Math 375: Introduction to Representation Theory
Study Guide for Exam 1

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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.

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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

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Maintained by ynaqvi and last modified 09/27/16