Chapter 7: Group actions, symmetric group and Cayley’s theorem | Essence of Group Theory

[@mathemaniac]

This time, when I say "a few days", it really means a short time. It should be exactly 1 week from now, which is way more frequent than my recent schedule... will strive to make more videos in the summer! Also, log your math levels on the Google form: forms.gle/QJ29hocF9uQAyZyH6 This will help me make better videos by making them more catered for the viewers!

[@samuelefraizzoli1070]

I'm so glad and grateful that there is such wonderful material on the Internet. All your videos are of exceptional quality and a source of thoughtful inspiration, even if the topic is more or less known (I'm a physicist). Best regards Samuele

[@12jgy]

It's kinda sad that everything at some point eventually ends, but putting that aside, this was a very interesting series, for me at least, it motived myself to actually start studying abstract algebra, so thanks for that! As for suggestions, I would love seeing something like a "'Essence of Ring Theory" as a follow-up for this series, I personally love ring, even more than group, so I'd like to see your take on them.

[@mathemaniac]

Ring theory is more "algebraic" in the sense that it is much less visualisable than group theory, but I might find a way. I am not too sure whether I could make a unique enough video though. Anyway, thanks for enjoying the video series! Glad to see I motivated someone to study abstract algebra!

[@12jgy]

@@mathemaniac Ah, fair enough, the only way I personally could think of doing any sort of visualization for rings would be as endomorphisms of abelian groups(R-modules, kinda of analogous to what you did with groups using group actions) or something using Cayley diagrams(I've seen them used to help visualize fields, so I would hope you could extend it to general rings, though I'm not sure if one could do that). As for another topic that you might be able to cover, some topology might be nice, specially algebraic topology, or maybe some differential geometry, stuff like differential forms, Stoke's Theorem, Riemannian manifolds and some tensor stuff.

[@mathemaniac]

I will try, but please also understand that I want to make these videos a bit more accessible, so if I can find a way to introduce the topics that you mentioned in a way that an amateur can understand, and does not oversimplify, I will make videos on those topics. Tensors may be easier to visualise, and the concept easier to grasp, so I will consider that topic first.

[@12jgy]

​@@mathemaniac It's fine, don't fell pressured to make any of this, just throwing some ideas on the wall to see if there's anything you can use, if it works, great, if it doesn't, oh well, what can we do. Anyway, thanks a lot for your attention,looking forward to the next videos (That one on Covid2019 using stochastic processes sounds very interresting!)

[@PunmasterSTP]

Yes, I would love to see something like that as well!

[@michellehobdari8920]

Thank you so much for this series, the animations were very clear and well explained - it saved my essay!

[@mathemaniac]

Glad to help!

[@PunmasterSTP]

I'm just curious, what was your essay about, and how'd the rest of the class go?

[@gooomaaal]

very insightful. for first time I can distinguish between symmetry and permutation.

[@mathemaniac]

Glad to help!

[@PunmasterSTP]

Yeah that really cleared things up for me as well.

[@hamzakeurti8203]

This series was so enjoyable to watch, any chance you would elaborate on group actions on some future chapter? And relationship to manifolds?

[@mathemaniac]

Thanks for the compliment. Currently no plan to expand the series, since I am concentrating on the series on complex analysis, and also I want to be able to have some sort of "unique" insight into the topics I cover, and I just don't have that for any topics beyond this other than those very close to a normal lecture / textbook, which is why I stopped making more videos in this series in the first place.

[@nainamat6861]

Your explanations in the videos are excellent ! Plzz make videos on types of group actions 😊

[@mathemaniac]

Thanks! I do consider the group theory series sort of finished - but never say never, maybe I can find another topic in group theory worth talking about and animating.

[@JoJoModding]

The symmetric group actually is a symmetry, namely all the ways you can permute the variables of symmetric functions in n variables while preserving the function value.

[@mathemaniac]

Perhaps I should say symmetric group has nothing to do with "isometry", which is the sort of symmetry that we have been looking at throughout this series.

[@hyperduality2838]

The image (co-domain) is a copy, equivalent or dual to the factor group (domain) - the 1st isomorphism theorem. Isomorphism is dual to homomorphism. Injective is dual to surjective synthesizes bijective or isomorphism.

[@barbietripping]

1:35 thank you for reassuring my understanding regarding swapping the vertices. Is there a link to topology to be made here, since the resulting figure has a different number of regions?

[@mathemaniac]

It does not have anything to do with the topology, and I am only saying that to justify that not all permutations of the vertices are symmetries, but there really is just 1 (path-)connected region, which only becomes 2 (path-)connected regions when you remove that point in the middle. It might be a good counting problem to see how many resulting regions there are for different permutations, but that's not really the focus of the video.

[@barbietripping]

@@mathemaniac Thank you!

[@noahsegal2805]

Would it be possible to make a video explaining the Abel-Ruffini theorem?

[@mathemaniac]

It is quite a stretch (an understatement) from the group theory series, especially since I haven't even talked about alternating groups on this channel. But this suggestion is surely on my idea list.

[@PunmasterSTP]

@@mathemaniac I think if you put your mind to it, you'd definitely be...Abel to do it. 😎

[@pabdavinchi]

Thank you!

[@tac0519]

Thank you for your Video. In Japan, there are few books which has a lot of pictures because the price should be about 2,000 to 3,000 yen, the students can buy it. They usually take class of Math. So, it is not good for me to study by myself. I don't know why they sell the kind of books in store but only in university. Anyway, picture is important to understand. Thank you.

[@toaj868]

Is the homomorphism from the group of symmetries to the symmetric group always an injection?

[@mathemaniac]

I assume you are referring to the homomorphism induced by a group action? It depends on what you are acting on: like here, if we are talking about the set of vertices, then sure - the homomorphism is indeed an injection. If we are instead talking about the action of the rotational group of symmetries of octagon (C_8) on PAIRS of OPPOSITE VERTICES, then the homomorphism is not an injection - the 180 degree rotation would also be in the kernel. If you are referring to the canonical homomorphism as in Cayley's theorem, then yes, the homomorphism is ALWAYS an injection, because as said in the video, the action of g maps the identity to g; the action of h maps the identity to h, so as long as g is not h, then they have different actions!

[@yBazo82]

this is good

[@matteobenati6917]

Just beautiful

[@depressedguy9467]

Nice but what above kernal

[@LemoUtan]

How might one go about animating essences of this thing called 'group character' I wonder?

[@manikantasripathi755]

HISS, please reduce it...