# Icosahedral Group and Classification of PSL(2,Z)-Orbits of Real Quadratic Fields

Authors: Tianlan Chen; Muhammad Nadeem Bari; Muhammad Aslam Malik; Hafiz Muhammad Afzal Siddiqui; Jia-Bao Liu
Journal: Journal of Mathematics
 (2020)

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## Abstract

Reduced numbers play an important role in the study of modular group action on thePSL2,ℤ-subset of Qm/Q. For this purpose, we define new notions of equivalent, cyclically equivalent, and similar G-circuits in PSL2,ℤ-orbits of real quadratic fields. In particular, we classify PSL2,ℤ-orbits of Qm/Q=∪k∈NQ∗k2m containing G-circuits of length 6 and determine that the number of equivalence classes of G-circuits of length 6 is ten. We also employ the icosahedral group to explore cyclically equivalence classes of circuits and similar G-circuits of length 6 corresponding to each of these ten circuits. This study also helps us in classifying reduced numbers lying in the PSL2,ℤ-orbits.


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## Body

## 1. Introduction

Letn=k2m, where k∈N and m is a square-free positive integer. Now, we define the following set Q∗n=a+n/c:a,b=a2−n/c,c∈Z,c≠0anda,b,c=1 and its subset as Qred∗n=γ∈Q∗n:γ>1and−1<γ¯<0. Then, Qm/Q=∪k∈NQ∗k2m.Forγ=a+n/c∈Q∗n, its algebraic conjugate γ¯=a−n/c has different signs; then, γ is said to be an ambiguous number, that is, γ is an ambiguous number if and only if a2<n1. A quadratic irrational number γ is said to be reduced if γ>1 and −1<γ¯<0. It is obvious from the definition that every reduced number is ambiguous. Note that if γ is a reduced number, then γ¯,−γ¯, and −γ are the ambiguous numbers but not reduced [1]. The modular group PSL2,ℤ is the group of all linear fractional transformations z⟶sz+r/uz+v with sv−ru=1, where s,r,u,andv are integers.This group can be presented asG=x,y:x2=y3=1, where x:z⟶−1/z,y:z⟶z−1/z.Throughout this paper,pq denotes the number of partitions of q, whereas Dn stands for the dihedral group of order 2n and Sn stands for the symmetric group of order n!. It is easy to see that Sn=n−1!/2×2n=An−1×Dn.Coset diagrams for theG-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 G, the 3-cycles of y are denoted by three vertices of a triangle permuted anticlockwise by y, and the two vertices which are interchanged by x are joined by an edge.In [2], it was proved that the ambiguous numbers in the orbit γG,γ∈Q∗n, form a single closed path (called G-circuit or simply circuit), and it is the only circuit contained in the coset diagram for the orbit γG. The number of disjoint orbits is γG, where γ∈Q∗n is equal to the number of circuits in the coset diagram under the action of G on Q∗n. Thus, it becomes interesting to classify the circuit.Ifl1′, l3′, l5′, …, l2t−1′ and l2′, l4′, l6′, …, l2t′ are two sequences of positive integers, then by a circuit l1′,l2′,l3′,l4′,…,l2t−1′,l2t′, we shall mean the circuit in which l2i−1′ triangles have one vertex outside the circuit and l2i′ triangles have one vertex inside the circuit, where 1≤i≤t.This circuit induces an elementg=xy2l1′xyl2′xy2l3′…xy2l2t−1′xyl2t′ and fixes a particular vertex of a triangle lying on the circuit. Throughout this paper, G-circuit (resp. G-orbit) will be simply denoted by circuit (resp. orbit). The concept of the circuit grew out of the study of group action on Qm∪∞ and the study of G-subsets Q∗n=a+n/c:a,b=a2−n/c,c∈Z,c≠0anda,b,c=1.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 transitiveG-subsets (called orbits) of Q∗n.


## 2. Preliminaries

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,G-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 circuitsl1′,l2′,l3′,l4′,…,l2t−1′,l2t′ and l1,l2,l3,l4,l5,l6,…,l2t−1,l2t are said to be equivalent iff l1θ′,l2θ′,l3θ′,l4θ′,…,l2t−1θ′,l2tθ′=l1,l2,l3,l4,l5,l6,…,l2t−1,l2t, where θ∈S2t. That is, the circuits are equivalent to l1′,l2′,l3′,l4′,…,l2t−1′,l2t′ if and only if they are obtained just by permuting the entries l1′, l2′, l3′, …, l2t′. 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 circuitsl1′,l2′,l3′,l4′,…,l2t−1′,l2t′ and l1,l2,l3,l4,l5,l6,…,l2t−1,l2t are said to be cyclically equivalent if and only if the circuit l1θ′,l2θ′,l3θ′,l4θ′,…,l2t−1θ′,l2tθ′=l1,l2,l3,l4,l5,l6,…,l2t−1,l2t, where θ∈Dt. Notation for cyclically equivalent is “∼c.” 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 circuitsl1′,l2′,l3′,l4′,…,l2t−1′,l2t′ and l1,l2,l3,l4,l5,l6,…,l2t−1,l2t are said to be similar if and only if l1θ′,l2θ′,l3θ′,l4θ′,…,l2t−1θ′,l2tθ′=l1,l2,l3,l4,l5,l6,…,l2t−1,l2t, where θ∈Ct=135…2t−1,246…2t. Notation for similarly equivalent is “∼s.” It is easy to see that being similar circuits is an equivalence relation.
It is interesting to note that the orbit containing circuitl1′,l2′,l3′,l4′,…,l2t−1′,l2t′ has exactly 2l1′+l2′+l3′+⋯+l2t−1′+l2t′ ambiguous numbers, while this circuit consists of only t number of reduced numbers. Thus, studying orbits with the help of reduced numbers is fruitful and economical. Throughout this paper, l1′, l2′, l3′, l4′, l5′, and l6′ denote distinct positive integers, and the expression d+d2+4ef/2f is replaced by d/h+d2+4ef/h2/2f/h when d,2e,2f=h>1. Classification of nonequivalent circuits and cyclically equivalent circuits plays a significant role in determining the orbits of Qm/Q because with this, the task of finding orbits is simplified.
In [3], Aslam and Asim found G-subsets of Q∗k2m⊆Qm/Q=∪k∈NQ∗k2m. Since G-subsets may or may not be transitive, it becomes interesting to explore transitive G-subsets called orbits. Reduced quadratic irrational numbers and types of G-circuits with length four by modular group and the orbits of Q∗p, p = 3 (mod 4) under the action of modular group have been studied in [4, 5].


## 3. Materials and Methods

The following results of [6–8] are used in the sequel.Lemma 1 (see [7]).
IfγG has a circuit l1′,l2′,l3′,l4′,l5′,l6′, then l2′,l3′,l4′,l5′,l6′,l1′,l1′,l6′,l5′,l4′,l3′,l2′, and l6′,l5′,l4′,l3′,l2′,l1′ are the circuits of −γG, −γ¯G, and γ¯G, respectively.Lemma 2 (see [7]).
Ifl1′,l2′,l3′,l4′,l5′,l6′ is the circuit contained in the orbit γG, then γG,−γG, −γ¯G, and γ¯G are all distinct.Lemma 3 (see [8]).
For a given sequence of positive integersl1′, l2′, l3′, …, l2t′, there does not exist a circuit which has a period of length 2t′, where t′ divides t.Lemma 4 (see [6]).
The number of different arrangements ofq objects of which q1 is alike, q2 is alike, q3 is alike, …, qr is alike is q!/q1!q2!q3!…qr!, where q1+q2+q3+⋯+qr=q.


