Visual Group Theory, Lecture 1.2: Cayley graphs

Visual Group Theory, Lecture 1.2: Cayley graphs

In this lecture, we introduce a visual way to "map out" a group using an object called a Cayley graph. This concept is a useful visualization tool, but it is often omitted entirely from traditional Abstract Algebra classes.

[alanhere]

I've found so much stuff on this sort of topic to be unintuitive and unaccessible as anything, as it were. This is all such a relief, I'm so thankful.

[OlivierSuire]

I prefer "commutative" to "Abelian" because it contains a hint to what the term actually means.
This habit of naming abstract math concepts by the name of their inceptor is a form of jargon.

The proof of uniqueness of identities makes use of associativity, but that axiom was not made explicit before.

[AlexJLim]

Im taking an honors abstract algebra course and your material has been a fantastic supplement in terms of helping to aid intuition. Thank you so much for putting this on the internet!

[anyad]

Hi,
I went to the course webpage and I see there are homework that we can do. Can you provide also solutions so we can know if we did it right? Anyway, thanks for this lecture!

[YT]

This is brilliant, I'm hooked. (: I wish my introduction to group theory would have been like this course. Unfortunately the training in elementary group theory I got was keen on abstraction for it's own sake, which actually obfuscated my grasp of key concepts back then.

[MrGiuse]

Hello Prof. Maccauley. It's NOV 2021 and I just found these lectures of BLESSING for studying. THANK YOU

[robharwood]

Interestingly, the 'Big Book' for the much simpler 2x2x2 Rubik's Cube *could* be practically made. I wonder if anyone has actually made it already? Could be!
Maybe that would be a good, concrete example of a complete description of an intuitively-understandable group, if you ever return to these lectures in the future.

[maxpercer]

note that, a 'horizontal flip' is a reflection about a vertical axis through the center
and a 'vertical flip' is a reflection about a horizontal axis through the center

[niuhaihui]

At about 5'26", it should be pointed out that it's possibly true that both are "Closer to solved", or "Farther to solved". The reason is that at this given configuration, a 180 degree rotation may be the best action towards the final solution.

[Mrpallekuling]

I love teachers who wants others to understand smth. Great course!
PS. Abel is pronounced "AH-bəl". Regards from Denmark (Danish is similar to Norwegian)

[pragalbhawasthi]

5:28 A minor doubt. In that configuration, if the best move happens to be the one that turns a face by 180°, then both moves (clockwise, and anti-clockwise) would take us closer to the solved configuration. Wouldn't that be true?

[maxpercer]

The non-square rectangle symmetry group, under the operation composition,
are the set of actions (or set of functions, also called mappings)
G = { e, h, v, r_180°}, generated by < h, v>
It has 1 cyclic subgroup { e, r_180°} = <r_180°>,
not counting the trivial subgroup {e}.

[vytah]

15:35 The largest order of an element of the Rubik's cube group is 1260 for elements like RU²DBD⁻¹. An easier example is a simple FR, which has the order 105 and can be very easily verified by hand.

[jadudar]

It’s actually rather nice to honor mathematicians, especially influential ones, since they give us so much and often receive nothing in return during their frequently curtailed lives.
It’s easy enough to learn the word “abelian” in a group theoretical context. “Commutative” is more general in its usage.

[mateiacd]

At 15:08 I would have preferred to see h*h=e, v*v=e, r*r=e meaning that by doing each action twice, we get back to the initial configuration

[HanZhang]

For the action 31:40, why can't the absolute value function in real numbers be an action that is an identity action for some configurations, but not for others?

[timelordyunt]

These videos are really great. Thanks!

[RatanLal-kbsc]

sir, I like the way you represent the topic of group presentations. I am having a problem in understanding the "canonical form of elements in group presentation{basically in free groups}. I need your help,   if you have some lectures on that topic or some literature that you can suggest me. thank you.

[DanielEstrada]

Question 1: Why don't you allow flips around the diagonal in the rectangle game? Fix two corners, swap the other two, similar to the flip in the triangle game. Would these generators form a different group, or not a group at all?

Question 2: What would the group representation of the rectangle game be, if you DID change the orientation of numbers during flips and rotations?