MATH W4041:
 Intro to Modern Algebra I
 Spring 2010

Week 2: January 26, 28     up     previous week     next week

Reading

Our goal for this week is to understand Artin Chapter 2 through Section 6: Cosets.

Meanwhile, become familiar with every group of order 9 or less. What group homomorphisms are possible between these small examples? Which subgroups appear as kernels of such maps? Which do not? Between a given pair of small groups, how many homomorphisms are there? Include the case of maps from a group to itself. In this case, how many isomorphisms are there? (These are called automorphisms, and form the "automorphism group" of the group.)

Be able to prove, to your satisfaction (and mine, on a test), that for finite groups, the order of a subgroup always divides the order of the group that contains it.

Exercises

Assignments may optionally be handed in for review by our teaching assistant, at your convenience and hers. Please give them to me after class. Assignments do not have due dates, and will not affect the course grade. However, learning mathematics is very much like picking up a foreign language or practicing a sport, in that constant exposure in short intervals is far more effective that binge studying before exams. So please keep up! It will be easier on both of us.

If you work all assignments, you will understand the course material better than we are able to cover it in class. However, this is not the only approach. Think of assignments as "structured study". If you find yourself instead engaging in "unstructured play" with examples, all the better. A good start would be to rework examples from class, either rewriting one's notes or as much as possible without the benefit of notes. This is how one typically studies, taking a graduate course. Mathematical research (or research in any domain) is in part a particularly vivid form of daydreaming and play. It would be helpful to start encouraging (or reverting to) such thought patterns now.

Problem Session

I'm offering a problem session this weekend: