Visual Group Theory, Lecture 1.5: Multiplication tables

Much like how children learn multiplication tables of the positive integers in grade school, one can create a multiplication table of any finite group. In this lecture, we begin by looking more closely at inverses in group. Next, we see some examples of multiplication tables and prove some basic properties. We conclude by introducing a new group of size 8: the (unit) Quaternions.

[justinbrown]

Why do pirates like the group D3? Because 11:23

[darkseid]

Thank you so much sir. I have watched entire group theory lectures online but never really understood what it really meant because I never understood it visually. But now I have started watching your play list and now I am starting to understand the visual understanding of everything in group theory.
Once again thank you very much for providing such quality knowledge for free. 😍🙏😊

[shawon]

14:45 I think a small thing was overlooked. When we remove r³ = e condition, the two group presentation on the top are not same anymore. The diagram drawn in the video is actually the diagram of the 2nd definition. I drew the diagram for first group presentation and confirmed it's a different group with -8 elements {e, r, r², r³, f, fr, fr², fr³}-

Edit: So turns out, there are actually 6 elements and the group is D₃. So the definition < r, f | f²=e, rf=fr² > is actually D₃. r³=e follows from the other 2 relations. Neat.

[NicolasMiari]

10:00 The existence of inverses (used in the proof) precludes the possibility of two different elements "multiplying into the same result", i.e. "collapsing", much in the same way the existence of an inverse function precludes a mapping from being many-to-one.

[conradgarcia]

I'm doing a refresher course on group theory and i found your videos as the best--visual, intuitive, step by step and exciting. i'm having a hard time following Cayley diagrams though coz they don't show the standard positions of the answers. and its even more indecipherable in text books coz they don't offer any explanation. so I'm forced to make a simplified table of my own to complement Cayley's example
1 x 1= 1
1 x 2 = 2
2 x 2 = 2
3 x 3 - 0

[scitwi]

10:20 It could be turned into a proof by contradiction by assuming in the beginning that `b` and `c` are different. Then the conclusion that `b` and `c` contradicts that assumption.
I wonder if this could work the other way around for other proofs by contradiction – whether they could be restated as direct proofs by reversing this trick ;>

[nathanj]

It seems like rf = fr^2 does not apply to the frieze group, but rf = fr^(-1) does. In the original group r^2 = r^(-1). So, how do we account for/explain this?

[stevexm]

From the exercise for finding inverses using the Caley diagram, the fourth example (3:56) where (r^2f)^-1 = r^2f, its own inverse; from the diagram isn’t it also true that the inverse is simply equal to fr, implying the relation r^2f = fr ?

[bonbonpony]

I'm not sure if it is correct to say that inverses follow the same arrows backwards. There might be a system in which you simply cannot do the reverse action _directly_ (due to physical limitations), but you can still sorta "undo" it by performing other actions and restoring the state that the system had before doing that one-directional action. A simple example of that is a wheel that can only turn clockwise, with some points marked on it in equal distances. Let's see that you have just 3 points, for simplicity. Performing one 120° turn changes it from the "solved" state, and you cannot simply "undo" this action by rotating the wheel backwards, because it can only turn clockwise. But you can continue turning it, and after two more 120° turns you will be back in the "solved" state that preceded this particular action. So there is no _single_ action that could reverse it, but there is a _composite_ action (which could be done with one move) that can undo it: the 240° rotation clockwise. This is not exactly "following the arrow in reverse", it's more like following some _other_ arrows to ultimately come to the same place.

[manisitach]

Can a Quaternion group be formed using two generators, say i and j only?