# Mobius Group Generated by Two Elements of Order 2, 4, and Reduced Quadratic Irrational Numbers

Authors: Dilshad Alghazzawi; M. Haris Mateen; M. Aslam Malik; P. Hammachukiattikul; Mohammed S. Abdo
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The construction of circuits for the evolution of orbits and reduced quadratic irrational numbers under the action of Mobius groups have many applications like in construction of substitution box (s-box), strong-substitution box (s.s-box), image processing, data encryption, in interest for security experts, and other fields of sciences. In this paper, we investigate the behavior of reduced quadratic irrational numbers (RQINs) in the coset diagrams of the setQ′′m=η/s:η∈Q∗m,s=1,2 under the action of group H=<x′,y′:x′2=y′4=1>, where m is square free integer and Q∗m=a′+m/c′,a′,a′2−m/c′c′=1,c′≠0. We discuss the type and reduced cardinality of the orbit Q′′p. By using the notion of congruence, we give the general form of reduced numbers (RNs) in particular orbits under certain conditions on prime p. Further, we classify that for a reduced number r whether −r,r¯,−r¯ lying in orbit or not. AMS Mathematics subject classification (2010): 05C25, 20G401.


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## Body

## 1. Introduction and Preliminaries

Groups are very helpful algebraic structures, carrying other algebraic structures on them. In abstract algebra, almost all typical structures are illustration of groups. The significance of groups was derived from their action on special structures or spaces. Cryptography is the technique of converting secret knowledge into information and a type of data pretending to reach its terminus without leaking data safely. Modern cryptography is classified into several branches. Although there are two main research fields such that symmetric and public key cryptography, the public and private keys are used in public key cryptography. The same keys are used at both ends to encrypt and decrypt data/information in symmetric key cryptography. It is well known that the substitution box is a standout in symmetric key cryptography. Shahzad et al. investigated the efficient technique for the construction of anS-box by using action of a PSL2,Z. For constructing an S box, the vertices of the coset diagram are considered in a special way. In this way, the generated S box is highly safe and also closely meeting the optimal values of the standard S-box. In [1–6], the construction of substitution boxes based on coset graphs under the action of modular group PSL2,Z has been discussed. In this piece of work, we investigate the structure of coset graphs under the action of modular group H. This work will be more helpful for construction of strong substitution boxes.TheH-circuits of the set upon which the groups act are the equivalence classes of group action. Group H can be written in the form of relations and generators as <x′,y′:x′2=y′4=1>.Assume thatm is nonsquare integer, then Qm=s+tm:s,t∈Q. In 1878, Cayley was the first who introduced the technique of analysis of the groups through graphs. To investigate the action of infinite groups generated by finite elements on the infinite field by the coset diagram was first introduced by Higman in 1978. A number β=s+tm∈Qm is said to be ambiguous number (AN) if β and β¯ have opposite signs. If β=s+tm is not ambiguous, then it is either totally positive or negative. The real quadratic irrational (RQI) numbers of the form a′+m/c′, where a′,a′2−m/c′,c′=1 and c′ is nonzero integer, make the set represented as Q∗m. A RQI number β=a′+m/c′ is known as RQIN if β>0 and −1<β¯<0. In this paper, we will denote the reduced number by r. If there are k reduced numbers, then they are denoted by r1,r2,r3,⋯rk. For β∈Q′′m, in the orbit of βH, the count of RQINs is called the reduced length (RL), which is denoted by βHred. These numbers are very less in Q′′m and play a significant part in the circuit of an orbit. A circuit