189-456A: Honors Algebra 3

Blog

In this blog, I will add whatever comments come up during our conversations or in office hours and that complement what I have said in the class. The blog will only be updated if something relevant comes up but any interesting comment or idea that you send my way will find its way here (with appropriate attribution).

Lectures 1 and 2, on Wednesday August 28 and Friday August 30 We discussed the axioms for a group and motivated the group concept with the principle that groups are an abstract way to characterize the symmetries of a (mathematical) object.
The relevant reading for this week and the coming ones are chapters 3 to 6 of Judson, and Chapter 2 of Garrett. The latter is more faithful to the way I will present the material in the class, while the former provides a more gentle exposition at a somewhat more leisurely pace.

Lectures 3,4, on the week of September 4-6. (Chapter 6 of Judson, and Chapter 2 of Garrett.) We discussed the notion of a group action on a set $X$, and further introduced the basic examples of the action of $G$ on itself by left multiplication, by right multiplcation, and by conjugation.

On Friday, we described the notion of coset spaces and observed that all $G$-sets are isomorphic to a space $G/H$ of cosets.

Monday, September 16.
The lecture was cancelled that day because I was out of town. To make up for it I am going to post two roughly 30 minute videos, which you can watch any time at your leisure.

The first is about the class equation and exceptional outer automorphism of $S_6$, which works out the conjugacy classes in $S_6$ and how the exceptional outer automorphism that you constructed in assignment$1$ acts on these conjugacy classes.

The second, which is not yet posted, will discuss a question discussed in your current assignment. If two groups $H_1$ and $H_2$ are conjugate, then the sets $G/H_1$ and $G/H_2$ are isomorphic as $G$ sets and every element of $G$ acting on each has the same number of fixed points. What of the converse?