Research on some particular simple groups by using elementary methods

Zhiyin Shao Guangdong Guangya High School Ruiheng Xie Guangdong Guangya High School Ziqi Zhao Guangdong Guangya High School

S.-T. Yau High School Science Awarded Papers mathscidoc:1608.35141

Dongrun-Yau Science Award, 2013
It is well known that group is a set with an algebra operation, which is often denoted by G . A simple group is the group which has only the 2 trivial normal subgroups. The importance of the simple groups may analogue to the prime numbers in the number theory. The classification of the finite simple groups was the central problem of 20th century’s’ algebra. In this research report, we use Sylow theorem, Burnside theorem, group action and some other elementally group theory methods to obtain some criteria of simple group . By using these criteria we determine the possible simple groups whose order less than 700. Generalizes the results in the reference [1],[2], [3],[4].We conclude that Theorem A group G of order less or equal than 700 could not be simple except for |G|∈{60,168,360,504,660 and all primes}.
finite group subgroup normal subgroup simple group Sylow theorem group action normalizer centralizer transitive permutation presentation N /C theorem
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  title={Research on some particular simple groups by using elementary methods},
  author={Zhiyin Shao, Ruiheng Xie, and Ziqi Zhao},
  booktitle={Dongrun-Yau Science Award},
Zhiyin Shao, Ruiheng Xie, and Ziqi Zhao. Research on some particular simple groups by using elementary methods. 2013. In Dongrun-Yau Science Award.
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