### 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 lengthq. Given the positive integer q, we say that the sequence of positive integers q1,q2,q3,q4,…,qr−1,qr with q1≥q2≥q3≥q4≥⋯≥qr−1≥qr constitutes a partition of q if q1+q2+q3+⋯+qr=q.Theorem 1.
The number of equivalence classes of lengthq>2 is exactly pq−1, where pq denotes the partition of q.Proof.
Letpq denote the partition of q, and we are looking in determining all equivalence classes of equivalent circuits of length q. For a given circuit of length q, to find all equivalence classes of equivalent circuits of length q, we adopt partitions of q1+q2+q3+⋯+qr=q in the sense that q1 entries are alike, q2 entries are alike, q3 entries are alike, …, qr entries are alike, where r≥2. We get nonequivalent circuits l1′,l1′,l1′,…,l1′,l2′,l2′,…,l2′,…,lr′,lr′,…,lr′ of length q corresponding to each partition of q, where li′ repeats qi times, 1≤i≤r.
Here,r≠1 because if q=q1, then the circuit corresponding to this partition is l1′,l1′,l1′,…,l1′, where l1′ repeats q times. This circuit of length q>2 is not possible by Lemma 3. So, distinct classes of the equivalent circuit of length q>2 are exactly pq−1.Remark 1.
Forq=2, there are precisely two partitions, namely, 2 and 1 + 1. Circuits of length 2 corresponding to these partitions are l1′,l1′ and l1′,l2′.Note 1.
We can get all other circuits which are equivalent tol1′,l2′,l3′,…,lr′ by just permuting l1′,l2′,l3′,…,lr′.
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.
Letl1′,l2′,l3′,l4′,l5′,l6′ be different positive integers. Then, by Theorem 1, we have ten nonequivalent circuits, namely, l1′,l2′,l3′,l4′,l5′,l6′,l1′,l1′,l2′,l3′,l4′,l5′,l1′,l1′,l1′,l2′,l3′,l4′,l1′,l1′,l2′,l2′,l3′,l4′,l1′,l1′,l2′,l2′,l3′,l3′,l1′,l1′,l1′,l2′,l2′,l3′,l1′,l1′,l1′,l1′,l2′,l3′,l1′,l1′,l1′,l1′,l2′,l2′,l1′,l1′,l1′,l1′,l1′,l2′,andl1′,l1′,l1′,l2′,l2′,l2′, corresponding to the number p6−1, 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 l1′,l1′,l1′,l1′,l1′,l1′. 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 isETi, and the number of circuits equivalent to Ti is denoted by ETi. Similarly, the notation for cyclically equivalent classes is ETic, and ETic denotes the number of circuits cyclically equivalent to Ti. The number of distinct orbits corresponding to ETi is denoted by OTi. Furthermore, each cyclically equivalent class ETic 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 thatSn=Dnφ:φ∈An−1. In our case, S6=D6φ:φ∈A5. We know that there are 6!=720 arrangements of six different numbers l1′,l2′,l3′,l4′,l5′,l6′ taken all at a time, and so circuits corresponding to these arrangements are l1θ′,l2θ′,l3θ′,l4θ′,l5θ′,l6θ′ for each θ∈S6. Hence the proof.Corollary 3.
Ifl1′,l2′,l3′,l4′,l5′,l6′ are distinct positive integers, then there exist 60 cyclically equivalent classes ET1c.Proof.
Letl1′,l2′,l3′,l4′,l5′,l6′ be different positive integers. It is well known that the icosahedral group is isomorphic to alternating group A5. Now, the cyclically equivalent classes ET1c are obtained by El1φ′,l2φ′,l3φ′,l4φ′,l5φ′,l6φ′c for each φ∈A5. There are exactly 60 cyclically equivalent classes, namely, El1′,l2′,l3′,l4′,l5′,l6′c,El2′,l1′,l4′,l3′,l5′,l6′c,El2′,l1′,l5′,l4′,l3′,l6′c,El4′,l3′,l2′,l1′,l5′,l6′c,El5′,l3′,l2′,l4′,l1′,l6′c,El3′,l4′,l1′,l2′,l5′,l6′c,El5′,l4′,l3′,l2′,l1′,l6′c,El5′,l2′,l4′,l3′,l1′,l6′c,El4′,l2′,l5′,l1′,l3′,l6′c,El2′,l1′,l3′,l5′,l4′,l6′c,El3′,l2′,l1′,l5′,l4′,l6′c,El3′,l5′,l1′,l4′,l2′,l6′c,El4′,l5′,l3′,l1′,l2′,l6′c,El1′,l4′,l5′,l2′,l3′,l6′c,El1′,l5′,l4′,l3′,l2′,l6′c,El2′,l4′,l3′,l1′,l5′,l6′c,El2′,l5′,l3′,l4′,l1′,l6′c,El3′,l2′,l4′,l1′,l5′,l6′c,El3′,l2′,l5′,l4′,l1′,l6′c,El4′,l2′,l3′,l5′,l1′,l6′c,El2′,l3′,l1′,l4′,l5′,l6′c,El3′,l1′,l2′,l4′,l5′,l6′c,El4′,l1′,l3′,l2′,l5′,l6′c,El5′,l1′,l3′,l4′,l2′,l6′c,El4′,l2′,l1′,l3′,l5′,l6′c,El1′,l4′,l2′,l3′,l5′,l6′c,El1′,l2′,l4′,l5′,l3′,l6′c,El1′,l3′,l5′,l4′,l2′,l6′c,El1′,l2′,l5′,l3′,l4′,l6′c,El5′,l2′,l3′,l1′,l4′,l6′c,El1′,l3′,l4′,l2′,l5′,l6′c,El1′,l5′,l3′,l2′,l4′,l6′c,El1′,l4′,l3′,l5′,l2′,l6′c,El5′,l2′,l1′,l4′,l3′,l6′c,El1′,l5′,l2′,l4′,l3′,l6′c,El2′,l3′,l4′,l5′,l1′,l6′c,El2′,l4′,l5′,l3′,l1′,l6′c,El3′,l4′,l2′,l5′,l1′,l6′c,El3′,l5′,l4′,l2′,l1′,l6′c,El4′,l3′,l5′,l2′,l1′,l6′c,El4′,l5′,l2′,l3′,l1′,l6′c,El3′,l1′,l4′,l5′,l2′,l6′c,El4′,l1′,l5′,l3′,l2′,l6′c,El4′,l3′,l1′,l5′,l2′,l6′c,El5′,l3′,l4′,l1′,l2′,l6′c,El3′,l4′,l5′,l1′,l2′,l6′c,El4′,l1′,l2′,l5′,l3′,l6′c,El5′,l4′,l2′,l1′,l3′,l6′c,El5′,l1′,l4′,l2′,l3′,l6′c,El2′,l3′,l5′,l1′,l4′,l6′c,El3′,l5′,l2′,l1′,l4′,l6′c,El3′,l1′,l5′,l2′,l4′,l6′c,El5′,l3′,l1′,l2′,l4′,l6′c,El2′,l5′,l1′,l3′,l4′,l6′c,El2′,l5′,l4′,l1′,l3′,l6′c,El5′,l1′,l2′,l3′,l4′,l6′c,El1′,l3′,l2′,l5′,l4′,l6′c,El5′,l4′,l1′,l3′,l2′,l6′c,El2′,l4′,l1′,l5′,l3′,l6′c, and El4′,l5′,l1′,l2′,l3′,l6′c.Corollary 4.
Ifl1′,l2′,l3′,l4′,l5′,l6′ are distinct positive integers, then each cyclically equivalent class contains 12 cyclically equivalent circuits.Proof.
Circuits cyclically equivalent to circuitl1′,l2′,l3′,l4′,l5′,l6′ are l1θ′,l2θ′,l3θ′,l4θ′,l5θ′,l6θ′ for each θ∈D6 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 Table1 are cyclically equivalent. Moreover, by Lemma 1, l1′,l2′,l3′,l4′,l5′,l6′∼sl3′,l4′,l5′,l6′,l1′,l2′∼sl5′,l6′,l1′,l2′,l3′,l4′ is the circuit contained in γG,l2′,l3′,l4′,l5′,l6′,l1′∼sl4′,l5′,l6′,l1′,l2′,l3′∼sl6′,l1′,l2′,l3′,l4′,l5′ is the circuit contained in −γG,l1′,l6′,l5′,l4′,l3′,l2′∼sl5′,l4′,l3′,l2′,l1′,l6′∼sl3′,l2′,l1′,l6′,l5′,l4′ is the circuit contained in −γ¯G, and l6′,l5′,l4′,l3′,l2′,l1′∼sl4′,l3′,l2′,l1′,l6′,l5′∼sl2′,l1′,l6′,l5′,l4′,l3′ is the circuit contained in γ¯G.
From Table1, 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 ET1 are 12×60=720 which equals to ET1.
Table2 is of considerable utility because it provides us with the exact number of circuits of length 6 and hence the number of G-orbits of Q∗n.
The following corollaries are an immediate consequence of Theorem1.Table 1
Cyclically equivalent circuits of length 6.
θl1θ′,l2θ′,l3θ′,l4θ′,l5θ′,l6θ′e=123456l1′,l2′,l3′,l4′,l5′,l6′α=123456l2′,l3′,l4′,l5′,l6′,l1′α2=135246l3′,l4′,l5′,l6′,l1′,l2′α3=142536l4′,l5′,l6′,l1′,l2′,l3′α4=153264l5′,l6′,l1′,l2′,l3′,l4′α5=165432l6′,l1′,l2′,l3′,l4′,l5′β=162534l6′,l5′,l4′,l3′,l2′,l1′αβ=1524l5′,l4′,l3′,l2′,l1′,l6′α2β=142356l4′,l3′,l2′,l1′,l6′,l5′α3β=1346l3′,l2′,l1′,l6′,l5′,l4′α4β=124536l2′,l1′,l6′,l5′,l4′,l3′α5β=3526l1′,l6′,l5′,l4′,l3′,l2′Table 2
Enumeration of orbits.
Classes of equivalent circuits of length 6ETiNumber of classes of cyclically equivalent circuitsOTiRegarding distinct orbitsET172060240γG,γ¯G,−γG,−γ¯GET236030120γG,γ¯G,−γG,−γ¯GET31681456γG,γ¯G,−γG,−γ¯G1224γG=−γ¯G,−γG=γ¯GET41201040γG,γ¯G,−γG,−γ¯GET548416γG,γ¯G,−γG,−γ¯G1836γG=−γ¯G,−γG=γ¯G1836γG=−γ¯G,−γG=γ¯G612γG=−γ¯G,−γG=γ¯GET648416γG,γ¯G,−γG,−γ¯G1224γG=−γ¯G,−γG=γ¯GET72428γG,γ¯G,−γG,−γ¯G612γG=−γ¯G,−γG=γ¯GET81214γG,γ¯G,−γG,−γ¯G612γG=−γ¯G,−γG=γ¯GET9612γG=−γ¯G,−γG=γ¯G311γG=−γ¯G=−γG=γ¯G612γG=−γ¯G,−γG=γ¯GET10612γG=−γ¯G,−γG=γ¯GCorollary 5.
There are 6 equivalent circuits of length 6 in which 5 numbers are alike and one number is different.Proof.
By Lemma4, the number of equivalent circuits of length 6 in which 5 numbers are alike is 6!/5!1!=6. These 6 circuits are l1′,l1′,l1′,l1′,l1′,l2′,l1′,l1′,l1′,l1′,l2′,l1′,l1′,l2′,l1′,l1′,l1′,l1′,l2′,l1′,l1′,l1′,l1′,l1′,l1′,l1′,l1′,l2′,l1′,l1′, and l1′,l1′,l2′,l1′,l1′,l1′. Clearly, all these types are cyclically equivalent as well. Moreover, by Lemma 1, l1′,l1′,l1′,l1′,l1′,l2′∼sl1′,l2′,l1′,l1′,l1′,l1′∼sl1′,l1′,l1′,l2′,l1′,l1′ is the circuit contained in γG=−γ¯G, and l2′,l1′,l1′,l1′,l1′,l1′∼sl1′,l1′,l1′,l1′,l2′,l1′∼sl1′,l1′,l2′,l1′,l1′,l1′ is the circuit contained in γ¯G=−γG. In this situation, there is only one cyclically equivalent class, namely, El1′,l1′,l1′,l1′,l1′,l2′c.Corollary 6.
There are 15 equivalent circuits of length 6 in which 4 numbers are alike and 2 numbers are alike.Proof.
By Lemma4, the number of equivalent circuits of length 6 in which 4 numbers are alike and 2 numbers are alike is 6!/4!2!=15. These possible 15 circuits are l1′,l1′,l1′,l1′,l2′,l2′,l1′,l1′,l1′,l2′,l1′,l2′,l1′,l1′,l1′,l2′,l2′,l1′,l1′,l1′,l2′,l1′,l1′,l2′,l1′,l1′,l2′,l1′,l2′,l1′,l1′,l1′,l2′,l2′,l1′,l1′,l2′,l1′,l1′,l1′,l1′,l2′,l1′,l2′,l1′,l1′,l1′,l2′,l1′,l2′,l1′,l1′,l2′,l1′,l1′,l2′,l1′,l2′,l1′,l1′,l1′,l2′,l2′,l1′,l1′,l1′,l2′,l2′,l1′,l1′,l1′,l1′,l2′,l1′,l1′,l1′,l2′,l1′,l2′,l1′,l2′,l1′,l1′,l1′,andl2′,l1′,l1′,l2′,l1′,l1′, which are equivalent circuits.
In the aforementioned circuits,l1′,l1′,l1′,l1′,l2′,l2′,l1′,l1′,l1′,l2′,l2′,l1′,l1′,l2′,l2′,l1′,l1′,l1′,l2′,l2′,l1′,l1′,l1′,l1′,l1′,l1′,l2′,l2′,l1′,l1′, and l2′,l1′,l1′,l1′,l1′,l2′ are cyclically equivalent. Moreover, by Lemma 1, l1′,l1′,l1′,l1′,l2′,l2′∼sl1′,l1′,l2′,l2′,l1′,l1′∼sl2′,l2′,l1′,l1′,l1′,l1′ is the circuit contained in αG=α¯G, and l2′,l1′,l1′,l1′,l1′,l2′∼sl1′,l2′,l2′,l1′,l1′,l1′∼sl1′,l1′,l1′,l2′,l2′,l1′ is the circuit contained in −α¯G=−αG.
Also,l2′,l1′,l1′,l2′,l1′,l1′,l1′,l1′,l2′,l1′,l1′,l2′,andl1′,l2′,l1′,l1′,l2′,l1′ are cyclically equivalent. Moreover, by Lemma 1, l2′,l1′,l1′,l2′,l1′,l1′∼sl1′,l1′,l2′,l1′,l1′,l2′∼sl1′,l2′,l1′,l1′,l2′,l1′ is the circuit contained in βG=β¯G=−βG=−β¯G.
