Analysis on the Properties of a Permutation Group

The structures of the subgroups play an important role in the study of the nature of symmetric groups. We calculate the 11300 subgroups of the permutation group S7 by group-theoretical approach. The analytic expressions for the numbers of subgroups are obtained. The subgroups of the permutation group S7 are all represented in an alternative way for further analysis and applications.


Introduction
The study of permutation groups is of significance for the development of group-theoretical approach [1, 2, 3]. On one hand, each row in the multiplication table of a finite group shows a permutation of group elements such that every finite group is a subgroup of a permutation group. On the other, the analysis of the tensor indices requires the theory of Young's permutation operators. The survey of the structures of the subgroups is meaningful to understand the properties of the permutation groups. Many researches have discussed the computing the subgroups of permutation groups [4]. Since the order of the permutation group S n , g = | | = ! , increases rapidly with the increase of the number n, it is generally considered that it would be quite difficult to calculate the subgroups of a permutation group by grouptheoretical method when the number n is getting larger. The subgroups of S 7 mainly come from the computer program [5,6,7]. However, we believe that if we can calculate the subgroups of S n through the group-theoretical method, then we can not only be independent of the computer program, but also we can study the properties of the subgroups of S n with analytic methods, provide the explanation for the pretty huge numbers of subgroups of different orders. It also might be possible that the research indicates some useful information for the simplification of the computer programs.
In the following section, we will make preliminary sketches of various non-isomorphic groups for a few finite groups. Then, we will analyze the properties of the group S 7 and calculate the subgroups of the permutation group S 7 in section 3. In this section, the analytic expressions for the numbers of subgroups are also presented by the numbers of group elements in the classes. We discuss its possible applications with the results and represent the subgroups in an alternative way in section 4. We conclude in the final section after pointing out various directions for future investigations.

Preliminaries
In the finite group, one may try to know how many nonisomorphic groups of a given order of n. Generally, the answer to this question is not yet given. Here, we present all non-isomorphic groups of orders less than 14. That is what we need to calculate the subgroups of the group S 7 .
The Lagrange's theorem [8,9] states that for any finite group G, the order of every subgroup H of G divides the order of G. It implies that every group of prime order is cyclic. If the order of the finite group is a prime number, g = 2, 3, 5, 7, 11, or 13, it can only be the cyclic group, denoted by = , R, ⋯ , ，where R is a generator, = , and E is the identity. If the order of a finite group is g = 2n (n = 2, 3, 5, 7), where n is a prime number, it can only be either the cyclic group C 2n or the dihedral group D n .
If the order of the group is 8, there are five non-isomorphic groups [10,11]. The first is the cyclic group C 8 . The second is the dihedral group D 4 , where two generators can be denoted by R and S 0 , satisfying = = . The third is an Abelian group, = × , where the generators satisfy = = and RS 0 =S 0 R. The fourth is also a commutative group, = × , and the generators satisfy = = = . The fifth is a quaternion group Q 8 , the generators satisfy = = .
There are two non-isomorphic groups of order 9. One is the cyclic group C 9 . The other is a direct product of two cyclic groups, × , where the generators satisfy = = ,, and RS=SR.
If the order of the group is 12, there are five nonisomorphic groups. The first is a cyclic group C 12 . The second is the dihedral group D 6 where the generators satisfy = = . The third group is denoted by T or A 4 where the generators T 2 and R 1 satisfy ( ) = = .
The fourth is denoted by Q, and the generators is chosen as R and S, satisfying = = . The fifth is the group C 6h and the generators satisfy = = and RS 0 =S 0 R.

