# Course Review: MATH 322

#### Introduction to Group Theory

“We say groups have a rich structure because it keeps us mathematicians in business”

Text: Basic Algebra I: Second Edition

Prof:  Dr Vinayak Vatsal

Nike is a humorous professor who cares about his students. The homework he set was also useful in getting a mastery of the material in the course. Unfortunately, the textbook was made up of large, listless walls of text that prove half a dozen theses arbitrarily before deciding to declare that a theorem worthy of a name has been proven. Thus it cannot be readily used as a reference book and is difficult to read. The classes were significantly better, but from time to time also suffered from a lack of direction and interest. Group theory, on the other hand, is a fascinating field and the proofs often use a lot of creativity.

Difficulty

I found the material in this course difficult, but since it was an introductory class, I felt I had enough time to master the relevant concepts. Unfortunately, the relatively few iClickers were weighted 10% of the grade and the exams were not difficult but punished small mistakes a lot. Also, the final exam used a peculiar notation and seemed almost entirely about symmetric groups and had no questions about abstract group theory. As a result, the average was quite low in this course.

Key Concepts

Groups

Subgroups

Action on groups

Orbit-Stabilizer

Class equation

Finite Abelian groups

Isomorphism Theorems

Homomorphism

Free group and generators

Hard Concepts

Action on groups: Useful to get some practice with a couple of these actions before one can start seeing the action that occurs in problems

Proving isomorphisms/finding a bijection: Often one can use the obvious or ‘canonical’ map, other times you need to do some clever

Counting elements of a certain order in abelian groups: Can make careless mistakes, double check your answer!

Conclusion

An interesting topic, the homework problems were fun, wished the textbook and exams were structured differently.