Group Theory

Syllabus:

The recommended books are: Artin, Algebra, and Dummit & Foote, Abstract Algebra. There is no required textbook, since we will not be following either. It is strongly recommended to buy at least one (better both). We will cover material from outside the books as well. Other books that are standard in the subject are Hungerford, Fraleigh, and Lang.

Office Hours: Mondays 2pm.

There will be two exams, a midterm on October 3 (in class) and a final on December 5.

Homeworks are due on Fridays at 4pm at the department desk on the second floor of Wexler, and to be addressed to Spencer Cvitanov. I encourage you to work in groups. Working in groups is much more enjoyable, and seeing other perspectives and even mistakes can be very instructive, but please write up your solutions in your own words, and state who you worked with. Please do not consult any online sources other than Wikipedia.

The TA is Spencer Cvitanov. His office hours are Wednesdays 11-1 in Wexler 303 MC Squared.

The course will consist of three parts.

1. Bare bones part: We will cover most of chapter two of Artin/the first three chapters of Dummit and Foote. The topics include groups, subgroups, morphisms, quotient groups, some elementary examples such as cyclic groups and the symmetry groups, the notion of abelian and non-abelian, cosets, Lagrange's theorem. We will at least define semi-direct products.

2. More group theory and the relation to linear algebra: Tentatively, this consists of chapters 7.6, 7.7, 7.8 (the Sylow theorems and classification of groups of small order), and material from chapter 10 (representations) in Artin, as well as semidirect products (not covered in Artin). In Dummit& Foote, this would be roughly chapter 4.5, 5.3, and 5.5 (which is semidirect products), and chapter 18.

3. We will dive deeper into chapter 10 of Artin, in particular character theory, and give many examples of character tables for the finite groups we have seen in the course. We will look at some explicit examples of elementary representations, such as the regular representation, the representation theory for the dihedral groups, a few alternating and symmetric groups, and Schur's lemma. The material is also covered in Dummit and Foote, but in a more abstract way (chapter 18).