The Theorem of Cayley and Matrices

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 Γ 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: Γ 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 Γ Γ, Γ 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, γ 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 (ksubgroups, 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 Γ γ 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 kClifford 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 Γ n the group table of the finite group . However, it is not an Γ easy task to find a symmetric group of S with k

matrix group of N=2. Our method is then used for the and all the remaining elements in the matrix group can be Γ Γ analysis of the finite matrix group in Section 3. The got from the explicit form (Eq. (5)) of and . Then, Γ Γ explicit form of the permutation group corresponding to the becomes group of N=4 are obtained in Section 4. The results are Γ ={1, , -1, - , , - , - , } discussed in detail in Section 5. We conclude in the final Γ 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 Γ 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 Γ 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 Γ k multiplication table of the group is calculated as follows. will continue to use our method to investigate the permutation Γ group related to other Γ 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 σ , σ and σ , satisfying the relations, -1 - 1 - - 1 2 3 -1 -1 - 1 - - (7) - - 1 -1 - - = 1 + ∑ , - - 1 - -1 - - - 1 - -1 are a set of γ matrices in dimension 3 with the Euclidean - - - -1 1 - metric signature. The matrix group is the set of three γ Γ - - - -1 1 matrices and all their products, Denote the eight elements {1, , -1, - , , - , - , (8) Γ = ±1, ±γ, ±γ, ±γ, ±γγ, ±γγ, ±γγ, ±γγγ. }, in the group by the digits {1, 2, 3, 4, 5, 6, 7, 8} Γ With a similar analysis for the matrix group , the Cayley's respectively as shown in the first row of the group table, then, Γ theorem states that the matrix group of order 16 is in accordance with the Cayley's theorem, and appear Γ 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. Γ 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 Γ 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 Γ 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 − 7= 1 283645. between these two groups, ≈ D . Hence, this isomorphism Γ 4 As noted, it is not difficult to come to the remaining gives a one-to-one correspondence of the elements from to Γ 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 Γ

Γ = ±1, ±γ, ±γ, ±γ, ±γγ, ±γγ, ±γγ, ±γγγ = , 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 Γ = 1 2345678, order 16 is isomorphic to a subgroup of the permutation group

S8. Though the correspondence chosen in Eq. (9) is not unique, = 1 2345678, for example, (20) = 1 2345678.

=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 = 12345678,

= 12345678, =2 1 56, =3 1 245867, (16) = 12345678, =8 1 275436, or = 16253748,

= 18273645, =2 1 78, =5 1 263847, (21) (17) = 14235867, =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. It is worth mentioning that the case of the matrix group = 15263847, Γ of N=2 can be taken as that of N=4m-2 when m=1. Since = 17283546, = leads to = − , and = , Γ can be written as = 13245768, (22) Γ = ±1,±γ, ±γ, ±γγ, ±γγγ, ±γγ, ∓γγ, ∓γ = 12 34 56 78 . In consideration of the square of (a, b=1, 2, 3, 4 and a =±1, ±γ , ±γ , ±γ γ , ±i, ±iγ , ±iγ , ±iγ γ . (18) ≠ b), The results in Eqs. (6), (14) and (18) can be used to check the relation ≈ { , i } through direct calculations on the Γ Γ Γ = = = = production of the permutations. This method might also be (23) generalized to understand the general properties of the gamma = 2= 1 345678 = −1, matrix groups, ≈ { , i }. the explicit form for - (a=1, 2, 3, 4) becomes Γ ΓΓ

4. Matrix Group − = ,

The set of products of the four γ matrices forms the matrix − = 1 2345678, group , Γ − = 1 2345678,

Γ = ±1, ±γ, ±γ, ±γ, ±γ, ±γγ, ±γγ, ±γγ, ±γγ, ±γγ, ±γγ, (24) − = 12345578. (19) ±γγγ, ±γγγ, ±γγγ, ±γγγ, ±γγγγ. One may also check the results for the products of - (a, In order to express implicitly the symmetric pattern of the b=1, 2, 3, 4 and a ≠ b), formulas, the second half of the digits from 1 to 16 are denoted − = 12345678, by the letters A, B, C, D, E, F, G and H respectively for convenience. Based on careful analysis, the explicit form of − = 12345678, the four γ matrices are taken as follows,

− = 12345678,

=2 1 345678, − = 15263847, 18 Xiao-Yan Gum and Jian-Qiang Sun: The Theorem of Cayley and Γ Matrices

− = 17283546, 5. Discussions (25) The symmetric groups isomorphic to the matrix groups − = 13245768, Γ have been discussed when the value of N is even or odd. It is and the products of - (a, b, c=1, 2, 3, 4 and a ≠ b ≠ c), found that the matrix group of order 8 is isomorphic to a Γ subgroup of the permutation group S and the matrix group 4 Γ − = 16253748, of order 16 is isomorphic to a subgroup of the permutation group S8. As is well known, up to isomorphism, there are five − = 18273645, different finite groups of order 8. The first is the cyclic group

