmetry group of the icosahedron? I've learned with this video that they are not subgroups, but their symmetries are all among the symmetries of the icosahedron. If so, how to call this relationship? Or is this a property not of the groups themselves, but of the geometrical realization of it? Thanks
I am not sure what you mean by symmetry groups of faces. Do you mean the stabilisers of faces, i.e. symmetries fixing a particular face. Those are subgroups because composing any two symmetries that fix face A still fixes face A, the inverses of the symmetries fix A, and the identity of course fix A.
I guess what you meant is the conjugacy classes which are coincidentally *collections of stabilisers of faces/edges/vertices*. Once you consider a collection of subgroups, it might not be a subgroup anymore.
So you are saying that there is no general quintic???!!!
0:30 Orbit stabilizer
Is your native language Cantonese by any chance?
Golden
My boy using the premiere function
I simply want to test this out, and see what effect it has on my video performance haha.
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