made of vertices of a square and edges existing in H-orbits of Q′′m, under the Mobius group H in coset diagram. If p10,q11,r12,p20,q21,r22⋯pk0,qk1,rk2 is the type of a circuit, then it makes an element of group h=x′y′p1,x′y′2q1,x′y′3r1,x′y′p2,x′y′2q2,x′y′3r2,⋯x′y′pk,x′y′2qk,x′y′3rk of H. This h fixes some element exists in this circuit.In [7, 8], Mushtaq and Aslam presented that there are only finite number of ambiguous numbers (ANS); in the coset diagram for the orbit of βH, the ambiguous numbers form unique closed path. A cost diagram is introduced in [7, 8] to investigate the action of an infinite group H on the projective line over real quadratic field (RQF). Malik and Zafar [9] investigated the properties of RQI numbers under the action of H. Zafar and Malik [10, 11] investigated the type and ambiguous lengths of the orbit of Q′′p. Farkhanda and Qamar discussed the real quadratic irrational and action of M=<x′,y′:x′2=y′6=1>. Razaq et al. [12, 13] investigated the circuits of length 4 in PSL2,Z, group theoretic construction of highly nonlinear substitution box, and its applications in image encryption. Ali and Malik [14, 15] discussed the classification of PSL2,Z-circuits and investigated the RQIN and types of G-circuits with length four and six. Chen et al. [16] investigate reduced numbers which play an important role in the study of modular group action on the PSL2,Z-subset. For more studies of group action on various field, we recommend reading of [17, 18]. The application of group theory and group action is obvious to encryption, physics, and mechanics to construct models and their structures [5, 19–21]. Mateen et al. [22–27] investigated the structure of power digraphs associated with the congruence xn≡ymodm, the partitioning of a set into two or more disjoint subsets of equal sums, and the symmetry of complete graphs and, moreover, investigated the importance of power digraphs in computer science. Alolaiyan et al. [28] discussed the homomorphic copies in coset graphs for the modular group.The major contributions of this paper are given below.(1)
This paper presents a graphical study of the action of a Mobius groupH on the real quadratic field (RQF)(2)
We discuss the classification ofH-circuits and find the numbers that play vital role in the structure of H-circuits(3)
We investigate the RQINs and the types ofH-circuits with different length(4)
We give the number of reduced numbers and their general form in different orbits for different values ofp under a certain condition on p by using the concept of congruencesTheorem 1. [29].
If<b1,b2,b2,⋯,bk> is symmetric continued fraction (CF) and <b1,b2,b2,⋯,bk>=R+M/s, then M=R2+S2.Theorem 2. [9].
The setℚ′′m=η/s:η∈ℚ∗m,s=1,2 is unchanged under the action of H.Theorem 3. [10].
Letm≡1mod8. Then, ℚ′′m splits into four H-subsets. In particular, m/1H, m/−1H, 1+m/2H, and 1+m/4H are at least four H-orbits of ℚ′′m.Theorem 4. [9].
Letm≡3mod8. Then, ℚ′′m splits into three H-subsets. In particular, m/1H, m/−1H, and 1+m/2H are at least three H-orbits of ℚ′′m.Lemma 5.
Every RQIN inℚ′′m is ambiguous number.Theorem 6. [29].
If<b1,b2,b2,⋯,bk> is symmetric continued fraction and if <b1,b2,b2,⋯,bk>=R+M/S, then M=R2+S2.Theorem 7. [9].
The setℚ′′m=η/s:η∈ℚ∗m,s=1,2 is unchanged under the action of H.Theorem 8. [10].
Letm≡1mod8. Then, ℚ′′m splits into four H-subsets. In particular, m/1H, m/−1H, 1+m/2H, and 1+m/4H are at least four H-orbits of ℚ′′m.Theorem 9. [9].
Letm≡3mod8. Then, ℚ′′m splits into three H-subsets. In particular, m/1H, m/−1H, and 1+m/2H are at least three H-orbits of ℚ′′m.Lemma 10.
Every RQIN inℚ′′m is an ambiguous number.Lemma 11. [14].
β=a′+m′/c′ is an ambiguous number if and only if c′<0 and b′=a′2−m/c′>0 or b′=a′2−m/c′<0, and c′>0.Remark 12. [9].
Letβa′,b′,c′∈Q∗n′ and m∈ℕ. Then,
(1)
x′y′mβ = β+m = y′3x′−mβ.(2)
y′x′mβ = β/1−2mβ = x′y′3−mβ.(3)
hmβ = β1∈βH.Remark 13.
It should be noted here that for a reduced numberβ=a′+m′/c′, we have a′>0,c′>0, and b′<0.


