Visual Group Theory, Lecture 1.4: Group presentations

Visual Group Theory, Lecture 1.4: Group presentations

We begin this lecture by learning how to take a Cayley diagram and label its nodes with the elements of a group. Such a labeled diagram can function as a "group calculator". It leads to the notion of a "group presentation", which is a convenient way to describe a group by listing a generating set, and a collection of relations that they generators satisfy.

[ykge]

For the last group at 18', using associativity, ba=a^2b=(aa)b=a(ab)=a(b^2a)=abba=(ab)(ba), so ab=e. So a=b^-1. Hence e=ab=b^2a=(bb)a=b(ba)=b. And a=e as well.

[hellshulk]

These videos are very very helpful. Thanks for putting up these.

[ragedorder]

This is great! Looking forward to your upcoming book!

[edvogel]

Thank you for these well illustrated and paced lectures.

[henlofrens]

is abelianality then always denoted in the relations part of the group representation?

[nathanielvirgo]

8:50 I don't get it. How do we know that these relations uniquely define this group? I.e. how do we know there isn't some other group that also has these relations, but also some other ones?

[manhtuannguyen]

Hello Professor Macauley, Isn't it r is 120 degree clockwise rotation?
Could you look at 8.26, at the outer circle of flipped object? Because it appears to me that the flipped object is rotated counter-clockwise.
Could you explain why or point out where I have been wrong?
You made good lectures. Thank you.

[wesleysuen]

9:25 I bet you wanted to write rf=fr^2 instead.

[gleedads]

There were two occasions in this lecture where you let a = e and b b = e. But doesn't this violate one of the central tenets of set theory that every element of the set must be distinct?