## Math 375: Introduction to Representation Theory

Study Guide for Exam 1

The first midterm exam consists of two parts. The first part will be held in class on **Monday, October 3**, during our regular class time. This is a closed book, 50 minute exam. The second part will be a take-home portion, which you will receive after completing the in-class part. You may use class notes and the textbooks on reserve for the take-home component, but no other outside resources are permitted. This part will be due at the beginning of class on **Friday, October 7**.

The following is a list of topics intended to help you organize
the material we have covered in class as you study for your exam. It is only
intended to serve as a guideline, and may not explicitly mention everything that
you need to study.

Know definitions, examples and associated theorems for the following concepts:

- Group actions and G-sets
- GL(V), the general linear group of a vector space V
- Group representations
- degree
- trivial
- faithful
- equivalent
- permutation representation
- regular representation

- 𝔽[G]-modules
- correspondence with representations
- homomorphisms and isomorphisms
- submodules
- irreducibility
- complete reducibility

- Maschke's Theorem
- Schur's Lemma
- Representations of abelian groups
- Decomposition of the regular ℂ[G]-module
- Hom(V,W), the space of homomorphisms of ℂ[G]-modules

Maintained by ynaqvi
and last modified 09/27/16