Rank of Unit Group of Burnside Ring: Linear character and Elementary Abelian 2 Groups

The work of Week 5 led to some new results that would improve the efficiency of original function and gave me the confidence about the direction of my work. However, finding all the normalizers of index 2 subgroups is still a very computationally heavy process in GAP. After doing a lot of literature review and talking to Dr. Carman, I think linear character and elementary abelian 2 group can be the breakthrough. [Read more…]

Rank of Unit Group of Burnside Ring: Improving the Function

The resulting code from Week 4 was very significant. However, it is not ideal due to its inefficiency. I mentioned in the previous post that the inefficiency was caused by the calculation of unnecessary lines. Unfortunately, this cannot be changed by simply improving on the previous code, because the idea of using table of marks and solving for each line does not discriminate between necessary and unnecessary equations. Thus, I decided to give up on previous method of using table of marks and vector space. The goal of Week 5 is to find a new direction to approach this problem.

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Rank of Unit Group of Burnside Ring: Calculation by Table of Marks

After studying group action, orbit & stabilizer theorem, and isomorphism class during the second week, I began to have a clearer picture of how to approach calculating the rank of the unit groups of Burnside rings. My goal for the third week is to find the methodology of calculating the rank of the unit group of a Burnside ring by hand, which would later shed some light on my later work of automating the process in GAP. During the third week, I read two research paper, which are “On the Unit Groups of Burnside Rings” by Tomboyish Yoshida, and “An Algorithms for the Unit Group of the Burnside Ring of A Finite Group” by Robert Boltje and Gotz Sniffer. There were very complicated parts of the paper that I could not fully comprehend, so I mainly focused on understanding the parts that are more crucial to my project. I also proved below propositions.

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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)

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Rank of Unit Group of Burnside Ring: Learning GAP

    The research objective is to find the mathematical theory of the size of the unit groups of Burnside rings. Specifically, it focuses on automating the algorithms of calculating the size of unit group of Burnside ring by employing GAP (Groups, Algorithms, Programming). After the first meeting with Dr. Carman, I wrote several things on my to-do list for the first week of research. As a large amount of the computational work of this research project will require the assistance of GAP, the main goal of the first week is to learn to use GAP by gathering relevant materials and reading tutorials and documents.

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