The Theorem of Cayley and Matrices
Xiao-Yan Gu^{1}^{, *}, Jian-Qiang Sun^{2}
^{1}Department of Physics, East China University of Science and Technology, Shanghai, China
^{2}College of Information Science and Technology, Hainan University, Haikou, China
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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
Abstract: In this article, the connections between symmetric groups and the matrix groups are investigated for exploring 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
1. Introduction
It is well known that Cayley's theorem is one of the most important results in group theory [1-3]. The theorem shows that if G is a finite group of order n, then G is isomorphic to a subgroup of S_{n}. This is a classic and intriguing result [4,5]. With this in hand, if we can fully understand the structure and properties of S_{n} and its subgroups, then we will automatically understand the structure and properties of this finite group. However, the symmetric group S_{n} of all the permutations of n objects has order n!, trying to use S_{n} to answer any questions about G means working with a group that factorially larger. Due to the sheer size of S_{n}, this becomes problematic. Some questions may arise from the investigation for the further applications of Cayley's theorem [6,7]. For instance, is it possible that G is isomorphic to a subgroup of S_{k} where k<n?
Consider the symmetry group of the equilateral triangle, D_{3}. The multiplication table shows that D_{3} is a finite group of the 6 group elements. These elements may be represented as permutations of {1, 2, 3, 4, 5, 6} according to the Cayley's theorem. For example, the rotation through 2π/3 can be represented to
(1)
On the other hand, if we label the vertices of the triangle with the numbers 1, 2, and 3, then the elements may be represented as permutations of {1, 2, 3}. With one certain labeling, we would get the rotation through 2π/3 to (123). We are acquainted with this fact that the dihedral group D_{3} of order 6 is isomorphic to the symmetric group S_{3} of all permutations of 3 objects. We now wonder whether one can find the symmetric group of S_{k} (k<n) corresponding to some other specific finite groups G of order n and have the explicit connections between them.
The groups of particular interest, discussed in this article, are matrix groups. matrices, also known as the Dirac matrices, play a highly significant role in mathematics and physics [8-11]. As a set of matrices satisfying special anticommutation relations [12],
(2)
N matrices and all its products generate a finite matrix group . Analogous sets of matrices can be defined in any dimension and signature of the metric. In five space-time dimensions, the four matrices above together with the fifth matrix generate the Clifford algebra.
According to the familiar theorem of Cayley, the connections between and S_{n} is fairly straightforward from the group table of the finite group . However, it is not an easy task to find a symmetric group of S_{k} with k<n corresponding to the group of order n.
In the following section, we briefly describe the comparison of different connections between the symmetric group and the matrix group of N=2. Our method is then used for the analysis of the finite matrix group in Section 3. The explicit form of the permutation group corresponding to the group of N=4 are obtained in Section 4. The results are discussed in detail in Section 5. We conclude in the final section after pointing out various directions for future investigations.
2. Matrix Group
Two matrices and satisfying the anti-commutation relations Eq. (2) and all their possible products form the Γ matrix group, ={1, , -1, -, , -, -, }. The multiplication table of the group is calculated as follows.
Table 1. The group table of .
