Groups, symmetries, Cayley tables and graphs | Group theory episode 2

#abstractalgebra #grouptheory

Groups are perfect for describing the symmetries of shapes and other objects. In this video, we look at many examples, including their Cayley tables and Cayley graphs. We explore the rotations and reflections of a square, and we discover a curious internal block structure. This will lead to a discussion about subgroups and quotient groups in upcoming videos.

Here are some interesting links for you to explore:

Good introduction to groups, abstracted from number systems. Extremely nice visual style. Proves that the neutral element and inverses are unique.
Talks about subgroups and "shifted copies". This is a rare gem. Go watch it!

Intro to group theory, symmetries, abstraction, and the monster group.

Takes its time to fill in the Cayley table of the D3 group (the symmetries of a triangle).

Symmetry in physics. Good introduction, though it does not go very deep. And it's confusing from time to time. It connects circle symmetries, particles, forces, generators, complex numbers, and more.

A long and very detailed series about group theory. If you plan to study higher math, this one is a must-watch. It will take time, but you will learn most of the details you need.

0:00 Introduction
1:08 Symmetries
3:49 Inverses
7:06 The rotations of a square
7:57 Cayley table
10:57 Isomorphic groups
13:42 Cayley graph
15:33 The reflections of a square
17:41 Orientation
20:18 Generators
23:18 Other polygons
24:46 What's next?

This video is published under a CC Attribution license

[angeldude]

Hold on a moment! The blocks that you divided the square's group into at the end; if you treat each sub-block as a single element, the result looks suspiciously like the diagram between the 4 basis quaternions. Considering the individual elements, it looks a lot like the Geometric Algebra Cl(2), with r2 as -1, which is why ignoring it and looking just at the blocks is just the transition between basis elements (which is equivalent to the quaternions not including sign). Also worth mentioning is that the upper left 2x2 block-of-sub-blocks would then be the basis ℂomplex numbers, which forms the even-subalgebra of Cl(2) (aka Cl(0, 1)), meanwhile joining each of the other two sub-blocks with upper left sub-block gives two copies of Cl(1) (aka the split-complex numbers).

It's always fun to find when two finite groups are actually the same. I think the more general instance of these blocks of sub-blocks, and Geometric Algebra's subalgebras, would be factoring into the "simple" groups.

[oddlyspecificmath]

_By the way..._ By 5 minutes in, something you said made me write a tiny little proof (my first, actually) for something I didn't realize needed one. Thank you.

[apolo]

Great video. I'm excited to see how this series evolves!

[patrickgambill]

This video is wonderful and the animations are beautiful! What tool are you using for making the animations?

[kirilica]

I really enjoyed the video, thank you

[entangledkittens]

First off, great video; looking forward to the rest of the series. Though, I'm a little confused by how you use the word symmetry here. You showed that a plain square with the same color on both sides can be rotated 0, 90, 180, or 270 degrees or flipped accross four axes and it will look exactly the same as before. But if the square is plain then shouldn't all of these transformations be equivalent to the identity? It seems that we were only able to differentiate between these transformations when you added your logo onto the square and colored the two sides differently; let's call that the colored square.
The only way I can wrap my head around this is that the set of symmetries of the plain square form a group on the colored square, but these transformations are NOT symmetries of the colored square. Did I understand that correctly?

[egilsandnes]

I _really_ like this series.

Also, I find it very funny that you "missed" the L while talking about four digit clocks at 12:40. :'D

Great video, i gained a much better intuitive understanding. Thanks!

[claudiotonelli]

Hi, i'm Claudio from Italy: first of all compliment for your fantastic video!!! All very clear. I'd like to ask you a shareware/freeware software to manage groups and do exercises. thank's a lot and i hope you continue to make great videoes.

[grantofat]

You said in your previous video that there must only be one neutral element, but doesn't rotating by 0, 360, 720... degrees count as having more than one neutral element? To me they all do the same (which is nothing), but they are still different.

[carterthaxton]

At 14:00 I found this diagram unintuitive, because the arrows themselves depict a geometrical rotation that is NOT what is depicted by each colored square.

I believe the arrows are just intended to show that there is a cyclic sequence of rotation operations, but they fact that they are drawn as 90 degree rotations (and which are not the same as the 90 degree rotation represented by the group operation) was confusing.