International Journal of Discrete Mathematics 2016; 1(1): 15-19 http://www.sciencepublishinggroup.com/j/dmath doi: 10.11648/j.dmath.20160101.13 The Theorem of Cayley and Matrices Xiao-Yan Gu1, *, Jian-Qiang Sun2 1Department of Physics, East China University of Science and Technology, Shanghai, China 2College of Information Science and Technology, Hainan University, Haikou, China Email address: [email protected] (Xiao-Yan Gu), [email protected] (Jian-Qiang Sun) *Corresponding author To cite this article: Xiao-Yan Gu, Jian-Qiang Sun. The Theorem of Cayley and Matrices. International Journal of Discrete Mathematics. Vol. 1, No. 1, 2016, pp. 15-19. doi: 10.11648/j.dmath.20160101.13 Received: October 31, 2016; Accepted: November 17, 2016; Published: December 27, 2016 In this article, the connections between symmetric groups and the matrix groups are investigated for exploring Abstract: the application of Cayley’s theorem in finite group theory. The exact forms of the permutation groups isomorphic to the groups , and are obtained within the frame of the group-theoretical approach. The results are analyzed in detail and compared , with that from Cayley's theorem. It shows that the orders of the symmetric groups in present formulas are less than the latter. Various directions for future investigations on the research results have been pointed out. Keywords: Permutation Group, Isomorphic, Matrices, Cayley's Theorem, Quaternion Group represented as permutations of {1, 2, 3}. With one certain 1. Introduction labeling, we would get the rotation through 2π/3 to (123). We It is well known that Cayley's theorem is one of the most are acquainted with this fact that the dihedral group D3 of important results in group theory [1-3]. The theorem shows order 6 is isomorphic to the symmetric group S3 of all that if G is a finite group of order n, then G is isomorphic to a permutations of 3 objects. We now wonder whether one can find the symmetric group of Sk (k<n) corresponding to some subgroup of Sn. This is a classic and intriguing result [4, 5]. With this in hand, if we can fully understand the structure and other specific finite groups G of order n and have the explicit connections between them. properties of Sn and its subgroups, then we will automatically understand the structure and properties of this finite group. The groups of particular interest, discussed in this article, are matrix groups. matrices, also known as the Dirac However, the symmetric group Sn of all the permutations of n matrices, play a highly significant role in mathematics and objects has order n!, trying to use Sn to answer any questions about G means working with a group that factorially larger. physics [8-11]. As a set of matrices satisfying special anticommutation relations [12], Due to the sheer size of Sn, this becomes problematic. Some questions may arise from the investigation for the further , = + = 2 , 1 ≤ , ≤ , (2) applications of Cayley's theorem [6, 7]. For instance, is it N matrices and all its products generate a finite matrix possible that G is isomorphic to a subgroup of Sk where k<n? group . Analogous sets of matrices can be defined in any Consider the symmetry group of the equilateral triangle, D3. The multiplication table shows that D3 is a finite group of the 6 dimension and signature of the metric. In five space-time group elements. These elements may be represented as dimensions, the four matrices above together with the fifth permutations of {1, 2, 3, 4, 5, 6} according to the Cayley's matrix generate the Clifford algebra. theorem. For example, the rotation through 2π/3 can be According to the familiar theorem of Cayley, the connections between and S is fairly straightforward from represented to n the group table of the finite group . However, it is not an easy task to find a symmetric group of S with k<n 1 2 3 4 5 6 (1) k = 123 456 . corresponding to the group of order n. 2 3 1 5 6 4 On the other hand, if we label the vertices of the triangle In the following section, we briefly describe the comparison of different connections between the symmetric group and the with the numbers 1, 2, and 3, then the elements may be 16 Xiao-Yan Gum and Jian-Qiang Sun: The Theorem of Cayley and Matrices matrix group of N=2. Our method is then used for the and all the remaining elements in the matrix group can be analysis of the finite matrix group in Section 3. The got from the explicit form (Eq. (5)) of and . Then, " explicit form of the permutation group corresponding to the becomes group of N=4 are obtained in Section 4. The results are ={1, , -1, - , , - , - , } discussed in detail in Section 5. We conclude in the final " " " " section after pointing out