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
1 | γ_{1}γ_{2} | -1 | -γ_{1}γ_{2} | γ_{1} | -γ_{2} | -γ_{1} | γ_{2} | |
1 | 1 | γ_{1}γ_{2} | -1 | -γ_{1}γ_{2} | γ_{1} | -γ_{2} | -γ_{1} | γ_{2} |
γ_{1}γ_{2} | γ_{1}γ_{2} | -1 | -γ_{1}γ_{2} | 1 | -γ_{2} | -γ_{1} | γ_{2} | γ_{1} |
-1 | -1 | -γ_{1}γ_{2} | 1 | γ_{1}γ_{2} | -γ_{1} | γ_{2} | γ_{1} | -γ_{2} |
-γ_{1}γ_{2} | -γ_{1}γ_{2} | 1 | γ_{1}γ_{2} | -1 | γ_{2} | γ_{1} | -γ_{2} | -γ_{1} |
γ_{1} | γ_{1} | γ_{2} | -γ_{1} | -γ_{2} | 1 | -γ_{1}γ_{2} | -1 | γ_{1}γ_{2} |
-γ_{2} | -γ_{2} | γ_{1} | γ_{2} | -γ_{1} | γ_{1}γ_{2} | 1 | -γ_{1}γ_{2} | -1 |
-γ_{1} | -γ_{1} | -γ_{2} | γ_{1} | γ_{2} | -1 | γ_{1}γ_{2} | 1 | -γ_{1}γ_{2} |
γ_{2} | γ_{2} | -γ_{1} | -γ_{2} | γ_{1} | -γ_{1}γ_{2} | -1 | γ_{1}γ_{2} | 1 |
Denote the eight elements {1, , -1, -, , -, -, }, in the group by the digits {1, 2, 3, 4, 5, 6, 7, 8} respectively as shown in the first row of the group table, then, in accordance with the Cayley's theorem, and appear
(3)
The rest in the group can be got from the multiplication rule. The matrix group of order 8 is isomorphic to the subgroup of the permutation group S_{8}. These are results directly from the Cayley's theorem.
On the other hand, notice that the order of the elements in the group are respectively: 1, 4, 2, 4, 2, 2, 2, 2, which are the same as the elements in the dihedral group D_{4} of the symmetry group of a square. This is a helpful hint. Label the vertices of the square with the number 1, 2, 3, and 4, then the elements in the group D_{4} may be represented as permutations of 1, 2, 3, 4.
(4)
where R is a π/2 rotation about the center of the square and S_{0}, S_{1}, S_{2}, S_{3} are the reflections about four symmetry axes respectively. It is not difficult to prove that there exists an isomorphism between these two groups, ≈ D_{4}. Hence, this isomorphism gives a one-to-one correspondence of the elements from to D_{4}, the explicit form for (a=1, 2) is as follows
(5)
and all the remaining elements in the matrix group can be got from the explicit form (Eq. (5)) of and . Then, becomes
={1, , -1, -, , -, -, }
(6)
It means that the matrix group of order 8 is also isomorphic to a subgroup of the permutation group S_{4}. The results in Eq. (3) and Eq. (5) present different connections between the symmetric group S_{k} and the matrix group . The comparison shows that the number of the different objects in the S_{k} in our results is only half of that from the theorem. We will continue to use our method to investigate the permutation group related to other matrix groups in the following.
3. Matrix Group
The Pauli matrices σ_{1}, σ_{2} and σ_{3}, satisfying the relations,
(7)
are a set of γ matrices in dimension 3 with the Euclidean metric signature. The matrix group is the set of three γ matrices and all their products,
(8)
With a similar analysis for the matrix group , the Cayley's theorem states that the matrix group of order 16 is isomorphic to the subgroup of the permutation group S_{16} directly from the multiplication table.
What is interesting is that how to find a permutation group S_{k }(k<16 ) whose subgroup is isomorphic to the matrix group of order 16 through investigation. Choose the correspondence between the γ matrices and the permutations as follows,
(9)
the products of γ matrices can be written as
(10)
Notice that the square of (a, b=1, 2, 3 and a≠ b) is
(11)
the explicit forms for -(a=1, 2, 3) are given by
(12)
As noted, it is not difficult to come to the remaining products of the γ matrices,
(13)
In this way, all elements are accounted for. Hence, can be written as
(14)
The calculated result means that the matrix group of order 16 is isomorphic to a subgroup of the permutation group S_{8}. Though the correspondence chosen in Eq. (9) is not unique, for example,
(15)
or
(16)
or
(17)
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 of N=2 can be taken as that of N=4m-2 when m=1. Since leads to , and , can be written as
(18)
The results in Eqs. (6), (14) and (18) can be used to check the relation ≈ {, i } through direct calculations on the production of the permutations. This method might also be generalized to understand the general properties of the gamma matrix groups, ≈ {, i }.
