Visual Group Theory, Lecture 5.1: Groups acting on sets

Visual Group Theory, Lecture 5.1: Groups acting on sets

When we first learned about groups as collections of actions, there was a subtle but important difference between actions and configurations. This is the tip of the iceberg of a more general and powerful concept of a group action. Many deep results in group theory have clever proofs using a seemingly related group action. Formally, a group action arises when there is a homomorphism from G to the group Perm(S) of permutations of a set S. One can think of this as a "group switchboard": every element has a button that permutes things in S, with the requirement that "pressing the a-button followed by the b-button is the same as pressing the ab-button". We also see an alternative way to formalize this, and examine the subtle difference between left and right group actions, with plenty of visual examples to motivate the concepts.

[algebraicgirl]

Timemarks for anyone who needs it.

00:00 Overview
02:26 Actions vs configurations
04:16 Action diagrams
10:39 A "group switchbord"
17:39 Left actions vs right actions (an annoyance we can deal with)
28:03 Cayley diagrams as group actions
31:05 Next lecture...

[scug-jnmy]

Literally 2 minutes into the video and my brain is already exploding with realization. This video is amazing, thank you so much for making it!

[chrislombardi]

LIghtbulb moment. I've been challenged by a professor to understand a set of objects called "quilt block designs". They are 4x4 arrays of blocks where each block is filled with one of 6 designs (white square, black square, and a black/white right triangle in one of its rotations). In trying to understand these designs in group theoretic terms, I've had this fundamental confusion that's been bugging me. It seemed there were two different modes of understanding-- in terms of the symmetries of the square, and the symmetries of each individual block. After this lecture, I think I see now that I have the group D4 acting upon a set of permutations of D4 itself. Cool problem and I'm still a bit confused, but now I have a clear path forward. Thanks for making these lectures available to the world!

[beyondsyllabus]

Wouldn't have picked up group theory again if it weren't for these set of lectures.

[carlosraventosprieto]

this was AMAZING

thank you so much you are an incredible teacher! i have my exam on monday and this video helped me a lot in terms of the ides behind it.

greetings from spain

[sahhaf]

I understand phi(a)phi(b) as: you first apply a, then apply b..

[Mrpallekuling]

19:43 The standard way to write conjugate (and this has also been done in the previous videos) is aHa^(-1)
Why complicate things to also write it as a(-1)Ha here?

[rasraster]

You point out the issue with the product of some left actions being the reverse product, but how do you note that in notation like (ab).s?

[sahhaf]

I guess there are very serious problems with your phi-notation... around @23:26 you have tried to correct it, but it is still inconsistent with what you have said before..

[samiaakram]

sir group action onreal line aur space pr kesy define ho ga??
kindly guide me...

[sahhaf]

this "little annoyance" is the most complicated and ununderstandable part of the whole lecture series....
@22:37, why theta(g)theta(h)s=theta(gh)s ??? Shouldn't it be theta(g)theta(h)s=theta(hg)s, as we apply first g, then h