## Summary Post

What a great summer spent!! At the end of the research project, I am so thankful for this opportunity to conduct research with Dr. Carman on the mathematical analysis of the rank of unit group of Burnside ring, which is a topic in abstract algebra that I’ve always been very interested in. I am so lucky to have a mentor like Dr. Carman who is so responsible and helpful. He discussed many ideas with me and taught me about many mathematical concepts. I am very grateful that Charles Center provided me with this precious opportunity to learn and grow in my first research project. During the seven weeks, I learned to plan a research, to do literature review, and to solve the problems we encountered along the way. In this process, I not only gained a lot of knowledge about Burnside ring and abstract algebra but also learned to think more critically as a researcher. There were several challenging parts in this research and I leanred that doing research is not as simple as taking a course. For example, in week 3 I spent two days finishing the first version of the code in GAP. I was so happy when it could finally generate the correct output, but it could not run for larger group. My original plan was to improve based on the existing algorithm. However, I found that the origin of the inefficiency was caused by its mathetical basis, which means that it can only be improved by finding an entirely new way of approaching the problem. It was a little frustrating when I had to give up the old results and codes and start from the begining, but I learned to think as a researcher.

## 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.

## 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…]