Some Computational Problems Of Low Order Symmetriacal Group

The symmetric group not only on mathematics but also on physics and chemistry hasimportant applications. In the actual engineering mathematics, permutation group isusually researched. Researching the subgroups of symmetric group and its structure is animportant subject in the computational group theory. Especially in biology, physics andchemistry, the structure of the subgroups bring out actual significance for researchers.By using the GAP (Groups, Algorithms, Programming– a System for ComputationalDiscrete Algebra), researching the properties of finite group is an indispensible method.Some results on S10 are obtained. In particular, it contains 1593 conjugacy classes ofsubgroups and 29594446 subgroups. The cyclic subgroups consist of 42 conjugacy classes.The solvable subgroups contain 1418 conjugacy classes. The nilpotent subgroups have 531conjugacy classes. The supersolvable subgroups are divided into 923 conjugacy classes.Also the quantity of the same order subgroups is counted out. The quantity of subgroups inevery conjugate subgroup class is achieved. A representative of every conjugacy class ofsubgroups is presented. All of these generalize the results due to H. Benwen, etc.We also compute some results about conjugacy classes of elements in the symmetricgroup S60. Specifically, we give the individual representative and the corresponding classlength for each of conjugacy classes. The order set of fixed-point-free permutations is alsoobtained. These extend the known results due to J. Bamberg, etc., on the strong Goldbach'sConjucture.When computing the permutation groups by computers, the low order of permutationgroup is not difficult. But with the increase of group orders, many properties of subgroupswill have a great change. After the research, we found that the key step of computing thefinite group is lowing the group order. A big question is divided into some slightly smallquestions, and by this way we can reduce a complex question into some of slightly easyquestions.