The Properties of the Subgroups of S 7
Before we get started with the subgroups of S 7 , we need to be clear about the number of group elements in S 7 , ! = | " | = 7! = 5040 , and the number of its conjugate classes, g c =15. The 15 classes [α], the number of elements n [α] in the classes, the order of the elements and one representative element in each class are presented in Table 1. In the following, we will analyze the subgroups of S 7 of different orders for non-isomorphic subgroups. Although there are a considerable number of subgroups, it will be shown that the number can be connected with the numbers of group elements in the classes by analytical expressions.
As can be seen, the subgroups of order 2 will take the form of {E, (12)} or {E, (12) (34)}, or {E, (12) (34) (56)}. Where there is an element of order 2, there is a subgroup of order 2. According to the number of elements in each classes in Table  1, if we denote the total number of cyclic subgroups of order 2 by N (2), then it is equal to the number of all elements of order 2, (1) Therefore, there are 231 cyclic subgroups of order 2 in the permutation group S 7 .
The subgroups of order 3 will look like {E, (123), (132)} or {E, (123) (456), (132) (465)}. If the total number of subgroups of order 3 is denoted by N(3), then the connection between N(3) and the number of elements in the classes is That is, there are 175 cyclic subgroup of order 3 in the group S 7 .
The subgroups of order 4 of the group S 7 need to be analyzed carefully. According to the preliminaries, there are two non-isomorphic groups of order 4, a cyclic group and an inversion group. Notice that the table 1 indicates that elements of order 4 are included in the class [4] The preliminaries indicate that there are five nonisomorphic groups of order 8. It can be found that there is no cyclic subgroup of order 8 or subgroup which is isomorphic to the quaternion group. There are 1050 dihedral subgroup There is no cyclic subgroup C 9 in the permutation group There are no subgroups of order 11 or 13 in the group S 7 according to the Lagrange's theorem. The subgroups, whose order are more than 14, can be deduced from the results what we have gotten and the analysis of the structures of the subgroups. Due to the length limit, we take the subgroup of order 16 as example. Refer to the subgroups of order 8 which are isomorphic to the group D 4 , we find that the subgroups of order 16 are isomorphic to D 4h , such as {E, (1234), (13) Similarly, the subgroups of order 18 can be referred to the subgroups of order 9, the subgroups of order 21 can be referred to the subgroups of order 7, the subgroups of order 144 can be referred to the subgroups of order 72, etc., and the analytical expressions can all be get through careful analysis. The subgroups of orders larger than 240 can be get directly from the analysis of the properties of the permutation group. Calculate the number of 6-combinations of 7, C " = 7, then we know that there are 7 subgroups of order 720 which are isomorphic to S 6 , correspondingly, there are 7 subgroups of order 360 which are isomorphic to A 6 . We have 1 subgroup of order 2520, which is denoted by A 7 , composed by all even permutations of S 7 . On the basis of the calculation, it is found that when we calculate the subgroups of order m of the permutation group S n , we can make full use of the results about the subgroups of order less than m of S n and the results of subgroups of order m of S (n-1) .

Discussions
In general, a group can be described by the generators. Since the choice of the generators is not unique, researchers have different choices. A group can also be described by giving all of the elements, whereas it appears to be redundant, especially when the order of the group is large. These two methods have been used in previous calculations. Here, for the convenience of analysis, we represent the subgroups in an alternative way, as shown in Table 2.
The subgroups of the permutation group S 7 are represented in the form of [8 ]  The presented research would widen the application of the Cayley's theorem [12,13,14]. The theorem stated that every finite group of order n is isomorphic to a subgroup of a permutation group S n . The order of the corresponding permutation subgroup, directly from the Cayley's theorem, is usually the same as the order of S n , i.e., n!. To study a group of order n! would be more difficult than to study a group of order n. Generally, it is not always an easy task to find the corresponding permutation subgroup with the same order as an arbitrary finite group. Now, we have obtained 11300 subgroups of S 7 . The expressions of these subgroups can reveal how many classes are included and how many elements there are in each class. These results would be quite useful in the study of the structures and properties of finite groups.
It is known that there are many non-isomorphic groups for a given order n and there are also several expressions consist of different classes for the isomorphic groups. When we choose the most appropriate form for a finite group, it should be noticed that the group elements of same order in different classes might have different meanings in the applications. Take for instance, the elements of order 2 in the octahedron group O. It can be found that the elements in the class [2]   To summarize, on the basis of theoretical calculation and analysis, there are 11300 subgroups of the permutation group  Table 2.

Conclusions
In this article, we calculate the 11300 subgroups of S 7 by group-theoretical approach and represent all the subgroups in an alternative way for further analysis and applications. Although the total number of the subgroups of S 7 is considerable large, we provide an explanation by several analytical formulae of N(m) and n α , where N(m) denotes the number of subgroups of order m and n α is the number of elements in the class [α]. The research shows the power of the group-theoretical approach and will be quite useful in analyzing the properties of the permutation groups. It will also be helpful in the study of the finite groups with the familiar theorem of Cayley. Further, how to apply this method to simplify the computer program is also an interesting subject to study in the future.