Chapter 2: Orbit-Stabiliser Theorem | Essence of Group Theory

An intuitive explanation of the Orbit-Stabilis(z)er theorem (in the finite case). It emerges very apparently when counting the total number of symmetries in some tricky but easy way. This video series continues to develop your intuition towards some fundamental concepts and results in Group theory.

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[mathemaniac]

There is usually also something called Burnside's lemma closely related to this, but this is usually more important in some arguments when proving results in group theory.

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[DanielKRui]

I've been through 2 graduate algebra courses, and it took until this video to realize the Orbit-Stabilizer theorem was so simple/intuitive: it's just understanding a group action by understanding its inverse: its inverse can be broken down into (1) first move x back to itself (x fixed/chosen before), and then (2) then there's exactly one group action left, that fixes x, and moves everyone else back. Just a simple 2 step process. And fantastic special case/example of the dihedral case! It's a perfect instructive and intuitive animation for the Orbit-Stabilizer theorem.

[shyj]

Thank you very much for this — it’s helped me a lot. I’m currently writing a project for my third year undergraduate mathematics degree on enumeration of 4x4 sudoku grids, and reducing the 288 possible enumerations down using group theory. I knew it reduces down to 2 but I struggled to understand why, and when I was pointed in the direction of the orbit-stabiliser theorem I often struggled to understand it properly. You’ve saved the day. Thank you!

[JeffreyMarshallMilne]

Very well explained. This helped me understand orbit stabilizer theorem A LOT.

[catsii]

Thank you so much!! Been having trouble understanding this section in my uni course but your description of stabilizers finally made everything click for me! Had to pause the video and take a minute and just go "OH MY GOD I GET IT NOW" haha

[KeerthiS-euqs]

the best one....it was so confusing to understand this theorem, this video made it so easy, thanks man.

[cowgomoo]

Awesome video. So glad I found this channel! I don't even have any math background and I am able to understand the series so far. Well explained for sure.

[radharanibhakta]

Thank you so much for making video on orbit stabilizer theorem in such a fashion.

[sheepphic]

I'm not exactly sure how the graphics package works, but at 0:31 the word "structure" is surrounded by two left quotation marks; if it's being rendered by LaTeX, you'll want to type it as ``structure'' instead (with the last two being apostrophes) to get the quotation marks to face the right way

[nervous]

Not sure if you're still replying for this video, but this is my first time studying group theory, and I'm gonna try to use my own words to summarize to see if I understand it correctly.

1. There are only 2 kinds of action. It either moves x, or it doesn't.
2. For every element x in X, there are certain amount of actions that uniquely change the x's position to different places, and there also are unique actions that not change x's position at all.
3. And any one of the former combined with the latter is also a unique action in G.
4. Hence, the product of the actions that would uniquely change x to different places and the unique actions that would not change x at all is all the actions you have in G.

I don't know of any math people can answer this question for me. Any feedback would be much appreciated, thanks ahead.🙏

[abhishekjoshi]

You sir, now have a loyal subscriber. GReat work

[pauselab]

so symmetry groups Aut(X), card(X)=n is using the orbit stabilizer multiple time where fixing an element is n and the orbit is the same question but of n-1. we get the relation |X_n| = n*|X_n-1| which is the definition of a factorial. for a proof, the orbit stabilizer clearly reaches every possible element of G as every permutation (multiplication by g in G) has to put x somewhere and then count what can happen to the other elements. It can't overcount neither because all permutations are different depending on where x is and the stabilizer.

[gbeziuk]

It's very inspiring to see the productive followers of the great 3blue1brown.

[parsimoniousdialog]

Certainly my favo urite math channel. Love u mathemaniac. icey =]

[zokalyx]

I have a question! When describing the orbit of the vertex 1 of the octagon, it seemed like we only considered rotations. But I can think of reflections that also fix the vertex! Why aren't these considered?

EDIT: I think my question boils down to: What are r_2, r_3, ..., r_n? There are multiple ways that move, for example, vertex 2 to vertex 1.

[BlueBeBlue]

In 10:55 you said this is only true for finite groups, so the orbit-stabiliser theorem doesn't work for circles, cause their |org_g(x)| is infinite, right? So my question is, can a infinite group have finite orbits?

[LastvanLichtenGlorie]

During the "translation to the right by 2 units" the number line went to the left. Is it just me? I apologize if it is just me. Maybe I'm having a dyslexic moment. By the way, excellent video and thank you very much for this production.

[meetjoshi]

If x goes to the same vertex through 2 different symmetries, then are those symmetries equal? I know this is a stupid and naïve question but it would be great if you replied. Thanks in advance!

[astroceleste]

could you not put text in the same space where go the captions? it is very annoying

[LucaFanciullini]

The animation are cool but he needs to be better explained. First of all, what do you mean by "fixing"? Couldn't I just do another symmetry operation and move 1?