Rank of Unit Group of Burnside Ring: Learning Group Action and Orbit & Stabilizer Theorem

This research project aims to automate the process of calculating the rank of the unit group of Burnside ring in GAP. After learning to use GAP during the first week, the new goal is to calculate the rank by hand. Based on a lot of literature review and talking to Dr. Carman, I wanted to first approach this problem by using the idea of table of marks, which requires profound understanding of group action, Burnside’s lemma, and isomorphism class, which were not covered in the abstract algebra class I took. I spent my second week learning about the concepts by both searching online and asking Dr. Carman during our meeting time. I mainly studied group action and orbit & stabilizer theorem, I later typed the essential statements in LaTex. ( See graphs below)

Group Action Notes

 

Below is the notes including both the propositions and examples.

 

Week2 part 1

 

 

isomorphism class