Group action and its application on classification of low order groups

Abstract

The classification of finite groups is an important topic in mathematics throughout history of mathematics. The topic of this paper is to use group action as a tool, to classify some special finite groups and some low order groups. First this paper introduces some concepts of group action. Then this paper states and proves some important theorems related to group action. For example, the Sylows theorem, which is very important in this paper. Research has found that, groups of specific order, such as groups whose order are 2p,p^2, pq(p, q are distinct prime numbers), p3(p is prime) can be classified using group action and the technique of semi-direct product, and groups whose order are no more than 15 are classified which can be seen as the special situations of the above ones. But in general, to make classification of a larger range of finite groups, more tools should be introduced.