various directions for future (6) investigations. , - ( 1234, 1324, 1432, 13, 1423, 24, 1234. It means that the matrix group of order 8 is also 2. Matrix Group isomorphic to a subgroup of the permutation group S4. The ! results in Eq. (3) and Eq. (5) present different connections Two matrices and satisfying the anti-commutation between the symmetric group S and the matrix group . The " k relations Eq. (2) and all their possible products form the Γ comparison shows that the number of the different objects in matrix group, ={1, , -1, - , , - , - , }. The the S in our results is only half of that from the theorem. We " " " " k multiplication table of the group is calculated as follows. will continue to use our method to investigate the permutation group related to other matrix groups in the following. Table 1. The group table of . # 1 2 3 4 5 6 7 8 3. Matrix Group 1 -1 - - - . " " " " 1 1 -1 - - - " " " " The Pauli matrices σ1, σ2 and σ3, satisfying the relations, -1 - 1 - - " " " " " -1 -1 - 1 - - " " " " (7) - - 1 -1 - - // = 1 + 0 134" 23 /3, " " " " " - - 1 - -1 " " " " " are a set of γ matrices in dimension 3 with the Euclidean - - - 1 - -1 " " " " - - - -1 1 - metric signature. The matrix group is the set of three γ " " " " " - - - -1 1 " " " " matrices and all their products, Denote the eight elements {1, , -1, - , , - , - , (8) " " " " = ±1, 5 ", 5 , 5 , 5 " , 5 " , 5 , 5 " . }, in the group by the digits {1, 2, 3, 4, 5, 6, 7, 8} With a similar analysis for the matrix group , the Cayley's respectively as shown in the first row of the group table, then, theorem states that the matrix group of order 16 is in accordance with the Cayley's theorem, and appear " isomorphic to the subgroup of the permutation group S16 1 2 3 4 5 6 7 8 directly from the multiplication table. = = 15283746, " 5 8 7 6 1 4 3 2 What is interesting is that how to find a permutation group Sk (k<16 ) whose subgroup is isomorphic to the matrix group 1 2 3 4 5 6 7 8 (3) of order 16 through investigation. Choose the = = 18273645. 8 7 6 5 4 3 2 1 correspondence between the γ matrices and the permutations The rest in the group can be got from the multiplication rule. as follows, The matrix group of order 8 is isomorphic to the subgroup of the permutation group S8. These are results directly from " =2 1 34, =5 1 263748, the Cayley's theorem. (9) On the other hand, notice that the order of the elements in =8 1 273546, the group are respectively: 1, 4, 2, 4, 2, 2, 2, 2, which are the products of γ matrices can be written as the same as the elements in the dihedral group D4 of the symmetry group of a square. This is a helpful hint. Label the " = 15263748, " = 18273546, vertices of the square with the number 1, 2, 3, and 4, then the (10) elements in the group D4 may be represented as permutations = 14235768, " = 13245768. of 1, 2, 3, 4. Notice that the square of (a, b=1, 2, 3 and a≠ b) is ， ， ， & = '(, ), ) , ) , *+ *" * * , " = " = =2 1 345678 = −1, (11) (4) the explicit forms for - (a=1, 2, 3) are given by =, ( 1234, 1324, 1432, 13, 1423, 24, 1234, where R is a π/2 rotation about the center of the square and S0, S1, −" =6 5 78, − =6 1 253847, S2, S3 are the reflections about four symmetry axes respectively. (12) It is not difficult to prove that there exists an isomorphism 6 =7 1 283645. between these two groups, ≈ D . Hence, this isomorphism 4 As noted, it is not difficult to come to the remaining gives a one-to-one correspondence of the elements from to products of the γ matrices, D , the explicit form for (a=1, 2) is as follows 4 (5) −" = 16253847, −" = 17283645, " =3 1 , =2 1 34, (13) − = 13245867, −" = 14235867. International Journal of Discrete Mathematics 2016; 1(1): 15-19 17 In this way, all elements are accounted for. Hence, can be written as = 51, 5 ", 5 , 5 , 5 " , 5 " , 5 , 5 " = (, 12345678, 1234, 5678, 15263748, 16253847, 18273546, 17283645, 15263748, 16253847, 18273546, 17283645, 14235768, 13245867, 13245768, 14235867 . (14) The calculated result means that the matrix group of =< 1 2=3>4&5(6?7@8A, order 16 is isomorphic to a subgroup of the permutation group S8. Though the correspondence chosen in Eq. (9) is not unique, ?= 1 2(3@4A5<6=7&8>, for example, (20) =A 1 2@3?4(5>6&7<8=. " =2 1 34, =7 1 283645, This leads to the products of two γ matrices, (a, b=1, (15) 2, 3, 4 and a ≠ b) =6 1 253847, or " = 1<2=3>4&5(6?7@8A, " = 1?2(3@4A5<6=7&8>, " 2= 1 56, =3 1 245867, (16) = 1A2@3?4(5>6&7<8=, 8= 1 275436, " or = 16253748<(=?>A&@, = 18273645<@=A>(&?, " =2 1 78, =5 1 263847, (21) (17) = 14235867<&=>(A?@, =4 1 235867, and the products of three matrices it can be found that the procedures for calculating the products of matrices are almost the same.