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: Converting Thoughts into Codes

The goal of Week 4 is to convert the calculation of the rank from work on paper into a GAP function. Even though I have experienced writing functions when I practiced with the GAP manual during Week 1, this was my first time writing an actual original function in GAP. The first step was to construct the table of marks of the input group. After trying to write lines of codes in GAP to simulate the construction of table of marks, I was very lucky to find out that GAP actually has built-in function which can be called by the command TableofMarks(group) that is similar to what I tried to do on paper. However, the table was reversed in direction comparing to what I did on paper, thus the second step was transposing the table of marks by using the function “TransposedMat”. The hardest part was simulating the process of generating the equations and finding the solutions, which was very easy by paper and pen but was hard to be generalized by codes. I first tried to code the process of equation solving by reproducing what I did by hand, but I found out it is a very inefficient process due to the excessive amount of “if” statements. After trying many other ways to code for this step, I found the simplest way to be thinking of the equations as vectors and finding the solution of the matrix. The final code is shown below:

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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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