These circuitsl1′,l1′,l1′,l2′,l1′,l2′,l1′,l1′,l2′,l1′,l2′,l1′,l1′,l2′,l1′,l2′,l1′,l1′,l2′,l1′,l2′,l1′,l1′,l1′,l1′,l2′,l1′,l1′,l1′,l2′,andl2′,l1′,l1′,l1′,l2′,l1′ are cyclically equivalent. Moreover, by Lemma 1, l1′,l1′,l1′,l2′,l1′,l2′∼sl1′,l2′,l1′,l2′,l1′,l1′∼sl1′,l2′,l1′,l1′,l1′,l2′ is the circuit contained in γG=−γ¯G, and l1′,l1′,l2′,l1′,l2′,l1′∼sl2′,l1′,l2′,l1′,l1′,l1′∼sl2′,l1′,l1′,l1′,l2′,l1′ is the circuit contained in −γG=γ¯G.
Table3 summarizes all the information.Table 3
Cyclically equivalent classes of T9.
ET9cNumber of classes of cyclically equivalent circuitsOT9Structure of orbitsEl1′,l1′,l1′,l1′,l2′,l2′c12γG=γ¯G,−γG=−γ¯GEl1′,l2′,l1′,l2′,l1′,l1′c12γG=−γ¯G,−γG=γ¯GEl1′,l1′,l2′,l1′,l1′,l2′c11γG=−γ¯G=−γG=γ¯GCorollary 7.
There are 18 equivalent circuits of length 6 in which 3 numbers are alike and 3 numbers are alike.Proof.
By Lemma4, the number of possible equivalent circuits of length 6 in which 3 numbers are alike and 3 numbers are alike is 6!/3!3!=20. These possible 20 circuits are given by l1′,l1′,l1′,l2′,l2′,l2′,l1′,l1′,l2′,l1′,l2′,l2′,l1′,l1′,l2′,l2′,l1′,l2′,l1′,l1′,l2′,l2′,l2′,l1′,l1′,l2′,l1′,l1′,l2′,l2′,l1′,l2′,l1′,l2′,l1′,l2′,l1′,l2′,l1′,l2′,l2′,l1′,l1′,l2′,l2′,l1′,l1′,l2′,l1′,l2′,l2′,l1′,l2′,l1′,l1′,l2′,l2′,l2′,l1′,l1′,l2′,l2′,l2′,l1′,l1′,l1′,l2′,l2′,l1′,l2′,l1′,l1′,l2′,l2′,l1′,l1′,l2′,l1′,l2′,l2′,l1′,l1′,l1′,l2′,l2′,l1′,l2′,l1′,l2′,l1′,l2′,l1′,l1′,l2′,l2′,l1′,l2′,l1′,l1′,l2′,l1′,l2′,l2′,l1′,l1′,l1′,l2′,l2′,l2′,l1′,l2′,l2′,l1′,l1′, and l2′,l1′,l2′,l1′,l1′,l2′ which are equivalent circuits. In the aforementioned circuits, l1′,l1′,l1′,l2′,l2′,l2′,l1′,l1′,l2′,l2′,l2′,l1′,l1′,l2′,l2′,l2′,l1′,l1′,l2′,l2′,l2′,l1′,l1′,l1′,l2′,l2′,l1′,l1′,l1′,l2′,andl2′,l1′,l1′,l1′,l2′,l2′ are cyclically equivalent. Moreover, by Lemma 1, l1′,l1′,l1′,l2′,l2′,l2′∼sl1′,l2′,l2′,l2′,l1′,l1′∼sl2′,l2′,l1′,l1′,l1′,l2′ is the circuit of αG=−α¯G, and l2′,l1′,l1′,l1′,l2′,l2′∼sl1′,l1′,l2′,l2′,l2′,l1′∼sl2′,l2′,l2′,l1′,l1′,l1′ is the circuit of α¯G=−αG. Also, circuits l1′,l1′,l2′,l2′,l1′,l2′,l1′,l2′,l1′,l1′,l2′,l2′,l2′,l2′,l1′,l2′,l1′,l1′,l2′,l1′,l1′,l2′,l1′,l2′,l2′,l2′,l1′,l1′,l2′,l1′,l1′,l1′,l2′,l1′,l2′,l2′,l1′,l2′,l2′,l1′,l1′,l2′,l2′,l1′,l2′,l2′,l1′,l1′,l1′,l2′,l2′,l1′,l2′,l1′,l2′,l1′,l2′,l1′,l1′,l2′,l2′,l1′,l1′,l2′,l2′,l1′,andl1′,l2′,l1′,l2′,l2′,l1′ are cyclically equivalent. Moreover, by Lemma 1, l1′,l1′,l2′,l2′,l1′,l2′∼sl1′,l2′,l1′,l1′,l2′,l2′∼sl2′,l2′,l1′,l2′,l1′,l1′ is the circuit of βG,l1′,l2′,l2′,l1′,l2′,l1′∼sl2′,l1′,l2′,l1′,l1′,l2′∼sl2′,l1′,l1′,l2′,l2′,l1′ is the circuit contained in −βG,l1′,l2′,l1′,l2′,l2′,l1′∼sl1′,l2′,l2′,l1′,l1′,l2′∼sl2′,l1′,l1′,l2′,l1′,l2′ is the circuit of −β¯G, and l2′,l1′,l2′,l2′,l1′,l1′∼sl2′,l2′,l1′,l1′,l2′,l1′∼sl1′,l1′,l2′,l1′,l2′,l2′ is the circuit contained in β¯G. Moreover, l1′,l2′,l1′,l2′,l1′,l2′ and l2′,l1′,l2′,l1′,l2′,l1′ are not possible by Lemma 3. Table 4 summarizes all the information.Table 4
Cyclically equivalent classes of T8.
ET8cNumber of classes of cyclically equivalent circuitsOT8Structure of orbitsEl1′,l1′,l1′,l2′,l2′,l2′c12γG=−γ¯G,−γG=γ¯GEl1′,l1′,l2′,l1′,l2′,l2′c14γG,γ¯G,−γG,−γ¯GCorollary 8.
There are 30 equivalent circuits of length 6 in which 4 numbers are alike and 2 numbers are different.Proof.
By Lemma4, the number of equivalent circuits of length 6 in which 4 numbers are alike and 2 numbers are different is 6!/4!1!1!=30. These 30 circuits are l1′,l1′,l1′,l1′,l2′,l3′,l1′,l1′,l1′,l2′,l1′,l3′,l1′,l1′,l1′,l2′,l3′,l1′,l1′,l1′,l1′,l3′,l1′,l2′,l1′,l1′,l1′,l1′,l3′,l2′,l1′,l1′,l3′,l2′,l1′,l1′,l1′,l1′,l3′,l1′,l2′,l1′,l1′,l1′,l3′,l1′,l1′,l2′,l1′,l1′,l1′,l3′,l2′,l1′,l1′,l1′,l2′,l3′,l1′,l1′,l1′,l1′,l2′,l1′,l3′,l1′,l1′,l1′,l2′,l1′,l1′,l3′,l1′,l2′,l1′,l1′,l1′,l3′,l1′,l2′,l1′,l1′,l3′,l1′,l1′,l2′,l1′,l3′,l1′,l1′,l1′,l2′,l3′,l1′,l1′,l1′,l1′,l3′,l1′,l1′,l1′,l2′,l1′,l3′,l1′,l1′,l2′,l1′,l1′,l3′,l1′,l2′,l1′,l1′,l1′,l3′,l2′,l1′,l1′,l1′,l2′,l1′,l1′,l1′,l1′,l3′,l2′,l1′,l1′,l1′,l3′,l1′,l2′,l1′,l1′,l3′,l1′,l1′,l2′,l1′,l3′,l1′,l1′,l1′,l2′,l3′,l1′,l1′,l1′,l1′,l3′,l1′,l1′,l1′,l1′,l2′,l3′,l1′,l1′,l1′,l2′,l1′,l3′,l1′,l1′,l2′,l1′,l1′,l3′,l1′,l2′,l1′,l1′,l1′,andl3′,l2′,l1′,l1′,l1′,l1′.
In the aforementioned circuits,l1′,l1′,l1′,l1′,l2′,l3′,l1′,l1′,l2′,l3′,l1′,l1′,l2′,l3′,l1′,l1′,l1′,l1′,l1′,l1′,l1′,l2′,l3′,l1′,l1′,l2′,l3′,l1′,l1′,l1′,l3′,l1′,l1′,l1′,l1′,l2′,l3′,l2′,l1′,l1′,l1′,l1′,l1′,l1′,l3′,l2′,l1′,l1′,l1′,l1′,l1′,l1′,l3′,l2′,l1′,l3′,l2′,l1′,l1′,l1′,l2′,l1′,l1′,l1′,l1′,l3′, and l1′,l1′,l1′,l3′,l2′,l1′ are cyclically equivalent. By Lemma 1, l1′,l1′,l1′,l1′,l2′,l3′∼sl1′,l1′,l2′,l3′,l1′,l1′∼sl2′,l3′,l1′,l1′,l1′,l1′ is the circuit of αG,l1′,l1′,l1′,l2′,l3′,l1′∼sl1′,l2′,l3′,l1′,l1′,l1′∼sl3′,l1′,l1′,l1′,l1′,l2′ is the circuit contained in −αG,l3′,l2′,l1′,l1′,l1′,l1′∼sl1′,l1′,l3′,l2′,l1′,l1′∼sl1′,l1′,l1′,l1′,l3′,l2′ is the circuit of α¯G, and l1′,l3′,l2′,l1′,l1′,l1′∼sl2′,l1′,l1′,l1′,l1′,l3′∼sl1′,l1′,l1′,l3′,l2′,l1′ is the circuit contained in −α¯G. Also, l1′,l1′,l1′,l3′,l1′,l2′,l1′,l3′,l1′,l2′,l1′,l1′,l1′,l2′,l1′,l1′,l1′,l3′,l1′,l1′,l3′,l1′,l2′,l1′,l3′,l1′,l2′,l1′,l1′,l1′,l2′,l1′,l1′,l1′,l3′,l1′,l1′,l2′,l1′,l3′,l1′,l1′,l1′,l3′,l1′,l1′,l1′,l2′,l1′,l1′,l1′,l2′,l1′,l3′,l2′,l1′,l3′,l1′,l1′,l1′,l3′,l1′,l1′,l1′,l2′,l1′,andl1′,l1′,l2′,l1′,l3′,l1′ are cyclically equivalent. By Lemma 1, l1′,l1′,l1′,l3′,l1′,l2′∼sl1′,l3′,l1′,l2′,l1′,l1′∼sl1′,l2′,l1′,l1′,l1′,l3′ is the circuit of βG,l1′,l1′,l3′,l1′,l2′,l1′∼sl3′,l1′,l2′,l1′,l1′,l1′∼sl2′,l1′,l1′,l1′,l3′,l1′ is the circuit of −βG,l1′,l2′,l1′,l3′,l1′,l1′∼sl1′,l3′,l1′,l1′,l1′,l2′∼sl1′,l1′,l1′,l2′,l1′,l3′ is the circuit contained in −β¯G, and l2′,l1′,l3′,l1′,l1′,l1′∼sl3′,l1′,l1′,l1′,l2′,l1′∼sl1′,l1′,l2′,l1′,l3′,l1′ is the circuit contained in β¯G. Also, l1′,l1′,l2′,l1′,l1′,l3′,l2′,l1′,l1′,l3′,l1′,l1′,l1′,l3′,l1′,l1′,l2′,l1′,l1′,l2′,l1′,l1′,l3′,l1′,l1′,l1′,l3′,l1′,l1′,l2′,andl3′,l1′,l1′,l2′,l1′,l1′ are cyclically equivalent. Moreover, by Lemma 1, l1′,l1′,l2′,l1′,l1′,l3′∼sl2′,l1′,l1′,l3′,l1′,l1′∼sl1′,l3′,l1′,l1′,l2′,l1′ is the circuit of γG=−γ¯G, and l1′,l2′,l1′,l1′,l3′,l1′∼sl1′,l1′,l3′,l1′,l1′,l2′∼sl3′,l1′,l1′,l2′,l1′,l1′ is the circuit of −γG=γ¯G.
Table5 summarizes all the information.Table 5
Cyclically equivalent classes of T7.
ET7cNumber of classes of cyclically equivalent circuitsOT7Structure of orbitsEl1′,l1′,l2′,l1′,l1′,l3′c12γG=−γ¯G, −γG=γ¯GEl1′,l1′,l1′,l1′,l2′,l3′c,El1′,l2′,l1′,l3′,l1′,l1′c28γG,γ¯G,−γG,−γ¯GCorollary 9.
There are 60 equivalent circuits of length 6 in which 3 numbers are alike, 2 numbers are alike, and 1 number is different.Proof.
By Lemma4, the number of equivalent length 6 in which 3 numbers are alike, 2 numbers are alike, and 1 number is different is 6!/3!2!1!=60. These circuits are l1′,l1′,l1′,l2′,l2′,l3′,l1′,l2′,l2′,l3′,l1′,l1′,l2′,l3′,l1′,l1′,l1′,l2′,l1′,l1′,l2′,l2′,l3′,l1′,l2′,l2′,l3′,l1′,l1′,l1′,l2′,l1′,l1′,l1′,l3′,l2′,l3′,l1′,l1′,l1′,l2′,l2′,l1′,l3′,l2′,l2′,l1′,l1′,l2′,l2′,l1′,l1′,l1′,l3′,l1′,l1′,l1′,l3′,l2′,l2′,l3′,l2′,l2′,l1′,l1′,l1′,l1′,l1′,l3′,l2′,l2′,l1′, and so on.
In the aforementioned circuits,l1′,l1′,l1′,l2′,l2′,l3′,l1′,l2′,l2′,l3′,l1′,l1′,l2′,l3′,l1′,l1′,l1′,l2′,l1′,l1′,l2′,l2′,l3′,l1′,l3′,l1′,l1′,l1′,l2′,l2′,l1′,l3′,l2′,l2′,l1′,l1′,l2′,l2′,l1′,l1′,l1′,l3′,l1′,l1′,l1′,l3′,l2′,l2′,l3′,l2′,l2′,l1′,l1′,l1′,l1′,l1′,l3′,l2′,l2′,l1′l2′,l1′,l1′,l1′,l3′,l2′, and l2′,l2′,l3′,l1′,l1′,l1′ are cyclically equivalent. By Lemma 1, l2′,l3′,l1′,l1′,l1′,l2′∼sl1′,l1′,l1′,l2′,l2′,l3′∼sl1′,l2′,l2′,l3′,l1′,l1′ is the circuit of αG,l1′,l1′,l2′,l2′,l3′,l1′∼sl2′,l2′,l3′,l1′,l1′,l1′∼sl3′,l1′,l1′,l1′,l2′,l2′ is the circuit contained in −αG,l1′,l3′,l2′,l2′,l1′,l1′∼sl2′,l2′,l1′,l1′,l1′,l3′∼sl1′,l1′,l1′,l3′,l2′,l2′ is the circuit of −α¯G,l3′,l2′,l2′,l1′,l1′,l1′∼sl2′,l1′,l1′,l1′,l3′,l2′∼sl1′,l1′,l3′,l2′,l2′,l1′ is the circuit contained in α¯G, and so on.
Table6 summarizes all the information.Table 6
Cyclically equivalent classes of T6.
ET6cNumber of classes of cyclically equivalent circuitsOT6Structure of orbitsEl1′,l3′,l2′,l2′,l1′,l1′c,El1′,l1′,l2′,l1′,l2′,l3′c,El1′,l2′,l2′,l1′,l3′,l1′c,El1′,l2′,l1′,l1′,l2′,l3′c416γG,γ¯G,−γG,−γ¯GEl1′,l1′,l1′,l2′,l3′,l2′c,El1′,l2′,l1′,l2′,l1′,l3′c24γG=−γ¯G, −γG=γ¯GCorollary 10.
There are 90 equivalent circuits of length 6 in which 2 numbers are alike, 2 numbers are alike, and 2 numbers are alike.Proof.
By Lemma4, the number of equivalent circuits of length 6 in which 2 numbers are alike, 2 numbers are alike, and 2 numbers are alike is 6!/2!2!2!=90. Some of these circuits are l1′,l1′,l2′,l2′,l3′,l3′,l2′,l2′,l3′,l3′,l1′,l1′,l3′,l3′,l1′,l1′,l2′,l2′,l1′,l2′,l2′,l3′,l3′,l1′,l2′,l3′,l3′,l1′,l1′,l2′,l2′,l2′,l3′,l1′,l1′,l3′,l3′,l1′,l1′,l2′,l2′,l3′,l1′,l3′,l3′,l2′,l2′,l1′,l3′,l2′,l2′,l1′,l1′,l3′,l2′,l1′,l1′,l3′,l3′,l2′,l3′,l3′,l2′,l2′,l1′,l1′,l2′,l3′,l1′,l1′,l3′,l2′,l2′,l2′,l1′,l1′,l3′,l3′,l1′,l1′,l3′,l3′,l2′,l2′,l1′,l1′,l3′,l2′,l2′,l3′,l1′,l3′,l2′,l2′,l3′,l1′,l3′,l2′,l2′,l3′,l1′,l1′,l3′,l1′,l1′,l3′,l2′,l2′, and so on.