− = 14235867, C8. The second is the dihedral group D4, where two generators can be denoted by R and S , satisfying . The (26) 0 = = − = 12345678. third is an Abelian group, , where the = ⨂ generators satisfy and RS =S R . The fourth is Similar to the previous statements, there are also other = = 0 0 also a commutative group, , and the generators options for the explicit form taken in Eq. (20) for the four γ = ⨂ satisfy . The fifth is a quaternion group Q , matrices, such as = = = 8 the generators satisfy . Since the symmetric = = groups corresponding to groups have been found, =2 1 34, Γ and Γ one might try to think if can be represented as the product Γ =5 1 263748, of finite group of order 8 and the cyclic group C2. Since there should be elements of order 8 in the cyclic group C , it is = 1 2345678, 8 natural that is not related with . It is verified that Γ C⨂C2 Γ (27) is also not the direct product or or = 1 2345678. D⨂C2 ⨂C2 ⨂C2 or 2 through multiplication of the permutations. In fact, Different from the groups and , it is interesting to find ⨂C Γ Γ it is finally found that can be represented as the that the product of all is a two-order element in the group Γ semi-product of and the group C , . Due to , 2 Γ = × Γ the expression

= − Γ = ±1,±γ, ±γ, ±γ, ±γγ, ±γγ, ±γγ, ±γγγ = 12345678, (28) = ±1, ±γ, ±γ, ±γγ, ±i, ±iγ, ±iγ, ±iγγ while Eq. (6) and Eq. (10) show that the product of all are the elements of order four. = 1, γγ, −γγ, γγ, −1, −γγ, γγ, −γγ, This formula is helpful in understanding the isomorphism γ, γγγ, γ, −γ, −γ, −γγγ, −γ, γ relations between Γ and Γ (Γ ≈ Γ ). Since this part can be viewed as the case of N=4m when m=1, this research might = 1, iγ, iγ, γγ, −1, −iγ, −iγ, −γγ, also be used to reveal the properties of the matrix groups, Γ (29) such as, ≈ . γ, i, −iγγ, −γ, −γ, −i, iγγ, γ Γ Γ Hence, the symmetric group whose subgroup corresponds therefore, Γ can be rewritten in the form to the gamma matrix group of order 32 is related to S in Γ 16 present results, while it is connected with S32 from the Cayley's theorem. The order of the symmetric group in our method is far less than the latter.

Γ = 1, iγ, iγ, γγ, −1, −iγ, −iγ, −γγ × 1, γ = , 14235768, 17283645, 15263748, 12345678, 13245867, 18273546, 16253847 × , 1234, = , 14235768, 17283645, 15263748, 12345678, 13245867, 18273546, 16253847, 1234, 13245768, 18273546, 16253847, 5678, 14235867, 17283645, 15263748 ≃ , 15263748, 17283645, 13245867, 12345678, 16253847, 18273546, 14235768, 1256, 16253748, 18275436, 14235768, International Journal of Discrete Mathematics 2016; 1(1): 15-19 19

3478, 15263847, 17283546, 13245867 ≃ , 17283645, 13245867, 15263748, 12345678, 18273546, 14235768, 16253847, 1278, 18273645, 14235867, 16253748, 3456, 17283546, 13245768, 15263847. (30)

It can be verified that interesting to find that the study of Γ matrix groups is is a group the set of 1, iγ, iγ, γγ, −1, −iγ, −iγ, −γγ meaningful to understand the properties of the quaternion isomorphic to the quaternion group Q8, which is an invariant group. These conclusions might be useful to study the Clifford subgroup of index two of . The cyclic group of is a Γ 1, γ algebra, Lorentz group and its representations. It is also hoped subgroup of and it is not a normal subgroup. That is, is Γ Γ to stimulate one to apply these results to other interesting the semi-product of these two subgroups. fields. It can also be found that though have γ, γ, and γ various forms in Eqs. (15)-(17), all the elements of order 4 in Acknowledgement the group Q8 remain unchanged. They are (1423)(5768), (1728)(3645), (1526)(3748), (1324)(5867), (1827)(3546) and This work was supported by the National Natural Science (1625)(3847) respectively. Further, it is noticed that Eq. (11) Foundation of China (Grant No. 11561018 and 11301183). leads to = = (31) References = = 12345678 = −1. If the corresponding relations between the permutations and [1] Cayley, A. (1854). On the theory of groups as depending on the symbolic equation n=1.-Part II. Phil. Magazine Ser., 7 (4), fundamental quaternion units are 40-47. (32) ↔ , ↔ , ↔ , [2] Nummela, E. C. (1980). Cayley’s Theorem for Topological Groups. Am. Math. Mon., 87 (3), 202-203. Equation (31) will immediately remind us of the famous formula of quaternion algebra [3] Crillya, T., Weintraubb, S. H., and Wolfsonc, P. R., (2016). Arthur Cayley, Robert Harley and the quintic equation: newly i = = = = −1. (33) discovered letters 1859–1863. Historia Mathematica, in press. This indicates that there is an isomorphism between the [4] Suksumran, T., Wiboonton, K. (2015). Isomorphism Theorems quaternion group and an invariant subgroup of index two in for Gyrogroups and L-Subgyrogroups. J. Geom. Symmetry Phys., 37, 67-83. the group . This conclusion could be meaningful in Γ understanding the properties of the quaternion group. It is [5] Childs, L. N., Corradino, J. (2007). Cayley's Theorem and known that the quaternion group play an important role in Hopf Galois structures for semidirect products of cyclic groups. mathematics and physics. Equations (31)-(33) might be an J. Algebra, 308 (1), 236-251. interesting starting point to study the quaternions and [6] Roman, S. (2012). Fundamentals of group theory: an advanced octonions. approach. New York: Birkhäuser. [7] Hamermesh, M. (1962). Group theory and its application to 6. Conclusions physical problems. Reading: Addison-Wesley.

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