Rank of Unit Group of Burnside Ring: Final Function

We spent the seventh week on coding for the final function. The coding process was very exhausting but Dr. Carman and I were so excited and happy to see our final function that can return the correct output (rank of unit group of B(G)) for an input group. After testing the function on different groups, we found that the the final function works for groups as large as S9 and M22, which is a much better result comparing to the first version which stopped working for S3.

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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: 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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