In th aforementioned circuits,l1′,l1′,l2′,l2′,l3′,l3′,l2′,l2′,l3′,l3′,l1′,l1′,l3′,l3′,l1′,l1′,l2′,l2′,l1′,l2′,l2′,l3′,l3′,l1′,l2′,l3′,l3′,l1′,l1′,l2′,l3′,l1′,l1′,l2′,l2′,l3′,l1′,l3′,l3′,l2′,l2′,l1′,l3′,l2′,l2′,l1′,l1′,l3′,l2′,l1′,l1′,l3′,l3′,l2′,l3′,l3′,l2′,l2′,l1′,l1′,l2′,l2′,l1′,l1′,l3′,l3′,andl1′,l1′,l3′,l3′,l2′,l2′ are cyclically equivalent. By Lemma 1, l1′,l1′,l2′,l2′,l3′,l3′∼sl2′,l2′,l3′,l3′,l1′,l1′∼sl3′,l3′,l1′,l1′,l2′,l2′ is the circuit of αG,l1′,l2′,l2′,l3′,l3′,l1′∼sl2′,l3′,l3′,l1′,l1′,l2′∼sl3′,l1′,l1′,l2′,l2′,l3′ is the circuit contained in −αG,l1′,l3′,l3′,l2′,l2′,l1′∼sl3′,l2′,l2′,l1′,l1′,l3′∼sl2′,l1′,l1′,l3′,l3′,l2′ is the circuit of −α¯G,l3′,l3′,l2′,l2′,l1′,l1′∼sl2′,l2′,l1′,l1′,l3′,l3′∼sl1′,l1′,l3′,l3′,l2′,l2′ is the circuit contained in α¯G, and so on.
Table7 summarizes all the information.Table 7
Cyclically equivalent classes of T5.
ET5cNumber of classes of cyclically equivalent circuitsOT5Structure of orbitsEl3′,l3′,l2′,l2′,l1′,l1′c,El1′,l3′,l2′,l3′,l2′,l1′c,El1′,l2′,l1′,l3′,l3′,l2′c,El1′,l3′,l2′,l2′,l1′,l3′c416γG,γ¯G,−γG,−γ¯GEl3′,l2′,l1′,l2′,l3′,l1′c,El1′,l3′,l2′,l3′,l1′,l2′c,El1′,l3′,l1′,l2′,l3′,l2′c312γG=−γ¯G,−γG=γ¯GEl1′,l3′,l2′,l2′,l3′,l1′c,El1′,l2′,l2′,l1′,l3′,l3′c,El1′,l1′,l2′,l3′,l3′,l2′c312γG=γ¯G,−γG=−γ¯GEl1′,l2′,l3′,l1′,l2′,l3′c12γG=−γG,γ¯G=−γ¯GCorollary 11.
There are 120 equivalent circuits of length 6 in which 3 numbers are alike and 3 numbers are different.Proof.
By Lemma4, the number of equivalent circuits of length 6 in which 3 numbers are alike and 3 numbers are different is 6!/3!1!1!1!=120. Some of these circuits are l1′,l1′,l1′,l2′,l3′,l4′,l1′,l2′,l3′,l4′,l1′,l1′,l3′,l4′,l1′,l1′,l1′,l2′,l1′,l1′,l2′,l3′,l4′,l1′,l2′,l3′,l4′,l1′,l1′,l1′,l4′,l2′,l3′,l1′,l1′,l1′,l3′,l2′,l4′,l1′,l1′,l1′,l4′,l1′,l1′,l1′,l2′,l3′,l1′,l4′,l3′,l2′,l1′,l1′,l3′,l2′,l1′,l1′,l1′,l4′,l1′,l1′,l1′,l4′,l3′,l2′,l4′,l3′,l2′,l1′,l1′,l1′,l2′,l4′,l1′,l1′,l1′,l3′,l2′,l1′,l1′,l1′,l4′,l3′,l1′,l1′,l4′,l3′,l2′,l1′,l1′,l1′,l1′,l3′,l2′,l4′,l1′,l1′,l3′,l2′,l4′,l1′,l1′,l3′,l2′,l4′,l1′,l1′,l4′,l1′,l1′,l1′,l3′,l2′,l1′,l4′,l2′,l3′,l1′,l1′, and so on.
In the aforementioned circuits,l1′,l1′,l1′,l2′,l3′,l4′,l1′,l2′,l3′,l4′,l1′,l1′,l3′,l4′,l1′,l1′,l1′,l2′,l1′,l1′,l2′,l3′,l4′,l1′,l2′,l3′,l4′,l1′,l1′,l1′,l4′,l1′,l1′,l1′,l2′,l3′,l1′,l4′,l3′,l2′,l1′,l1′,l3′,l2′,l1′,l1′,l1′,l4′,l1′,l1′,l1′,l4′,l3′,l2′,l4′,l3′,l2′,l1′,l1′,l1′,l2′,l1′,l1′,l1′,l4′,l3′,andl1′,l1′,l4′,l3′,l2′,l1′ are cyclically equivalent. By Lemma 1, l1′,l1′,l1′,l2′,l3′,l4′∼sl1′,l2′,l3′,l4′,l1′,l1′∼sl3′,l4′,l1′,l1′,l1′,l2′ is the circuit of αG,l1′,l1′,l2′,l3′,l4′,l1′∼sl2′,l3′,l4′,l1′,l1′,l1′∼sl4′,l1′,l1′,l1′,l2′,l3′ is the circuit contained in −αG,l1′,l4′,l3′,l2′,l1′,l1′∼sl3′,l2′,l1′,l1′,l1′,l4′∼sl1′,l1′,l1′,l4′,l3′,l2′ is the circuit of −α¯G,l4′,l3′,l2′,l1′,l1′,l1′∼sl2′,l1′,l1′,l1′,l4′,l3′∼sl1′,l1′,l4′,l3′,l2′,l1′ is the circuit contained in α¯G, and so on.
Table8 summarizes all the information.Table 8
Cyclically equivalent classes of T4.
ET4cNumber of classes of cyclically equivalent circuitsOT4Structure of orbitsEl1′,l4′,l3′,l2′,l1′,l1′c,El1′,l1′,l1′,l2′,l4′,l3′c,El1′,l2′,l4′,l1′,l3′,l1′c,El1′,l3′,l4′,l1′,l2′,l1′c,El1′,l2′,l4′,l1′,l1′,l3′c,El1′,l4′,l2′,l3′,l1′,l1′c,El1′,l2′,l1′,l1′,l4′,l3′c,El1′,l1′,l4′,l1′,l2′,l3′c,El1′,l3′,l2′,l1′,l1′,l4′c,El1′,l4′,l1′,l2′,l1′,l3′c1040γG,γ¯G,−γG,−γ¯GCorollary 12.
There are 180 equivalent circuits of length 6 in which 2 numbers are alike, 2 numbers are alike, and 2 numbers are different.Proof.
By Lemma4, the number of equivalent circuits of length 6 in which 2 numbers are alike, 2 numbers are alike, and 2 numbers are different is 6!/2!2!1!1!=180. Some these circuits are l1′,l1′,l2′,l2′,l3′,l4′,l2′,l2′,l3′,l4′,l1′,l1′,l3′,l4′,l1′,l1′,l2′,l2′,l1′,l2′,l2′,l3′,l4′,l1′,l2′,l3′,l4′,l1′,l1′,l2′,l4′,l1′,l1′,l2′,l2′,l3′,l1′,l4′,l3′,l2′,l2′,l1′,l3′,l2′,l2′,l1′,l1′,l4′,l2′,l1′,l1′,l4′,l3′,l2′,l4′,l3′,l2′,l2′,l1′,l1′,l2′,l2′,l1′,l1′,l4′,l3′,l1′,l1′,l4′,l3′,l2′,l2′,l1′,l1′,l2′,l3′,l4′,l2′,l1′,l1′,l2′,l4′,l2′,l3′,l1′,l1′,l2′,l4′,l3′,l2′,l1′,l1′,l3′,l2′,l2′,l4′,l1′,l1′,l3′,l2′,l4′,l2′,l1′,l1′,l3′,l4′,l2′,l2′,l1′,l1′,l4′,l2′,l2′,l3′, and so on.
In the aforementioned circuits,l1′,l1′,l2′,l2′,l3′,l4′,l2′,l2′,l3′,l4′,l1′,l1′,l3′,l4′,l1′,l1′,l2′,l2′,l1′,l2′,l2′,l3′,l4′,l1′,l2′,l3′,l4′,l1′,l1′,l2′,l4′,l1′,l1′,l2′,l2′,l3′,l1′,l4′,l3′,l2′,l2′,l1′,l3′,l2′,l2′,l1′,l1′,l4′,l2′,l1′,l1′,l4′,l3′,l2′,l4′,l3′,l2′,l2′,l1′,l1′,l2′,l2′,l1′,l1′,l4′,l3′,andl1′,l1′,l4′,l3′,l2′,l2′ are cyclically equivalent. Moreover, by Lemma 1, l1′,l1′,l2′,l2′,l3′,l4′∼sl2′,l2′,l3′,l4′,l1′,l1′∼sl3′,l4′,l1′,l1′,l2′,l2′ is the circuit of αG,l1′,l2′,l2′,l3′,l4′,l1′∼sl2′,l3′,l4′,l1′,l1′,l2′∼sl4′,l1′,l1′,l2′,l2′,l3′ is the circuit of −αG,l1′,l4′,l3′,l2′,l2′,l1′∼sl3′,l2′,l2′,l1′,l1′,l4′∼sl2′,l1′,l1′,l4′,l3′,l2′ is the circuit contained in −α¯G,l4′,l3′,l2′,l2′,l1′,l1′∼sl2′,l2′,l1′,l1′,l4′,l3′∼sl1′,l1′,l4′,l3′,l2′,l2′ is the circuit contained in α¯G, and so on.
Table9 summarizes all the information.Table 9
Cyclically equivalent classes of T3.
ET3cNumber of classes of cyclically equivalent circuitsOT3Structure of orbitsEl1′,l4′,l3′,l2′,l2′,l1′c,El1′,l1′,l2′,l2′,l4′,l3′c,El1′,l4′,l2′,l3′,l2′,l1′c,El1′,l3′,l2′,l4′,l2′,l1′c,El1′,l1′,l3′,l2′,l2′,l4′c,El1′,l1′,l2′,l3′,l4′,l2′c,El1′,l4′,l3′,l2′,l1′,l2′c,El1′,l2′,l1′,l2′,l4′,l3′c,El1′,l3′,l2′,l2′,l1′,l4′c,El1′,l2′,l1′,l3′,l2′,l4′c,El1′,l3′,l1′,l2′,l2′,l4′c,El1′,l4′,l2′,l1′,l3′,l2′c,El1′,l2′,l2′,l1′,l3′,l4′c,El1′,l3′,l2′,l1′,l2′,l4′c1456γG,γ¯G,−γG,−γ¯GEl1′,l2′,l4′,l2′,l1′,l3′c,El2′,l3′,l2′,l1′,l4′,l1′c24γG=−γ¯G,−γG=γ¯GCorollary 13.
There are 360 equivalent circuits of length 6 in which 2 numbers are alike and 4 numbers are different.Proof.
By Lemma4, the number of equivalent circuits of length 6 in which 2 numbers are alike and 4 numbers are different is 6!/2!1!1!1!=360. Some of these circuits are l1′,l1′,l2′,l3′,l4′,l5′,l2′,l3′,l4′,l5′,l1′,l1′,l4′,l5′,l1′,l1′,l2′,l3′,l1′,l2′,l3′,l4′,l5′,l1′,l3′,l4′,l5′,l1′,l1′,l2′,l3′,l2′,l1′,l1′,l5′,l4′,l5′,l1′,l1′,l2′,l3′,l4′,l1′,l5′,l4′,l3′,l2′,l1′,l4′,l3′,l2′,l1′,l1′,l5′,l2′,l1′,l1′,l5′,l4′,l3′,l5′,l4′,l3′,l2′,l1′,l1′,l1′,l1′,l5′,l4′,l3′,l2′, and so on.
In the aforementioned circuits,l1′,l1′,l2′,l3′,l4′,l5′,l1′,l2′,l3′,l4′,l5′,l1′,l2′,l3′,l4′,l5′,l1′,l1′,l3′,l4′,l5′,l1′,l1′,l2′,l4′,l5′,l1′,l1′,l2′,l3′,l5′,l1′,l1′,l2′,l3′,l4′,l1′,l5′,l4′,l3′,l2′,l1′,l5′,l4′,l3′,l2′,l1′,l1′,l4′,l3′,l2′,l1′,l1′,l5′,l3′,l2′,l1′,l1′,l5′,l4′,l2′,l1′,l1′,l5′,l4′,l3′andl1′,l1′,l5′,l4′,l3′,l2′ are cyclically equivalent.
Table10 summarizes all the information.
It was proved in [8] that any circuit l1′,l2′,l3′,l4′,l5′,l6′ of length 6 corresponds to the orbit contained in Q∗fl1′,l2′,l3′,l4′,l5′,l6′.Table 10
Cyclically equivalent classes of T2.
ET3cNumber of classes of the cyclically equivalent circuitsOT3Structure of orbitsEl1′,l1′,l2′,l3′,l4′,l5′c,El1′,l1′,l2′,l3′,l5′,l4′c,El1′,l1′,l2′,l4′,l3′,l5′c,El1′,l1′,l2′,l4′,l5′,l3′c,El1′,l1′,l2′,l5′,l3′,l4′c,El1′,l1′,l2′,l5′,l4′,l3′c,El1′,l1′,l3′,l2′,l5′,l4′c,El1′,l1′,l3′,l2′,l4′,l5′c,El1′,l1′,l3′,l4′,l2′,l5′c,El1′,l1′,l3′,l5′,l2′,l4′c,El1′,l1′,l4′,l2′,l3′,l5′c,El1′,l1′,l4′,l3′,l2′,l5′c,El1′,l4′,l1′,l2′,l3′,l5′c,El2′,l1′,l4′,l3′,l5′,l1′c,El2′,l1′,l5′,l4′,l3′,l1′c,El4′,l3′,l2′,l1′,l5′,l1′c,El3′,l4′,l1′,l2′,l5′,l1′c,El4′,l2′,l5′,l1′,l3′,l1′c,El2′,l1′,l3′,l5′,l4′,l1′c,El3′,l2′,l1′,l5′,l4′,l1′c,El5′,l3′,l1′,l2′,l4′,l1′c,El5′,l1′,l4′,l2′,l3′,l1′c,El3′,l5′,l2′,l1′,l4′,l1′c,El5′,l4′,l2′,l1′,l3′,l1′c,El2′,l5′,l4′,l1′,l3′,l1′c,El5′,l2′,l3′,l1′,l4′,l1′c,El5′,l1′,l3′,l4′,l2′,l1′c,El5′,l2′,l1′,l3′,l4′,l1′c,El5′,l3′,l1′,l4′,l2′,l1′c,El5′,l4′,l1′,l2′,l3′,l1′c30120γG,γ¯G,−γG,−γ¯GTheorem 2.
All the circuits inEl1′,l2′,l3′,l4′,l5′,l6′c correspond to the orbits contained in Q∗n1.Proof.
To prove this result, it is enough to show thatfl1′,l2′,l3′,l4′,l5′,l6′ is unchanged for θ∈D6, where fl1′,l2′,l3′,l4′,l5′,l6′=l1′l6′+l4′+l2′+l3′l6′+l4′+l5′l6′+l2′l1′l4′l3′+l6′l5′+l4′l5′l6′l3′+l1′+l1′l6′l3′l2′1+l4′l5′−l2′l5′+l3′−l5′l4′1+l2′l3′2+4l5′1+l1′l2′+l3′l4′+l1′l4′+l1′l2′l3′l4+l3′+l1′+l3′l2′l1′1+l6′l5′l4′+l2′+l2′l3′l4+l6′1+l2′l3′.
It is easy to see that after applyingθ∈D6 and simplifying, the expression for fl1′,l2′,l3′,l4′,l5′,l6′ is unaltered. Since there are 60 cyclically equivalent classes, we have sixty n's, namely, n1,n2,n3,⋯,n60.Example 4.
We verify the above results by considering the circuits1,2,3,4,5,6,2,3,4,5,6,1,1,6,5,4,3,2, and 6,5,4,3,2,1. These four circuits correspond to the orbits γG,−γG,−γ¯G, andγ¯G, and these four circuits are shown in Figures 1–4, respectively.
Now, in all figures containing 3 reduced numbers,E1,2,3,4,5,6c contains 12 reduced numbers.Figure 1
Shows the location of reduced numbers and describes that1,2,3,4,5,6∼s3,4,5,6,1,2∼s5,6,1,2,3,4.Figure 2
Shows the location of reduced numbers and describes that2,3,4,5,6,1∼s4,5,6,1,2,3∼s6,1,2,3,4,5.Figure 3
Shows the location of reduced numbers and describes that1,6,5,4,3,2∼s5,4,3,2,1,6∼s3,2,1,6,5,4.Figure 4
Shows the location of reduced numbers and describes that6,5,4,3,2,1∼s4,3,2,1,6,5∼s2,1,6,5,4,3.


