Further Thinking About Some Problems In Group Theory

In this paper, we mainly focus on two problems:one is on the structure of a group, and the second is about subgroups of a group.(1) For the first problem, we construct a specific decomposable group to illustrate the related questions, from which we know exactly what each direct product factor is, and further know the structure of the group. Towards this end, we start with the introduction of the direct product and the definition of a decomposable group, then investigate some properties of the direct product and introduce the definition of a quasi-prefect decomposable group, and finally we show some properties of it. We also investigate some properties of a prefect group, which will be used later. We conclude the problem with a concrete example to illustrate the process of constructing a specific decomposable group.(2) For the second problem, we mainly take the symmetric group S5 as an example to determine its subgroups. Having done some preparations, we analyze carefully the structure of all the group elements and find out all 156 subgroups of 19 types. Next, we classify the 156 subgroups by solvability. After introducing briefly the related definitions and some fundamental results to be used, we study some properties of the commutator group of a group in some special cases, from which we have a good understanding of the concept of quasi-prefect decomposable groups.