Group theory 2: Cayley's theorem

This is lecture 2 of an online mathematics course on group theory. It describes Cayley's theorem that every abstract group is the group of symmetries of something, and as examples shows the Cayley graphs of the Klein 4-group and the symmetric group on 3 points.

[lellosbarello]

Professor I am deeply grateful for these lessons. Having an introduction to group theory from such a prestigious mathematician is an honour and a little miracle of these modern times. Kindest regards from an electronic engineering student from Italy.

[thehopsful]

You have done a great service to humanity with these lectures, professor. Thank you very much.

[lukephillips]

Wow! I read about Caley's theorem in a textbook but I really had no idea what I was reading about. This lecture spells it out so well!

[rezaasad]

Thank you for these lessons professor. This is a great gift to humanity during the tough times of pandemic :)

[ManavAgarwal-nu]

Happy Holidays Professor Borcherds. Your videos made my holidays better.

[austingubbels]

Thank you so much Dr. Borcherds. This is someone who has a deep, curious, thorough, and intelligent understanding of what he is talking about. These lectures are so valuable because the words coming out of his mouth are remarkable and bear the semblance of having pondered these ideas for decades. Seek out the best, and learn from them.

[AkamiChannel]

I think these lectures are best for people who have already studied group theory. This is very difficult as an introduction. It is quite common that a master of a field forgets how difficult something was in the beginning.

[PunmasterSTP]

I'll need to take some time to digest this material, but you did an expert job explaining it. Thank you so much for uploading all these stellar videos!

[MegaSlimshady]

I highly appreciate these lectures! These are one of the best i have ever seen on the internet!

[ayshashahul]

Its quite beautiful that the construction of groups also ends up producing it as the symmetries of itself (by the left action)

[lhtemcl]

pretty sure... (12) lime green and (23) lemon yellow are mixed up @ 22:00.. sure theory of everything still holds. Thanks for creating this fantastic resource.

[unixguru]

Thank you for the awesome lectures! Can you kindly recommend a good textbook for exercises in group theory?

[ebenenspinne]

If I'm not missing something, at 16:00 there is a slight error. The alternative to rotating 180° shown here only works on squares, not on all rectangles.

[zoeb.]

Hi Professor Borcherds! At 27:17 of the video, when you show that g(s)=sg^{-1}, I'm confused about the step when the inverse of (hs) is taken during (gh)s=g(hs)=g(sh^{-1}). Should it be g(sh)^{-1}=gh^{-1}s^{-1} or is that a consequence of an earlier definition? You elucidate these concepts so well and have added a new dimension to my mathematical thought, thank you!

[rajdeepghosh]

For those of you wondering, what Professor Borcherds meant, when he said that left action as a permutation of S preserves right action, it means that since
g((h)s) = ghs = (g(h))s, we get more verbosely that the image of the right action of s on h( i.e- (h)s)) under left action of g (i.e- g((h)s)) is the right action of s on the image of h under left action of g (i.e- (g(h))s). Basically (h)s goes to (g(h))s under g, preserving the right action of s on h.

[josevidal]

Thank you so much professor! The approach is so clarifying ...

[meithecatte]

The way I like to think of Cayley's theorem is that all groups G are isomorphic to a subgroup of a permutation group S_n. Not really confusing anymore.

[mikeywatts]

Why is G a subgroup of the permutations of S (see 4:30)? I get that S is really G. And that the permutations of G form another group. But I can't see how to identify elements of an arbitrary group, G, with a permutation of the group. Thank you!

[TruthNerds]

8:09 "If any two elements of a group commute, then we say G is commutative."
Pendanticism … in formal logic using the English language, "any" is understood to mean existential quantification ("there is a … such that …").
What is needed here is universal quantification and the right word for that is "all", e.g. "all pairs of elements".

EDIT: I mean that would be clearer esp. to viewers trained in formal logic.

[koenigmagnus]

Either the description or the title of video should include "group action" as a topic.