Icosahedral symmetry - conjugacy classes and simplicity

How do we prove the rotational symmetries of icosahedron form a simple group? But wait, how do we prove *any* group is simple? The key to that involves the concept of conjugacy classes. This video explains intuitively why a normal subgroup has to be a union of conjugacy classes.

This video is a continuation of the summary of my previous video series, and it is highly recommended that you watch the entire video series before this video, because there are a lot of intuitions developed throughout the video series, like conjugation is simply viewing symmetries in different perspectives. It might not make sense if you have not heard of this intuition of conjugation before.

I haven't had the time to talk about centralisers and centre, which are strongly associated with the concept of conjugacy classes, because these two other concepts are less related to simplicity of the group under consideration. Maybe another video on these two concepts?

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[mathemaniac]

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!

[MrJaffjunior]

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).

[duckymomo]

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.

[johanneskunz]

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.

[Heuristicpohangtomars]

Rotational symmetry of Isosahedron category 5:38

[Alpasonic]

I watch some video from your channel time to time just to reach inner balance and sense that everything is just perfect :)

[余淼-eb]

Thanks so much for your sharing. Love your channel very much.

[aryamanmishra]

I have a hw problem of representation of A5 and it was hard for me to visualize icosahedron. Thank you.

[jimjam]

This comment is for the algorithm to boost this channel up!

[NovaWarrior]

Thank you so much for covering these things in an excellent way

[nicolasperez]

Thank you!! I needed that video for my math class

[sachs]

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

[alejrandom]

So you are saying that there is no general quintic???!!!

[danielsebald]

8:10 you can also merge the 72- and 144-degree rotations.

[tinfoilhomer]

Is your native language Cantonese by any chance?

[피클모아태산]

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.