## 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 lengthq. Given the positive integer q, we say that the sequence of positive integers q1,q2,q3,q4,…,qr−1,qr with q1≥q2≥q3≥q4≥⋯≥qr−1≥qr constitutes a partition of q if q1+q2+q3+⋯+qr=q.Theorem 1.
The number of equivalence classes of lengthq>2 is exactly pq−1, where pq denotes the partition of q.Proof.
Letpq denote the partition of q, and we are looking in determining all equivalence classes of equivalent circuits of length q. For a given circuit of length q, to find all equivalence classes of equivalent circuits of length q, we adopt partitions of q1+q2+q3+⋯+qr=q in the sense that q1 entries are alike, q2 entries are alike, q3 entries are alike, …, qr entries are alike, where r≥2. We get nonequivalent circuits l1′,l1′,l1′,…,l1′,l2′,l2′,…,l2′,…,lr′,lr′,…,lr′ of length q corresponding to each partition of q, where li′ repeats qi times, 1≤i≤r.
Here,r≠1 because if q=q1, then the circuit corresponding to this partition is l1′,l1′,l1′,…,l1′, where l1′ repeats q times. This circuit of length q>2 is not possible by Lemma 3. So, distinct classes of the equivalent circuit of length q>2 are exactly pq−1.Remark 1.
Forq=2, there are precisely two partitions, namely, 2 and 1 + 1. Circuits of length 2 corresponding to these partitions are l1′,l1′ and l1′,l2′.Note 1.
We can get all other circuits which are equivalent tol1′,l2′,l3′,…,lr′ by just permuting l1′,l2′,l3′,…,lr′.
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.
Letl1′,l2′,l3′,l4′,l5′,l6′ be different positive integers. Then, by Theorem 1, we have ten nonequivalent circuits, namely, l1′,l2′,l3′,l4′,l5′,l6′,l1′,l1′,l2′,l3′,l4′,l5′,l1′,l1′,l1′,l2′,l3′,l4′,l1′,l1′,l2′,l2′,l3′,l4′,l1′,l1′,l2′,l2′,l3′,l3′,l1′,l1′,l1′,l2′,l2′,l3′,l1′,l1′,l1′,l1′,l2′,l3′,l1′,l1′,l1′,l1′,l2′,l2′,l1′,l1′,l1′,l1′,l1′,l2′,andl1′,l1′,l1′,l2′,l2′,l2′, corresponding to the number p6−1, 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 l1′,l1′,l1′,l1′,l1′,l1′. 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 isETi, and the number of circuits equivalent to Ti is denoted by ETi. Similarly, the notation for cyclically equivalent classes is ETic, and ETic denotes the number of circuits cyclically equivalent to Ti. The number of distinct orbits corresponding to ETi is denoted by OTi. Furthermore, each cyclically equivalent class ETic 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 thatSn=Dnφ:φ∈An−1. In our case, S6=D6φ:φ∈A5. We know that there are 6!=720 arrangements of six different numbers l1′,l2′,l3′,l4′,l5′,l6′ taken all at a time, and so circuits corresponding to these arrangements are l1θ′,l2θ′,l3θ′,l4θ′,l5θ′,l6θ′ for each θ∈S6. Hence the proof.Corollary 3.
Ifl1′,l2′,l3′,l4′,l5′,l6′ are distinct positive integers, then there exist 60 cyclically equivalent classes ET1c.Proof.
Letl1′,l2′,l3′,l4′,l5′,l6′ be different positive integers. It is well known that the icosahedral group is isomorphic to alternating group A5. Now, the cyclically equivalent classes ET1c are obtained by El1φ′,l2φ′,l3φ′,l4φ′,l5φ′,l6φ′c for each φ∈A5. There are exactly 60 cyclically equivalent classes, namely, El1′,l2′,l3′,l4′,l5′,l6′c,El2′,l1′,l4′,l3′,l5′,l6′c,El2′,l1′,l5′,l4′,l3′,l6′c,El4′,l3′,l2′,l1′,l5′,l6′c,El5′,l3′,l2′,l4′,l1′,l6′c,El3′,l4′,l1′,l2′,l5′,l6′c,El5′,l4′,l3′,l2′,l1′,l6′c,El5′,l2′,l4′,l3′,l1′,l6′c,El4′,l2′,l5′,l1′,l3′,l6′c,El2′,l1′,l3′,l5′,l4′,l6′c,El3′,l2′,l1′,l5′,l4′,l6′c,El3′,l5′,l1′,l4′,l2′,l6′c,El4′,l5′,l3′,l1′,l2′,l6′c,El1′,l4′,l5′,l2′,l3′,l6′c,El1′,l5′,l4′,l3′,l2′,l6′c,El2′,l4′,l3′,l1′,l5′,l6′c,El2′,l5′,l3′,l4′,l1′,l6′c,El3′,l2′,l4′,l1′,l5′,l6′c,El3′,l2′,l5′,l4′,l1′,l6′c,El4′,l2′,l3′,l5′,l1′,l6′c,El2′,l3′,l1′,l4′,l5′,l6′c,El3′,l1′,l2′,l4′,l5′,l6′c,El4′,l1′,l3′,l2′,l5′,l6′c,El5′,l1′,l3′,l4′,l2′,l6′c,El4′,l2′,l1′,l3′,l5′,l6′c,El1′,l4′,l2′,l3′,l5′,l6′c,El1′,l2′,l4′,l5′,l3′,l6′c,El1′,l3′,l5′,l4′,l2′,l6′c,El1′,l2′,l5′,l3′,l4′,l6′c,El5′,l2′,l3′,l1′,l4′,l6′c,El1′,l3′,l4′,l2′,l5′,l6′c,El1′,l5′,l3′,l2′,l4′,l6′c,El1′,l4′,l3′,l5′,l2′,l6′c,El5′,l2′,l1′,l4′,l3′,l6′c,El1′,l5′,l2′,l4′,l3′,l6′c,El2′,l3′,l4′,l5′,l1′,l6′c,El2′,l4′,l5′,l3′,l1′,l6′c,El3′,l4′,l2′,l5′,l1′,l6′c,El3′,l5′,l4′,l2′,l1′,l6′c,El4′,l3′,l5′,l2′,l1′,l6′c,El4′,l5′,l2′,l3′,l1′,l6′c,El3′,l1′,l4′,l5′,l2′,l6′c,El4′,l1′,l5′,l3′,l2′,l6′c,El4′,l3′,l1′,l5′,l2′,l6′c,El5′,l3′,l4′,l1′,l2′,l6′c,El3′,l4′,l5′,l1′,l2′,l6′c,El4′,l1′,l2′,l5′,l3′,l6′c,El5′,l4′,l2′,l1′,l3′,l6′c,El5′,l1′,l4′,l2′,l3′,l6′c,El2′,l3′,l5′,l1′,l4′,l6′c,El3′,l5′,l2′,l1′,l4′,l6′c,El3′,l1′,l5′,l2′,l4′,l6′c,El5′,l3′,l1′,l2′,l4′,l6′c,El2′,l5′,l1′,l3′,l4′,l6′c,El2′,l5′,l4′,l1′,l3′,l6′c,El5′,l1′,l2′,l3′,l4′,l6′c,El1′,l3′,l2′,l5′,l4′,l6′c,El5′,l4′,l1′,l3′,l2′,l6′c,El2′,l4′,l1′,l5′,l3′,l6′c, and El4′,l5′,l1′,l2′,l3′,l6′c.Corollary 4.
Ifl1′,l2′,l3′,l4′,l5′,l6′ are distinct positive integers, then each cyclically equivalent class contains 12 cyclically equivalent circuits.Proof.
Circuits cyclically equivalent to circuitl1′,l2′,l3′,l4′,l5′,l6′ are l1θ′,l2θ′,l3θ′,l4θ′,l5θ′,l6θ′ for each θ∈D6 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 Table1 are cyclically equivalent. Moreover, by Lemma 1, l1′,l2′,l3′,l4′,l5′,l6′∼sl3′,l4′,l5′,l6′,l1′,l2′∼sl5′,l6′,l1′,l2′,l3′,l4′ is the circuit contained in γG,l2′,l3′,l4′,l5′,l6′,l1′∼sl4′,l5′,l6′,l1′,l2′,l3′∼sl6′,l1′,l2′,l3′,l4′,l5′ is the circuit contained in −γG,l1′,l6′,l5′,l4′,l3′,l2′∼sl5′,l4′,l3′,l2′,l1′,l6′∼sl3′,l2′,l1′,l6′,l5′,l4′ is the circuit contained in −γ¯G, and l6′,l5′,l4′,l3′,l2′,l1′∼sl4′,l3′,l2′,l1′,l6′,l5′∼sl2′,l1′,l6′,l5′,l4′,l3′ is the circuit contained in γ¯G.
From Table1, 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 ET1 are 12×60=720 which equals to ET1.
Table2 is of considerable utility because it provides us with the exact number of circuits of length 6 and hence the number of G-orbits of Q∗n.
The following corollaries are an immediate consequence of Theorem1.Table 1
Cyclically equivalent circuits of length 6.
θl1θ′,l2θ′,l3θ′,l4θ′,l5θ′,l6θ′e=123456l1′,l2′,l3′,l4′,l5′,l6′α=123456l2′,l3′,l4′,l5′,l6′,l1′α2=135246l3′,l4′,l5′,l6′,l1′,l2′α3=142536l4′,l5′,l6′,l1′,l2′,l3′α4=153264l5′,l6′,l1′,l2′,l3′,l4′α5=165432l6′,l1′,l2′,l3′,l4′,l5′β=162534l6′,l5′,l4′,l3′,l2′,l1′αβ=1524l5′,l4′,l3′,l2′,l1′,l6′α2β=142356l4′,l3′,l2′,l1′,l6′,l5′α3β=1346l3′,l2′,l1′,l6′,l5′,l4′α4β=124536l2′,l1′,l6′,l5′,l4′,l3′α5β=3526l1′,l6′,l5′,l4′,l3′,l2′Table 2
Enumeration of orbits.
Classes of equivalent circuits of length 6ETiNumber of classes of cyclically equivalent circuitsOTiRegarding distinct orbitsET172060240γG,γ¯G,−γG,−γ¯GET236030120γG,γ¯G,−γG,−γ¯GET31681456γG,γ¯G,−γG,−γ¯G1224γG=−γ¯G,−γG=γ¯GET41201040γG,γ¯G,−γG,−γ¯GET548416γG,γ¯G,−γG,−γ¯G1836γG=−γ¯G,−γG=γ¯G1836γG=−γ¯G,−γG=γ¯G612γG=−γ¯G,−γG=γ¯GET648416γG,γ¯G,−γG,−γ¯G1224γG=−γ¯G,−γG=γ¯GET72428γG,γ¯G,−γG,−γ¯G612γG=−γ¯G,−γG=γ¯GET81214γG,γ¯G,−γG,−γ¯G612γG=−γ¯G,−γG=γ¯GET9612γG=−γ¯G,−γG=γ¯G311γG=−γ¯G=−γG=γ¯G612γG=−γ¯G,−γG=γ¯GET10612γG=−γ¯G,−γG=γ¯GCorollary 5.