4. Matrix Group
The set of products of the four γ matrices forms the matrix group ,
(19)
In order to express implicitly the symmetric pattern of the formulas, the second half of the digits from 1 to 16 are denoted by the letters A, B, C, D, E, F, G and H respectively for convenience. Based on careful analysis, the explicit form of the four γ matrices are taken as follows,
(20)
This leads to the products of two γ matrices, (a, b=1, 2, 3, 4 and a ≠ b)
(21)
and the products of three matrices
(22)
In consideration of the square of (a, b=1, 2, 3, 4 and a ≠ b),
(23)
the explicit form for - (a=1, 2, 3, 4) becomes
(24)
One may also check the results for the products of - (a, b=1, 2, 3, 4 and a ≠ b),
(25)
and the products of - (a, b, c=1, 2, 3, 4 and a ≠ b ≠ c),
(26)
Similar to the previous statements, there are also other options for the explicit form taken in Eq. (20) for the four γ matrices, such as
(27)
Different from the groups and , it is interesting to find that the product of all is a two-order element in the group ,
(28)
while Eq. (6) and Eq. (10) show that the product of all are the elements of order four.
This formula is helpful in understanding the isomorphism relations between and ( ≈ ). Since this part can be viewed as the case of N=4m when m=1, this research might also be used to reveal the properties of the matrix groups, such as, ≈ .
Hence, the symmetric group whose subgroup corresponds to the gamma matrix group of order 32 is related to S_{16} in present results, while it is connected with S_{32} from the Cayley's theorem. The order of the symmetric group in our method is far less than the latter.
5. Discussions
The symmetric groups isomorphic to the matrix groups have been discussed when the value of N is even or odd. It is found that the matrix group of order 8 is isomorphic to a subgroup of the permutation group S_{4} and the matrix group of order 16 is isomorphic to a subgroup of the permutation group S_{8}. As is well known, up to isomorphism, there are five different finite groups of order 8. 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 . Since the symmetric groups corresponding to groups have been found, one might try to think if can be represented as the product of finite group of order 8 and the cyclic group C_{2}. Since there should be elements of order 8 in the cyclic group C_{8}, it is natural that is not related with _{2}. It is verified that is also not the direct product _{2} or _{2} or _{2} or _{2} through multiplication of the permutations. In fact, it is finally found that can be represented as the semi-product of and the group C_{2},_{ }. Due to the expression
(29)
therefore, can be rewritten in the form
(30)
It can be verified that is a group isomorphic to the quaternion group Q_{8}, which is an invariant subgroup of index two of . The cyclic group of is a subgroup of and it is not a normal subgroup. That is, is the semi-product of these two subgroups.
It can also be found that though have various forms in Eqs. (15)-(17), all the elements of order 4 in the group Q_{8 }remain unchanged. They are (1423)(5768), (1728)(3645), (1526)(3748), (1324)(5867), (1827)(3546) and (1625)(3847) respectively. Further, it is noticed that Eq. (11) leads to
(31)
If the corresponding relations between the permutations and fundamental quaternion units are
(32)
Equation (31) will immediately remind us of the famous formula of quaternion algebra
(33)
This indicates that there is an isomorphism between the quaternion group and an invariant subgroup of index two in the group . This conclusion could be meaningful in understanding the properties of the quaternion group. It is known that the quaternion group play an important role in mathematics and physics. Equations (31)-(33) might be an interesting starting point to study the quaternions and octonions.
6. Conclusions
In this article, we find the symmetric group S_{k} (k=n/2) corresponding to the matrix group of order n and provide the exact relations between them. This specific finite group is investigated for the value of N that is odd or even. Especially, when N is even, we studied separately the cases when N=4n-2 and when N=4n. This research indicates that the order of the symmetric group in our approach is less than that of the group directly from the Cayley's theorem.
The generalization of this method to matrix group when N is arbitrary number, even or odd, is straightforward. The properties of the symmetric groups can be used to check and understand the properties of the matrix groups which will widen the application of the familiar theorem of Cayley. It is interesting to find that the study of matrix groups is meaningful to understand the properties of the quaternion group. These conclusions might be useful to study the Clifford algebra, Lorentz group and its representations. It is also hoped to stimulate one to apply these results to other interesting fields.
Acknowledgement
This work was supported by the National Natural Science Foundation of China (Grant No. 11561018 and 11301183).
References