Research Article | Open Access
Tianlan Chen, Muhammad Nadeem Bari, Muhammad Aslam Malik, Hafiz Muhammad Afzal Siddiqui, Jia-Bao Liu, "Icosahedral Group and Classification of PSL(2, Z)-Orbits of Real Quadratic Fields", Journal of Mathematics, vol. 2020, Article ID 9568254, 10 pages, 2020. https://doi.org/10.1155/2020/9568254
Icosahedral Group and Classification of PSL(2, Z)-Orbits of Real Quadratic Fields
Reduced numbers play an important role in the study of modular group action on the -subset of . For this purpose, we define new notions of equivalent, cyclically equivalent, and similar -circuits in -orbits of real quadratic fields. In particular, we classify -orbits of containing G-circuits of length 6 and determine that the number of equivalence classes of -circuits of length 6 is ten. We also employ the icosahedral group to explore cyclically equivalence classes of circuits and similar -circuits of length 6 corresponding to each of these ten circuits. This study also helps us in classifying reduced numbers lying in the -orbits.
Let where and is a square-free positive integer. Now, we define the following set and its subset as Then,
For its algebraic conjugate has different signs; then, is said to be an ambiguous number, that is, is an ambiguous number if and only if A quadratic irrational number is said to be reduced if and It is obvious from the definition that every reduced number is ambiguous. Note that if is a reduced number, then , and are the ambiguous numbers but not reduced . The modular group is the group of all linear fractional transformations with where are integers.
This group can be presented as where
Throughout this paper, denotes the number of partitions of , whereas stands for the dihedral group of order and stands for the symmetric group of order It is easy to see that
Coset diagrams for the -orbit acting on the real quadratic field give some interesting information. A coset diagram is a graph consisting of vertices and edges. It depicts a permutation representation of the modular group the 3-cycles of are denoted by three vertices of a triangle permuted anticlockwise by , and the two vertices which are interchanged by are joined by an edge.
In , it was proved that the ambiguous numbers in the orbit , form a single closed path (called -circuit or simply circuit), and it is the only circuit contained in the coset diagram for the orbit The number of disjoint orbits is where is equal to the number of circuits in the coset diagram under the action of on Thus, it becomes interesting to classify the circuit.
If , , , …, and , , , …, are two sequences of positive integers, then by a circuit , we shall mean the circuit in which triangles have one vertex outside the circuit and triangles have one vertex inside the circuit, where
This circuit induces an element and fixes a particular vertex of a triangle lying on the circuit. Throughout this paper, -circuit (resp. -orbit) will be simply denoted by circuit (resp. orbit). The concept of the circuit grew out of the study of group action on and the study of -subsets
In this paper, we define what a circuit of specific length is and we classify nonequivalent circuits of length 6 so as to classify orbits containing these circuits. We also consider some of the elementary concepts associated with equivalent circuits, cyclically equivalent circuits, and similar circuits which are introduced to explore transitive -subsets (called orbits) of
In this section, necessary definition of an equivalent circuits will be given. The definition of an equivalent circuit, that is now standard and was in formulation since long ago. It was required of course, a definition that is as broad as possible, so that it would include all special cases of the various examples that are useful in group action, -subset and orbit. However, the definition was also required to be narrow enough that the standard theorems corresponding to these concepts like partition of positive integer q would help in general. The definition finally settled on may seem a bit abstract, but as we work through various ways of determining orbits, we will get a better feeling for what the concept means.
Definition 1. Two circuits and are said to be equivalent iff where That is, the circuits are equivalent to if and only if they are obtained just by permuting the entries , , , …, Notation for equivalent is It is easy to see that being equivalent of circuits is an equivalence relation. Thus, a property is possessed by one circuit that is also possessed by all equivalent circuits. Such properties which are preserved under equivalent are called equivalent properties or circuit invariant.
Two circuits and are said to be cyclically equivalent if and only if the circuit where Notation for cyclically equivalent is It is easy to see that being cyclically equivalent of circuits is an equivalence relation.
Two circuits are said to be similar if they represent the same orbit. That is, two circuits and are said to be similar if and only if where Notation for similarly equivalent is It is easy to see that being similar circuits is an equivalence relation.
It is interesting to note that the orbit containing circuit has exactly ambiguous numbers, while this circuit consists of only number of reduced numbers. Thus, studying orbits with the help of reduced numbers is fruitful and economical. Throughout this paper, , , , , , and denote distinct positive integers, and the expression is replaced by when Classification of nonequivalent circuits and cyclically equivalent circuits plays a significant role in determining the orbits of because with this, the task of finding orbits is simplified.