There are 6 equivalent circuits of length 6 in which 5 numbers are alike and one number is different.Proof.
By Lemma4, the number of equivalent circuits of length 6 in which 5 numbers are alike is 6!/5!1!=6. These 6 circuits are l1′,l1′,l1′,l1′,l1′,l2′,l1′,l1′,l1′,l1′,l2′,l1′,l1′,l2′,l1′,l1′,l1′,l1′,l2′,l1′,l1′,l1′,l1′,l1′,l1′,l1′,l1′,l2′,l1′,l1′, and l1′,l1′,l2′,l1′,l1′,l1′. Clearly, all these types are cyclically equivalent as well. Moreover, by Lemma 1, l1′,l1′,l1′,l1′,l1′,l2′∼sl1′,l2′,l1′,l1′,l1′,l1′∼sl1′,l1′,l1′,l2′,l1′,l1′ is the circuit contained in γG=−γ¯G, and l2′,l1′,l1′,l1′,l1′,l1′∼sl1′,l1′,l1′,l1′,l2′,l1′∼sl1′,l1′,l2′,l1′,l1′,l1′ is the circuit contained in γ¯G=−γG. In this situation, there is only one cyclically equivalent class, namely, El1′,l1′,l1′,l1′,l1′,l2′c.Corollary 6.
There are 15 equivalent circuits of length 6 in which 4 numbers are alike and 2 numbers are alike.Proof.
By Lemma4, the number of equivalent circuits of length 6 in which 4 numbers are alike and 2 numbers are alike is 6!/4!2!=15. These possible 15 circuits are l1′,l1′,l1′,l1′,l2′,l2′,l1′,l1′,l1′,l2′,l1′,l2′,l1′,l1′,l1′,l2′,l2′,l1′,l1′,l1′,l2′,l1′,l1′,l2′,l1′,l1′,l2′,l1′,l2′,l1′,l1′,l1′,l2′,l2′,l1′,l1′,l2′,l1′,l1′,l1′,l1′,l2′,l1′,l2′,l1′,l1′,l1′,l2′,l1′,l2′,l1′,l1′,l2′,l1′,l1′,l2′,l1′,l2′,l1′,l1′,l1′,l2′,l2′,l1′,l1′,l1′,l2′,l2′,l1′,l1′,l1′,l1′,l2′,l1′,l1′,l1′,l2′,l1′,l2′,l1′,l2′,l1′,l1′,l1′,andl2′,l1′,l1′,l2′,l1′,l1′, which are equivalent circuits.
In the aforementioned circuits,l1′,l1′,l1′,l1′,l2′,l2′,l1′,l1′,l1′,l2′,l2′,l1′,l1′,l2′,l2′,l1′,l1′,l1′,l2′,l2′,l1′,l1′,l1′,l1′,l1′,l1′,l2′,l2′,l1′,l1′, and l2′,l1′,l1′,l1′,l1′,l2′ are cyclically equivalent. Moreover, by Lemma 1, l1′,l1′,l1′,l1′,l2′,l2′∼sl1′,l1′,l2′,l2′,l1′,l1′∼sl2′,l2′,l1′,l1′,l1′,l1′ is the circuit contained in αG=α¯G, and l2′,l1′,l1′,l1′,l1′,l2′∼sl1′,l2′,l2′,l1′,l1′,l1′∼sl1′,l1′,l1′,l2′,l2′,l1′ is the circuit contained in −α¯G=−αG.
Also,l2′,l1′,l1′,l2′,l1′,l1′,l1′,l1′,l2′,l1′,l1′,l2′,andl1′,l2′,l1′,l1′,l2′,l1′ are cyclically equivalent. Moreover, by Lemma 1, l2′,l1′,l1′,l2′,l1′,l1′∼sl1′,l1′,l2′,l1′,l1′,l2′∼sl1′,l2′,l1′,l1′,l2′,l1′ is the circuit contained in βG=β¯G=−βG=−β¯G.
These circuitsl1′,l1′,l1′,l2′,l1′,l2′,l1′,l1′,l2′,l1′,l2′,l1′,l1′,l2′,l1′,l2′,l1′,l1′,l2′,l1′,l2′,l1′,l1′,l1′,l1′,l2′,l1′,l1′,l1′,l2′,andl2′,l1′,l1′,l1′,l2′,l1′ are cyclically equivalent. Moreover, by Lemma 1, l1′,l1′,l1′,l2′,l1′,l2′∼sl1′,l2′,l1′,l2′,l1′,l1′∼sl1′,l2′,l1′,l1′,l1′,l2′ is the circuit contained in γG=−γ¯G, and l1′,l1′,l2′,l1′,l2′,l1′∼sl2′,l1′,l2′,l1′,l1′,l1′∼sl2′,l1′,l1′,l1′,l2′,l1′ is the circuit contained in −γG=γ¯G.
Table3 summarizes all the information.Table 3
Cyclically equivalent classes of T9.
ET9cNumber of classes of cyclically equivalent circuitsOT9Structure of orbitsEl1′,l1′,l1′,l1′,l2′,l2′c12γG=γ¯G,−γG=−γ¯GEl1′,l2′,l1′,l2′,l1′,l1′c12γG=−γ¯G,−γG=γ¯GEl1′,l1′,l2′,l1′,l1′,l2′c11γG=−γ¯G=−γG=γ¯GCorollary 7.
There are 18 equivalent circuits of length 6 in which 3 numbers are alike and 3 numbers are alike.Proof.
By Lemma4, the number of possible equivalent circuits of length 6 in which 3 numbers are alike and 3 numbers are alike is 6!/3!3!=20. These possible 20 circuits are given by l1′,l1′,l1′,l2′,l2′,l2′,l1′,l1′,l2′,l1′,l2′,l2′,l1′,l1′,l2′,l2′,l1′,l2′,l1′,l1′,l2′,l2′,l2′,l1′,l1′,l2′,l1′,l1′,l2′,l2′,l1′,l2′,l1′,l2′,l1′,l2′,l1′,l2′,l1′,l2′,l2′,l1′,l1′,l2′,l2′,l1′,l1′,l2′,l1′,l2′,l2′,l1′,l2′,l1′,l1′,l2′,l2′,l2′,l1′,l1′,l2′,l2′,l2′,l1′,l1′,l1′,l2′,l2′,l1′,l2′,l1′,l1′,l2′,l2′,l1′,l1′,l2′,l1′,l2′,l2′,l1′,l1′,l1′,l2′,l2′,l1′,l2′,l1′,l2′,l1′,l2′,l1′,l1′,l2′,l2′,l1′,l2′,l1′,l1′,l2′,l1′,l2′,l2′,l1′,l1′,l1′,l2′,l2′,l2′,l1′,l2′,l2′,l1′,l1′, and l2′,l1′,l2′,l1′,l1′,l2′ which are equivalent circuits. In the aforementioned circuits, l1′,l1′,l1′,l2′,l2′,l2′,l1′,l1′,l2′,l2′,l2′,l1′,l1′,l2′,l2′,l2′,l1′,l1′,l2′,l2′,l2′,l1′,l1′,l1′,l2′,l2′,l1′,l1′,l1′,l2′,andl2′,l1′,l1′,l1′,l2′,l2′ are cyclically equivalent. Moreover, by Lemma 1, l1′,l1′,l1′,l2′,l2′,l2′∼sl1′,l2′,l2′,l2′,l1′,l1′∼sl2′,l2′,l1′,l1′,l1′,l2′ is the circuit of αG=−α¯G, and l2′,l1′,l1′,l1′,l2′,l2′∼sl1′,l1′,l2′,l2′,l2′,l1′∼sl2′,l2′,l2′,l1′,l1′,l1′ is the circuit of α¯G=−αG. Also, circuits l1′,l1′,l2′,l2′,l1′,l2′,l1′,l2′,l1′,l1′,l2′,l2′,l2′,l2′,l1′,l2′,l1′,l1′,l2′,l1′,l1′,l2′,l1′,l2′,l2′,l2′,l1′,l1′,l2′,l1′,l1′,l1′,l2′,l1′,l2′,l2′,l1′,l2′,l2′,l1′,l1′,l2′,l2′,l1′,l2′,l2′,l1′,l1′,l1′,l2′,l2′,l1′,l2′,l1′,l2′,l1′,l2′,l1′,l1′,l2′,l2′,l1′,l1′,l2′,l2′,l1′,andl1′,l2′,l1′,l2′,l2′,l1′ are cyclically equivalent. Moreover, by Lemma 1, l1′,l1′,l2′,l2′,l1′,l2′∼sl1′,l2′,l1′,l1′,l2′,l2′∼sl2′,l2′,l1′,l2′,l1′,l1′ is the circuit of βG,l1′,l2′,l2′,l1′,l2′,l1′∼sl2′,l1′,l2′,l1′,l1′,l2′∼sl2′,l1′,l1′,l2′,l2′,l1′ is the circuit contained in −βG,l1′,l2′,l1′,l2′,l2′,l1′∼sl1′,l2′,l2′,l1′,l1′,l2′∼sl2′,l1′,l1′,l2′,l1′,l2′ is the circuit of −β¯G, and l2′,l1′,l2′,l2′,l1′,l1′∼sl2′,l2′,l1′,l1′,l2′,l1′∼sl1′,l1′,l2′,l1′,l2′,l2′ is the circuit contained in β¯G. Moreover, l1′,l2′,l1′,l2′,l1′,l2′ and l2′,l1′,l2′,l1′,l2′,l1′ are not possible by Lemma 3. Table 4 summarizes all the information.Table 4
Cyclically equivalent classes of T8.
ET8cNumber of classes of cyclically equivalent circuitsOT8Structure of orbitsEl1′,l1′,l1′,l2′,l2′,l2′c12γG=−γ¯G,−γG=γ¯GEl1′,l1′,l2′,l1′,l2′,l2′c14γG,γ¯G,−γG,−γ¯GCorollary 8.
There are 30 equivalent circuits of length 6 in which 4 numbers are alike and 2 numbers are different.Proof.
By Lemma4, the number of equivalent circuits of length 6 in which 4 numbers are alike and 2 numbers are different is 6!/4!1!1!=30. These 30 circuits are l1′,l1′,l1′,l1′,l2′,l3′,l1′,l1′,l1′,l2′,l1′,l3′,l1′,l1′,l1′,l2′,l3′,l1′,l1′,l1′,l1′,l3′,l1′,l2′,l1′,l1′,l1′,l1′,l3′,l2′,l1′,l1′,l3′,l2′,l1′,l1′,l1′,l1′,l3′,l1′,l2′,l1′,l1′,l1′,l3′,l1′,l1′,l2′,l1′,l1′,l1′,l3′,l2′,l1′,l1′,l1′,l2′,l3′,l1′,l1′,l1′,l1′,l2′,l1′,l3′,l1′,l1′,l1′,l2′,l1′,l1′,l3′,l1′,l2′,l1′,l1′,l1′,l3′,l1′,l2′,l1′,l1′,l3′,l1′,l1′,l2′,l1′,l3′,l1′,l1′,l1′,l2′,l3′,l1′,l1′,l1′,l1′,l3′,l1′,l1′,l1′,l2′,l1′,l3′,l1′,l1′,l2′,l1′,l1′,l3′,l1′,l2′,l1′,l1′,l1′,l3′,l2′,l1′,l1′,l1′,l2′,l1′,l1′,l1′,l1′,l3′,l2′,l1′,l1′,l1′,l3′,l1′,l2′,l1′,l1′,l3′,l1′,l1′,l2′,l1′,l3′,l1′,l1′,l1′,l2′,l3′,l1′,l1′,l1′,l1′,l3′,l1′,l1′,l1′,l1′,l2′,l3′,l1′,l1′,l1′,l2′,l1′,l3′,l1′,l1′,l2′,l1′,l1′,l3′,l1′,l2′,l1′,l1′,l1′,andl3′,l2′,l1′,l1′,l1′,l1′.
