Subgroups of the Symmetric Group

We started our research with the intent on answering the following question: can we find a way to calculate all the subgroups of the symmetric group. This is easier said that done, as the number of subgroups for a symmetric group grows quickly with each successive symmetric group. This problem can actually be simplified to finding the subgroup conjugacy classes. So now we have the slightly different question of how to find all the conjugacy classes of a subgroup. One tool that we used to help answer this question is GAP. GAP, Groups, Algorithms, and Programming, is a programming system for computation discrete algebra. One of its commands is “ConjugacyClassesSubgroups” which, when given a symmetric group, calculates its subgroup conjugacy classes. However, if this were all we needed to calculate our conjugacy classes, our paper would be rather short. This command is designed to work for all groups not just the conjugacy groups, and runs rather slowly. In addition, after a certain point, GAP runs out of memory. So we needed to design another program to write into GAP in order to calculate the conjugacy classes of symmetric subgroups. Before we started our programming, the conjugacy classes of symmetric groups had been found up through S12. The number of these conjugacy classes were listed online on a very use website entitled “The On-Line Encyclopedia of Integer Sequences.” Through this, we were able to check our results up to S12. We were also able to discover the conjugacy classes up to S15, including confirming someone else’s results on the S14. Given more time and newer computers, we would be to find subgroup conjugacy classes for even larger symmetric groups. In order to calculate these conjugacy classes, we wrote the program which I will call “ConjClasses.” We will break our process into pieces. First, we will show that if we are given some subgroup of a symmetric group Sn that it will be conjugate to one of the groups in our list. Secondly, we will show that our list is duplicate-free; that is that there are no conjugate subgroups in our list. Say we are given some subgroup of the symmetric group Sn. We first find the length of its smallest orbit, and we call that length l. We conjugate such that this smallest orbit is n’s right-most points. This does not affect our results, as we will compute our subgroups up to conjugation. We then break our subgroup into two subgroups, one acting on k points and one acting on l points, where k = n − l. Let us call a subgroup of Sk G and the subgroup of Sl which acts on our smallest orbit H. We know that H is a transitive subgroup, as it only acts on one orbit, while G may be transitive or intransitive. It simplifies matters for us think of our symmetric group Sn in terms of transitive groups, as GAP has a library listing all the transitive groups up to S30. We knew that it was unlikely for us to figure out the subgroups for symmetric groups on more than 30 points, so this library was more than enough for our purposes. We can also think of intransitive groups as