In , Aslam and Asim found -subsets of Since -subsets may or may not be transitive, it becomes interesting to explore transitive -subsets called orbits. Reduced quadratic irrational numbers and types of G-circuits with length four by modular group and the orbits of , p = 3 (mod 4) under the action of modular group have been studied in [4, 5].
3. Materials and Methods
Lemma 1 (see ). If has a circuit then and are the circuits of , , and , respectively.
Lemma 2 (see ). If is the circuit contained in the orbit , then , , and are all distinct.
Lemma 3 (see ). For a given sequence of positive integers , , , …, , there does not exist a circuit which has a period of length where divides
Lemma 4 (see ). The number of different arrangements of objects of which is alike, is alike, is alike, …, is alike is , where
3.1. Classification of Circuits of Length Six
Circuits play an important role in the study of modular group acting on the quadratic field.
We start this section with a consideration of finding the nonequivalent relation of circuits in a specific length Given the positive integer we say that the sequence of positive integers with constitutes a partition of if
Theorem 1. The number of equivalence classes of length is exactly where denotes the partition of
Proof. Let denote the partition of , and we are looking in determining all equivalence classes of equivalent circuits of length For a given circuit of length to find all equivalence classes of equivalent circuits of length we adopt partitions of in the sense that entries are alike, entries are alike, entries are alike, …, entries are alike, where We get nonequivalent circuits of length corresponding to each partition of where repeats times,
Here, because if , then the circuit corresponding to this partition is where repeats times. This circuit of length is not possible by Lemma 3. So, distinct classes of the equivalent circuit of length are exactly
Remark 1. For there are precisely two partitions, namely, 2 and 1 + 1. Circuits of length 2 corresponding to these partitions are and
Note 1. We can get all other circuits which are equivalent to by just permuting
In the following theorems, we discuss equivalence classes of circuits, cyclically equivalence classes of circuits, equivalent circuits, cyclically equivalent circuits, and similar circuits of length 6.
Corollary 1. There are precisely ten nonequivalent circuits of length six.
Proof. Let be different positive integers. Then, by Theorem 1, we have ten nonequivalent circuits, namely, , corresponding to the number , namely, 1 + 1 + 1 + 1 + 1 + 1, 2 + 1 + 1 + 1 + 1, 3 + 1 + 1 + 1, 2 + 2 + 1 + 1, 2 + 2 + 2, 3 + 2 + 1, 4 + 1 + 1, 4 + 2, 5 + 1, and 3 + 3, respectively. The circuit corresponding to summand 6 is This circuit is not possible by Lemma 3. So, these are the only ten nonequivalent circuits of length 6.
The notation used in this paper for equivalence classes of circuits of length 6 is , and the number of circuits equivalent to is denoted by Similarly, the notation for cyclically equivalent classes is , and denotes the number of circuits cyclically equivalent to The number of distinct orbits corresponding to is denoted by Furthermore, each cyclically equivalent class is discussed in each corresponding relevant corollary.
Corollary 2. There are 720 equivalent circuits of length 6 in which all numbers are different.
Proof. It is well known that In our case, We know that there are arrangements of six different numbers taken all at a time, and so circuits corresponding to these arrangements are for each Hence the proof.
Corollary 3. If are distinct positive integers, then there exist 60 cyclically equivalent classes
Proof. Let be different positive integers. It is well known that the icosahedral group is isomorphic to alternating group Now, the cyclically equivalent classes are obtained by for each There are exactly 60 cyclically equivalent classes, namely, , and
Corollary 4. If are distinct positive integers, then each cyclically equivalent class contains 12 cyclically equivalent circuits.
Proof. Circuits cyclically equivalent to circuit are for each which is shown in Table 1. Similarly, we can find cyclically equivalent classes corresponding to the remaining circuits.
Circuits of length 6 which are given in Table 1 are cyclically equivalent. Moreover, by Lemma 1, is the circuit contained in is the circuit contained in is the circuit contained in and is the circuit contained in
From Table 1, it is easy to see that the effect of permutation on the circuit is the same as to change the places of the circuit accordingly. So, if the circuit in which at least two entries are the same, we change places of circuits according to permutation. As each circuit of length 6 contains 3 reduced numbers, each cyclically equivalent class contains 12 reduced numbers. Since there are 60 cyclically equivalent classes, each class contains 12 reduced numbers, so reduced numbers used in are which equals to
Table 2 is of considerable utility because it provides us with the exact number of circuits of length 6 and hence the number of -orbits of
The following corollaries are an immediate consequence of Theorem 1.