In the aforementioned circuits,l1′,l1′,l1′,l1′,l2′,l3′,l1′,l1′,l2′,l3′,l1′,l1′,l2′,l3′,l1′,l1′,l1′,l1′,l1′,l1′,l1′,l2′,l3′,l1′,l1′,l2′,l3′,l1′,l1′,l1′,l3′,l1′,l1′,l1′,l1′,l2′,l3′,l2′,l1′,l1′,l1′,l1′,l1′,l1′,l3′,l2′,l1′,l1′,l1′,l1′,l1′,l1′,l3′,l2′,l1′,l3′,l2′,l1′,l1′,l1′,l2′,l1′,l1′,l1′,l1′,l3′, and l1′,l1′,l1′,l3′,l2′,l1′ are cyclically equivalent. By Lemma 1, l1′,l1′,l1′,l1′,l2′,l3′∼sl1′,l1′,l2′,l3′,l1′,l1′∼sl2′,l3′,l1′,l1′,l1′,l1′ is the circuit of αG,l1′,l1′,l1′,l2′,l3′,l1′∼sl1′,l2′,l3′,l1′,l1′,l1′∼sl3′,l1′,l1′,l1′,l1′,l2′ is the circuit contained in −αG,l3′,l2′,l1′,l1′,l1′,l1′∼sl1′,l1′,l3′,l2′,l1′,l1′∼sl1′,l1′,l1′,l1′,l3′,l2′ is the circuit of α¯G, and l1′,l3′,l2′,l1′,l1′,l1′∼sl2′,l1′,l1′,l1′,l1′,l3′∼sl1′,l1′,l1′,l3′,l2′,l1′ is the circuit contained in −α¯G. Also, l1′,l1′,l1′,l3′,l1′,l2′,l1′,l3′,l1′,l2′,l1′,l1′,l1′,l2′,l1′,l1′,l1′,l3′,l1′,l1′,l3′,l1′,l2′,l1′,l3′,l1′,l2′,l1′,l1′,l1′,l2′,l1′,l1′,l1′,l3′,l1′,l1′,l2′,l1′,l3′,l1′,l1′,l1′,l3′,l1′,l1′,l1′,l2′,l1′,l1′,l1′,l2′,l1′,l3′,l2′,l1′,l3′,l1′,l1′,l1′,l3′,l1′,l1′,l1′,l2′,l1′,andl1′,l1′,l2′,l1′,l3′,l1′ are cyclically equivalent. By Lemma 1, l1′,l1′,l1′,l3′,l1′,l2′∼sl1′,l3′,l1′,l2′,l1′,l1′∼sl1′,l2′,l1′,l1′,l1′,l3′ is the circuit of βG,l1′,l1′,l3′,l1′,l2′,l1′∼sl3′,l1′,l2′,l1′,l1′,l1′∼sl2′,l1′,l1′,l1′,l3′,l1′ is the circuit of −βG,l1′,l2′,l1′,l3′,l1′,l1′∼sl1′,l3′,l1′,l1′,l1′,l2′∼sl1′,l1′,l1′,l2′,l1′,l3′ is the circuit contained in −β¯G, and l2′,l1′,l3′,l1′,l1′,l1′∼sl3′,l1′,l1′,l1′,l2′,l1′∼sl1′,l1′,l2′,l1′,l3′,l1′ is the circuit contained in β¯G. Also, l1′,l1′,l2′,l1′,l1′,l3′,l2′,l1′,l1′,l3′,l1′,l1′,l1′,l3′,l1′,l1′,l2′,l1′,l1′,l2′,l1′,l1′,l3′,l1′,l1′,l1′,l3′,l1′,l1′,l2′,andl3′,l1′,l1′,l2′,l1′,l1′ are cyclically equivalent. Moreover, by Lemma 1, l1′,l1′,l2′,l1′,l1′,l3′∼sl2′,l1′,l1′,l3′,l1′,l1′∼sl1′,l3′,l1′,l1′,l2′,l1′ is the circuit of γG=−γ¯G, and l1′,l2′,l1′,l1′,l3′,l1′∼sl1′,l1′,l3′,l1′,l1′,l2′∼sl3′,l1′,l1′,l2′,l1′,l1′ is the circuit of −γG=γ¯G.
Table5 summarizes all the information.Table 5
Cyclically equivalent classes of T7.
ET7cNumber of classes of cyclically equivalent circuitsOT7Structure of orbitsEl1′,l1′,l2′,l1′,l1′,l3′c12γG=−γ¯G, −γG=γ¯GEl1′,l1′,l1′,l1′,l2′,l3′c,El1′,l2′,l1′,l3′,l1′,l1′c28γG,γ¯G,−γG,−γ¯GCorollary 9.
There are 60 equivalent circuits of length 6 in which 3 numbers are alike, 2 numbers are alike, and 1 number is different.Proof.
By Lemma4, the number of equivalent length 6 in which 3 numbers are alike, 2 numbers are alike, and 1 number is different is 6!/3!2!1!=60. These circuits are l1′,l1′,l1′,l2′,l2′,l3′,l1′,l2′,l2′,l3′,l1′,l1′,l2′,l3′,l1′,l1′,l1′,l2′,l1′,l1′,l2′,l2′,l3′,l1′,l2′,l2′,l3′,l1′,l1′,l1′,l2′,l1′,l1′,l1′,l3′,l2′,l3′,l1′,l1′,l1′,l2′,l2′,l1′,l3′,l2′,l2′,l1′,l1′,l2′,l2′,l1′,l1′,l1′,l3′,l1′,l1′,l1′,l3′,l2′,l2′,l3′,l2′,l2′,l1′,l1′,l1′,l1′,l1′,l3′,l2′,l2′,l1′, and so on.
In the aforementioned circuits,l1′,l1′,l1′,l2′,l2′,l3′,l1′,l2′,l2′,l3′,l1′,l1′,l2′,l3′,l1′,l1′,l1′,l2′,l1′,l1′,l2′,l2′,l3′,l1′,l3′,l1′,l1′,l1′,l2′,l2′,l1′,l3′,l2′,l2′,l1′,l1′,l2′,l2′,l1′,l1′,l1′,l3′,l1′,l1′,l1′,l3′,l2′,l2′,l3′,l2′,l2′,l1′,l1′,l1′,l1′,l1′,l3′,l2′,l2′,l1′l2′,l1′,l1′,l1′,l3′,l2′, and l2′,l2′,l3′,l1′,l1′,l1′ are cyclically equivalent. By Lemma 1, l2′,l3′,l1′,l1′,l1′,l2′∼sl1′,l1′,l1′,l2′,l2′,l3′∼sl1′,l2′,l2′,l3′,l1′,l1′ is the circuit of αG,l1′,l1′,l2′,l2′,l3′,l1′∼sl2′,l2′,l3′,l1′,l1′,l1′∼sl3′,l1′,l1′,l1′,l2′,l2′ is the circuit contained in −αG,l1′,l3′,l2′,l2′,l1′,l1′∼sl2′,l2′,l1′,l1′,l1′,l3′∼sl1′,l1′,l1′,l3′,l2′,l2′ is the circuit of −α¯G,l3′,l2′,l2′,l1′,l1′,l1′∼sl2′,l1′,l1′,l1′,l3′,l2′∼sl1′,l1′,l3′,l2′,l2′,l1′ is the circuit contained in α¯G, and so on.
Table6 summarizes all the information.Table 6
Cyclically equivalent classes of T6.
ET6cNumber of classes of cyclically equivalent circuitsOT6Structure of orbitsEl1′,l3′,l2′,l2′,l1′,l1′c,El1′,l1′,l2′,l1′,l2′,l3′c,El1′,l2′,l2′,l1′,l3′,l1′c,El1′,l2′,l1′,l1′,l2′,l3′c416γG,γ¯G,−γG,−γ¯GEl1′,l1′,l1′,l2′,l3′,l2′c,El1′,l2′,l1′,l2′,l1′,l3′c24γG=−γ¯G, −γG=γ¯GCorollary 10.
There are 90 equivalent circuits of length 6 in which 2 numbers are alike, 2 numbers are alike, and 2 numbers are alike.Proof.
By Lemma4, the number of equivalent circuits of length 6 in which 2 numbers are alike, 2 numbers are alike, and 2 numbers are alike is 6!/2!2!2!=90. Some of these circuits are l1′,l1′,l2′,l2′,l3′,l3′,l2′,l2′,l3′,l3′,l1′,l1′,l3′,l3′,l1′,l1′,l2′,l2′,l1′,l2′,l2′,l3′,l3′,l1′,l2′,l3′,l3′,l1′,l1′,l2′,l2′,l2′,l3′,l1′,l1′,l3′,l3′,l1′,l1′,l2′,l2′,l3′,l1′,l3′,l3′,l2′,l2′,l1′,l3′,l2′,l2′,l1′,l1′,l3′,l2′,l1′,l1′,l3′,l3′,l2′,l3′,l3′,l2′,l2′,l1′,l1′,l2′,l3′,l1′,l1′,l3′,l2′,l2′,l2′,l1′,l1′,l3′,l3′,l1′,l1′,l3′,l3′,l2′,l2′,l1′,l1′,l3′,l2′,l2′,l3′,l1′,l3′,l2′,l2′,l3′,l1′,l3′,l2′,l2′,l3′,l1′,l1′,l3′,l1′,l1′,l3′,l2′,l2′, and so on.
In th aforementioned circuits,l1′,l1′,l2′,l2′,l3′,l3′,l2′,l2′,l3′,l3′,l1′,l1′,l3′,l3′,l1′,l1′,l2′,l2′,l1′,l2′,l2′,l3′,l3′,l1′,l2′,l3′,l3′,l1′,l1′,l2′,l3′,l1′,l1′,l2′,l2′,l3′,l1′,l3′,l3′,l2′,l2′,l1′,l3′,l2′,l2′,l1′,l1′,l3′,l2′,l1′,l1′,l3′,l3′,l2′,l3′,l3′,l2′,l2′,l1′,l1′,l2′,l2′,l1′,l1′,l3′,l3′,andl1′,l1′,l3′,l3′,l2′,l2′ are cyclically equivalent. By Lemma 1, l1′,l1′,l2′,l2′,l3′,l3′∼sl2′,l2′,l3′,l3′,l1′,l1′∼sl3′,l3′,l1′,l1′,l2′,l2′ is the circuit of αG,l1′,l2′,l2′,l3′,l3′,l1′∼sl2′,l3′,l3′,l1′,l1′,l2′∼sl3′,l1′,l1′,l2′,l2′,l3′ is the circuit contained in −αG,l1′,l3′,l3′,l2′,l2′,l1′∼sl3′,l2′,l2′,l1′,l1′,l3′∼sl2′,l1′,l1′,l3′,l3′,l2′ is the circuit of −α¯G,l3′,l3′,l2′,l2′,l1′,l1′∼sl2′,l2′,l1′,l1′,l3′,l3′∼sl1′,l1′,l3′,l3′,l2′,l2′ is the circuit contained in α¯G, and so on.
Table7 summarizes all the information.Table 7
Cyclically equivalent classes of T5.
ET5cNumber of classes of cyclically equivalent circuitsOT5Structure of orbitsEl3′,l3′,l2′,l2′,l1′,l1′c,El1′,l3′,l2′,l3′,l2′,l1′c,El1′,l2′,l1′,l3′,l3′,l2′c,El1′,l3′,l2′,l2′,l1′,l3′c416γG,γ¯G,−γG,−γ¯GEl3′,l2′,l1′,l2′,l3′,l1′c,El1′,l3′,l2′,l3′,l1′,l2′c,El1′,l3′,l1′,l2′,l3′,l2′c312γG=−γ¯G,−γG=γ¯GEl1′,l3′,l2′,l2′,l3′,l1′c,El1′,l2′,l2′,l1′,l3′,l3′c,El1′,l1′,l2′,l3′,l3′,l2′c312γG=γ¯G,−γG=−γ¯GEl1′,l2′,l3′,l1′,l2′,l3′c12γG=−γG,γ¯G=−γ¯GCorollary 11.
There are 120 equivalent circuits of length 6 in which 3 numbers are alike and 3 numbers are different.Proof.
By Lemma4, the number of equivalent circuits of length 6 in which 3 numbers are alike and 3 numbers are different is 6!/3!1!1!1!=120. Some of these circuits are l1′,l1′,l1′,l2′,l3′,l4′,l1′,l2′,l3′,l4′,l1′,l1′,l3′,l4′,l1′,l1′,l1′,l2′,l1′,l1′,l2′,l3′,l4′,l1′,l2′,l3′,l4′,l1′,l1′,l1′,l4′,l2′,l3′,l1′,l1′,l1′,l3′,l2′,l4′,l1′,l1′,l1′,l4′,l1′,l1′,l1′,l2′,l3′,l1′,l4′,l3′,l2′,l1′,l1′,l3′,l2′,l1′,l1′,l1′,l4′,l1′,l1′,l1′,l4′,l3′,l2′,l4′,l3′,l2′,l1′,l1′,l1′,l2′,l4′,l1′,l1′,l1′,l3′,l2′,l1′,l1′,l1′,l4′,l3′,l1′,l1′,l4′,l3′,l2′,l1′,l1′,l1′,l1′,l3′,l2′,l4′,l1′,l1′,l3′,l2′,l4′,l1′,l1′,l3′,l2′,l4′,l1′,l1′,l4′,l1′,l1′,l1′,l3′,l2′,l1′,l4′,l2′,l3′,l1′,l1′, and so on.
In the aforementioned circuits,l1′,l1′,l1′,l2′,l3′,l4′,l1′,l2′,l3′,l4′,l1′,l1′,l3′,l4′,l1′,l1′,l1′,l2′,l1′,l1′,l2′,l3′,l4′,l1′,l2′,l3′,l4′,l1′,l1′,l1′,l4′,l1′,l1′,l1′,l2′,l3′,l1′,l4′,l3′,l2′,l1′,l1′,l3′,l2′,l1′,l1′,l1′,l4′,l1′,l1′,l1′,l4′,l3′,l2′,l4′,l3′,l2′,l1′,l1′,l1′,l2′,l1′,l1′,l1′,l4′,l3′,andl1′,l1′,l4′,l3′,l2′,l1′ are cyclically equivalent. By Lemma 1, l1′,l1′,l1′,l2′,l3′,l4′∼sl1′,l2′,l3′,l4′,l1′,l1′∼sl3′,l4′,l1′,l1′,l1′,l2′ is the circuit of αG,l1′,l1′,l2′,l3′,l4′,l1′∼sl2′,l3′,l4′,l1′,l1′,l1′∼sl4′,l1′,l1′,l1′,l2′,l3′ is the circuit contained in −αG,l1′,l4′,l3′,l2′,l1′,l1′∼sl3′,l2′,l1′,l1′,l1′,l4′∼sl1′,l1′,l1′,l4′,l3′,l2′ is the circuit of −α¯G,l4′,l3′,l2′,l1′,l1′,l1′∼sl2′,l1′,l1′,l1′,l4′,l3′∼sl1′,l1′,l4′,l3′,l2′,l1′ is the circuit contained in α¯G, and so on.
