Chapter 4: Conjugation, normal subgroups and simple groups | Essence of Group Theory

A VERY important concept of group theory, but often taught without any intuition, so much that it often confuses a lot of people when they first learned it (including me). Conjugation can be visualised easily with a (literal) change of perspective! This video also lays the foundation for quotient groups, which gives rise to some unexpected relationship with number theory.

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#mathemaniac #math #conjugation #grouptheory #simplegroups #normal #abstractalgebra

[mathemaniac]

EDIT: ghg^(-1) being inside H does not automatically imply gHg^(-1) = H, but we also need the other way around, i.e. for all h in H, h can be expressed gh*g^(-1) for some h* in H. There isn't any issue with that if we are proving that H is a normal subgroup though (think about why).

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[marcocecchi]

I love this series!
I found 3b1b's essence of linear algebra really useful and i'm so happy someone else is using this kind of format on more arguments.
Keep up the good work

[blinded]

This helped me finalize my understanding of geometric algebra. Thank you from the bottom of my heart.

[WahranRai]

4:19 Put label / number on vertex hexagon to illustrate the conjiugaison etc...

[MRKS]

You’re good. Great animations. I’d focus more on motivation, narrative arcs. Just an idea. Keep going.

[reimannx]

Wonderful geometric insights beyond the algebriac structural relationships.

[_P_a_o_l_o_]

At 9:32, what is the element g, explictly? Under such symmetry g, the group has changed completely (in fact it now acts on a square instead of a cube). So I don't understand how it would make sense to even compare the subgroups H and gHg^{-1}, since they seem to be subgroups of different groups. I know how to algebraically prove that H is not normal in G, but I don't understand the intuition behind this example. Thanks for anyone who would like to help me.

[JoJoModding]

Hey, thank you for this series. However, I don't quite see why ghg^{-1} \in H \implies gHg^{-1}=H. Equality does not follow trivially, only subset.

[pauselab]

I think the most intuitive way to understand what a normal subgroup is is to think of them as a subgroup H where H acting on elements of g is the same as elements of g acting on H.

[samr]

Thank you so much! I have my first group theory exam tomorrow and this is a huge help! I wish my Professor knew how to animate things hahaha!

[TheTessatje]

Thanks Mathemaniac, this clarifies a lot.
One question: S3 (group of all symmetries of the triangle), how can we find its (normal) subgroup(s)? By Lagrange, the order of |H| \in {1, 2, 3, 6}, with {e} and {e, r, r2, f, rf, r2f} have orders 1 and 6. But for other subsets, e.g. {e, r, f}, why is this (not) a subgroup and why is (not) it normal?

[golden_smaug]

I need to develop more intuition! lol
I got a very square mindset after so many calculus jajaja

[monny]

Fuck me, I needed this when I was studying for my algebra course last year haha, I guess it doesn’t hurt to revise though. I was actually looking for stuff like this very hard on the internet, but I couldn’t obviously find them because well, they weren’t there yet

[sandeepjain]

I know, this might be a dumb question, but can anyone tell me, why anticlockwise rotation is same as clockwise rotation after changing perspective. Thanks in advance!

[hiltonmarquessantana]

Very great!! Many thanks for the video

[PunmasterSTP]

Normal subgroups? More like "No bad videos, yep!" Thanks again for putting together such a wonderful series! You really do explain so many things so much better than anything I've come across before.

[alejrandom]

I first came across this concept when solving rubik's cubes. Sometimes you have a sequence of moves that permutes 3 pieces, but not the 3 pieces that you want to permutes. So you do a setup, sequence, undo setup. This is a conjugation. Then, when I was watching 3b1b's video about diagonalization of a matrix, I realized it's the exact same concept, P^-1 DP. You can also use this to rotate the complex plane around any given point by translating, rotating, undoing translation. I love how group theory brings together all of this.

[paschalcharles]

The big problems in mathematics is lack of clarity

[theboombody]

The point you make around the 4:30 to 5:15 mark is very helpful. Definitely worth you repeating it. Thank you.