Table8 summarizes all the information.Table 8
Cyclically equivalent classes of T4.
ET4cNumber of classes of cyclically equivalent circuitsOT4Structure of orbitsEl1′,l4′,l3′,l2′,l1′,l1′c,El1′,l1′,l1′,l2′,l4′,l3′c,El1′,l2′,l4′,l1′,l3′,l1′c,El1′,l3′,l4′,l1′,l2′,l1′c,El1′,l2′,l4′,l1′,l1′,l3′c,El1′,l4′,l2′,l3′,l1′,l1′c,El1′,l2′,l1′,l1′,l4′,l3′c,El1′,l1′,l4′,l1′,l2′,l3′c,El1′,l3′,l2′,l1′,l1′,l4′c,El1′,l4′,l1′,l2′,l1′,l3′c1040γG,γ¯G,−γG,−γ¯GCorollary 12.
There are 180 equivalent circuits of length 6 in which 2 numbers are alike, 2 numbers are alike, and 2 numbers are different.Proof.
By Lemma4, the number of equivalent circuits of length 6 in which 2 numbers are alike, 2 numbers are alike, and 2 numbers are different is 6!/2!2!1!1!=180. Some these circuits are l1′,l1′,l2′,l2′,l3′,l4′,l2′,l2′,l3′,l4′,l1′,l1′,l3′,l4′,l1′,l1′,l2′,l2′,l1′,l2′,l2′,l3′,l4′,l1′,l2′,l3′,l4′,l1′,l1′,l2′,l4′,l1′,l1′,l2′,l2′,l3′,l1′,l4′,l3′,l2′,l2′,l1′,l3′,l2′,l2′,l1′,l1′,l4′,l2′,l1′,l1′,l4′,l3′,l2′,l4′,l3′,l2′,l2′,l1′,l1′,l2′,l2′,l1′,l1′,l4′,l3′,l1′,l1′,l4′,l3′,l2′,l2′,l1′,l1′,l2′,l3′,l4′,l2′,l1′,l1′,l2′,l4′,l2′,l3′,l1′,l1′,l2′,l4′,l3′,l2′,l1′,l1′,l3′,l2′,l2′,l4′,l1′,l1′,l3′,l2′,l4′,l2′,l1′,l1′,l3′,l4′,l2′,l2′,l1′,l1′,l4′,l2′,l2′,l3′, and so on.
In the aforementioned circuits,l1′,l1′,l2′,l2′,l3′,l4′,l2′,l2′,l3′,l4′,l1′,l1′,l3′,l4′,l1′,l1′,l2′,l2′,l1′,l2′,l2′,l3′,l4′,l1′,l2′,l3′,l4′,l1′,l1′,l2′,l4′,l1′,l1′,l2′,l2′,l3′,l1′,l4′,l3′,l2′,l2′,l1′,l3′,l2′,l2′,l1′,l1′,l4′,l2′,l1′,l1′,l4′,l3′,l2′,l4′,l3′,l2′,l2′,l1′,l1′,l2′,l2′,l1′,l1′,l4′,l3′,andl1′,l1′,l4′,l3′,l2′,l2′ are cyclically equivalent. Moreover, by Lemma 1, l1′,l1′,l2′,l2′,l3′,l4′∼sl2′,l2′,l3′,l4′,l1′,l1′∼sl3′,l4′,l1′,l1′,l2′,l2′ is the circuit of αG,l1′,l2′,l2′,l3′,l4′,l1′∼sl2′,l3′,l4′,l1′,l1′,l2′∼sl4′,l1′,l1′,l2′,l2′,l3′ is the circuit of −αG,l1′,l4′,l3′,l2′,l2′,l1′∼sl3′,l2′,l2′,l1′,l1′,l4′∼sl2′,l1′,l1′,l4′,l3′,l2′ is the circuit contained in −α¯G,l4′,l3′,l2′,l2′,l1′,l1′∼sl2′,l2′,l1′,l1′,l4′,l3′∼sl1′,l1′,l4′,l3′,l2′,l2′ is the circuit contained in α¯G, and so on.
Table9 summarizes all the information.Table 9
Cyclically equivalent classes of T3.
ET3cNumber of classes of cyclically equivalent circuitsOT3Structure of orbitsEl1′,l4′,l3′,l2′,l2′,l1′c,El1′,l1′,l2′,l2′,l4′,l3′c,El1′,l4′,l2′,l3′,l2′,l1′c,El1′,l3′,l2′,l4′,l2′,l1′c,El1′,l1′,l3′,l2′,l2′,l4′c,El1′,l1′,l2′,l3′,l4′,l2′c,El1′,l4′,l3′,l2′,l1′,l2′c,El1′,l2′,l1′,l2′,l4′,l3′c,El1′,l3′,l2′,l2′,l1′,l4′c,El1′,l2′,l1′,l3′,l2′,l4′c,El1′,l3′,l1′,l2′,l2′,l4′c,El1′,l4′,l2′,l1′,l3′,l2′c,El1′,l2′,l2′,l1′,l3′,l4′c,El1′,l3′,l2′,l1′,l2′,l4′c1456γG,γ¯G,−γG,−γ¯GEl1′,l2′,l4′,l2′,l1′,l3′c,El2′,l3′,l2′,l1′,l4′,l1′c24γG=−γ¯G,−γG=γ¯GCorollary 13.
There are 360 equivalent circuits of length 6 in which 2 numbers are alike and 4 numbers are different.Proof.
By Lemma4, the number of equivalent circuits of length 6 in which 2 numbers are alike and 4 numbers are different is 6!/2!1!1!1!=360. Some of these circuits are l1′,l1′,l2′,l3′,l4′,l5′,l2′,l3′,l4′,l5′,l1′,l1′,l4′,l5′,l1′,l1′,l2′,l3′,l1′,l2′,l3′,l4′,l5′,l1′,l3′,l4′,l5′,l1′,l1′,l2′,l3′,l2′,l1′,l1′,l5′,l4′,l5′,l1′,l1′,l2′,l3′,l4′,l1′,l5′,l4′,l3′,l2′,l1′,l4′,l3′,l2′,l1′,l1′,l5′,l2′,l1′,l1′,l5′,l4′,l3′,l5′,l4′,l3′,l2′,l1′,l1′,l1′,l1′,l5′,l4′,l3′,l2′, and so on.
In the aforementioned circuits,l1′,l1′,l2′,l3′,l4′,l5′,l1′,l2′,l3′,l4′,l5′,l1′,l2′,l3′,l4′,l5′,l1′,l1′,l3′,l4′,l5′,l1′,l1′,l2′,l4′,l5′,l1′,l1′,l2′,l3′,l5′,l1′,l1′,l2′,l3′,l4′,l1′,l5′,l4′,l3′,l2′,l1′,l5′,l4′,l3′,l2′,l1′,l1′,l4′,l3′,l2′,l1′,l1′,l5′,l3′,l2′,l1′,l1′,l5′,l4′,l2′,l1′,l1′,l5′,l4′,l3′andl1′,l1′,l5′,l4′,l3′,l2′ are cyclically equivalent.
Table10 summarizes all the information.
It was proved in [8] that any circuit l1′,l2′,l3′,l4′,l5′,l6′ of length 6 corresponds to the orbit contained in Q∗fl1′,l2′,l3′,l4′,l5′,l6′.Table 10
Cyclically equivalent classes of T2.
ET3cNumber of classes of the cyclically equivalent circuitsOT3Structure of orbitsEl1′,l1′,l2′,l3′,l4′,l5′c,El1′,l1′,l2′,l3′,l5′,l4′c,El1′,l1′,l2′,l4′,l3′,l5′c,El1′,l1′,l2′,l4′,l5′,l3′c,El1′,l1′,l2′,l5′,l3′,l4′c,El1′,l1′,l2′,l5′,l4′,l3′c,El1′,l1′,l3′,l2′,l5′,l4′c,El1′,l1′,l3′,l2′,l4′,l5′c,El1′,l1′,l3′,l4′,l2′,l5′c,El1′,l1′,l3′,l5′,l2′,l4′c,El1′,l1′,l4′,l2′,l3′,l5′c,El1′,l1′,l4′,l3′,l2′,l5′c,El1′,l4′,l1′,l2′,l3′,l5′c,El2′,l1′,l4′,l3′,l5′,l1′c,El2′,l1′,l5′,l4′,l3′,l1′c,El4′,l3′,l2′,l1′,l5′,l1′c,El3′,l4′,l1′,l2′,l5′,l1′c,El4′,l2′,l5′,l1′,l3′,l1′c,El2′,l1′,l3′,l5′,l4′,l1′c,El3′,l2′,l1′,l5′,l4′,l1′c,El5′,l3′,l1′,l2′,l4′,l1′c,El5′,l1′,l4′,l2′,l3′,l1′c,El3′,l5′,l2′,l1′,l4′,l1′c,El5′,l4′,l2′,l1′,l3′,l1′c,El2′,l5′,l4′,l1′,l3′,l1′c,El5′,l2′,l3′,l1′,l4′,l1′c,El5′,l1′,l3′,l4′,l2′,l1′c,El5′,l2′,l1′,l3′,l4′,l1′c,El5′,l3′,l1′,l4′,l2′,l1′c,El5′,l4′,l1′,l2′,l3′,l1′c30120γG,γ¯G,−γG,−γ¯GTheorem 2.
All the circuits inEl1′,l2′,l3′,l4′,l5′,l6′c correspond to the orbits contained in Q∗n1.Proof.
To prove this result, it is enough to show thatfl1′,l2′,l3′,l4′,l5′,l6′ is unchanged for θ∈D6, where fl1′,l2′,l3′,l4′,l5′,l6′=l1′l6′+l4′+l2′+l3′l6′+l4′+l5′l6′+l2′l1′l4′l3′+l6′l5′+l4′l5′l6′l3′+l1′+l1′l6′l3′l2′1+l4′l5′−l2′l5′+l3′−l5′l4′1+l2′l3′2+4l5′1+l1′l2′+l3′l4′+l1′l4′+l1′l2′l3′l4+l3′+l1′+l3′l2′l1′1+l6′l5′l4′+l2′+l2′l3′l4+l6′1+l2′l3′.
It is easy to see that after applyingθ∈D6 and simplifying, the expression for fl1′,l2′,l3′,l4′,l5′,l6′ is unaltered. Since there are 60 cyclically equivalent classes, we have sixty n's, namely, n1,n2,n3,⋯,n60.Example 4.
We verify the above results by considering the circuits1,2,3,4,5,6,2,3,4,5,6,1,1,6,5,4,3,2, and 6,5,4,3,2,1. These four circuits correspond to the orbits γG,−γG,−γ¯G, andγ¯G, and these four circuits are shown in Figures 1–4, respectively.
Now, in all figures containing 3 reduced numbers,E1,2,3,4,5,6c contains 12 reduced numbers.Figure 1
Shows the location of reduced numbers and describes that1,2,3,4,5,6∼s3,4,5,6,1,2∼s5,6,1,2,3,4.Figure 2
Shows the location of reduced numbers and describes that2,3,4,5,6,1∼s4,5,6,1,2,3∼s6,1,2,3,4,5.Figure 3
Shows the location of reduced numbers and describes that1,6,5,4,3,2∼s5,4,3,2,1,6∼s3,2,1,6,5,4.Figure 4
Shows the location of reduced numbers and describes that6,5,4,3,2,1∼s4,3,2,1,6,5∼s2,1,6,5,4,3.


## 4. Conclusion

The idea ofG-circuits of G-orbits on the real quadratic field by the modular group, which is given in this paper, is new and original. We have classified G-circuits into the distinct equivalence classes of equivalent circuits, and they are precisely pq−1 in number; particularly, for circuits of length 6, we have p6−1=10 equivalence classes of equivalent circuits, i.e., E6=∪i=110ETi. We further classify equivalence classes of equivalent circuits into cyclically equivalence classes and determine G-orbits corresponding to each cyclically equivalence class. All results are verified by suitable examples.



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