## 2. Properties of Reduced Quadratic Irrational Numbers inQ′′m

This section is devoted to study the behavior of reduced numbers.Lemma 14.
Ifr∈Q′′m is an RQIN, then x′r is an ambiguous number but not RQIN.Proof.
Letr=a′+m/c′ be a reduced quadratic irrational number such that b′<0,a′>0,andc′>0, where b′=a′2−m/c′. Then, by using the Mobius transformation x′r=−1/2r, we have x′r=−a′+m/2b′=a1+m/b1, where a1<0 and b1<0. Since b′<0 by using Remark 12, x′r is not RQIN.Theorem 15.
. Letp≡1 or 5mod8 such that p−1=s2. Then, the circuit of a reduced number r∈p+p/2H has the type 2p−12,p−1)0. Moreover, r¯, −r, and −r¯ each exists on the turning points of the circuit and not reduced.Proof.
r∈p+p/2H, r=p+p/2, y′3x′p−1−1r=−r¯, where p−1−1 is the number of squares inside the circuit. x′y′2x′−r¯=−r which shows that one circuit is lying between the inside and outside boundary of the circuit. x′y′x′p−1−2−r=−r¯, where p−1−1 is the number of squares inside the circuit. y′2r¯=r which implies that one of the squares is lying between the inside and outside boundary of the circuit.Theorem 16.
Forp≡5 or 1mod23 such that −1+p=s2 and r=p+p/2 be a reduced number, then r¯, −r, and −r¯ map onto the nonreduced number under the action of x′.Proof.
Letr=p+p−1/2 and −r=p+p/−2. By using linear fractional transformation x′:β=−1/β and Table 1, where β=a+m/c, x′−r=−p−1+p/c1=a1+m/c1 where a1=−p−1<0. By using Remark 12, it is not a reduced number. Similarly, x′−r is not a reduced number. r¯=−p−1+p/−2 and β=x′−r=−p−1+p/c′; c′=2b>0 where b>0. By Remark 12, hence β is not a RQIN.Theorem 17.
Letr=a′+m/c′∈Q′′m be a RQIN moved to 1/2a′+m/c′∈Q′′m under a Mobius transformation x′. Then,
(1)c′+ma′H∩a′+mc′H=Φ.Proof.
Supposea′+m/c′∈Q′′n be RQIN under Mobius transformation x′ moved to half of their conjugate, i.e., x′a′+m/c′=1/2a′−m/c′ by using Table 1, as m=a′2+c′2. By using Theorem 1, a′+m/c′ and −1/2a′+m/c′ have symmetric periodic part, since, in the form of continued fraction, every RQIN has unique description. In similar fashion, c′+m/a′ and −1/2a′−m/c′ with symmetric periodic parts are identical. By Lemma 5a′+m/c′ and c′+m/a′ are not identical. Hence, we conclude that
(2)c′+ma′H∩a′+mc′H=Φ.Lemma 18.
Letβ=a′+m/c′∈Q′′m which moves to half of their conjugate under the linear fractional transformation x′. Then, Q′′m has at least 2 distinct circuits
(3)c′+ma′Handa′+mc′H.
Example1 reflects Lemma 18.Example 1.
Supposem=13 and 2+m/3∈Q′′13 be reduced quadratic irrational transformed to half of their conjugate under the x′ transformation and 11,20,11,12,11,22,11,12 is the type of 2+m/3. 3+m/2∈Q′′13 be reduced quadratic irrational transformed to half of their conjugate under the x′ transformation and 21,1o,32,10,11,12,30 is the type of 3+m/2. It is easy to see that 2+m/3 and 3+m/2 are not equivalent, so that 2+m/3⋂3+m/2=ϕas shown in figures below.Table 1
Under the action of the groupH. The images of elements of Q∗m [11].
β=a′+mc′a′b′c′x′β−12β−a′−c′22b′y′β−12β+1−a′−c′−c′222a′+b′+c′y′2β−β+12β+1−3a′−2a′−c′2a′+b′+c′4a′+4b′+c′y′3β−2β+12β−a′+2b′4a′+4b′+c′222a′+b′+c′x′y′ββ+1a′+c′2a′+b′+c′cx′y′3ββ2β+1a′+2b′b′4a′+4b′+c′y′x′ββ1−2βa′−2b′b′−4a′+4b′+c′y′2x′β1−2β2−1+β3a′−2a′−c′−4a′+4b′+c′22−2a′+b′+c′y′3x′ββ−1a′−c′−2a′+b′+c′c′Figures1 and 2 reflect Lemma 18.Figure 1
Closed path of2+13/3H.Figure 2
Closed path of3+13/2H.


## 3. Reduced Length of theH-Circuits of Q′′P

The circuit generates an element of the formg=x′y′jk+1mk⋯x′y′j2+1m2x′y′j1+1m1 of H and fixes some vertex of a square on the closed orbit, and thus, the reduced length of closed orbit is the count of RNs in this closed circuit.Example 2.
The circuit of the type12,10,12,11,22,40,22,11,12,10,11,80 represents that the circuit generates an element k=x′y′8x′y′2x′y′x′y′3x′y′2x′y′32x′y′2x′y′32x′y′2x′y′3x′y′x′y′3 of H and fixes the vertex r1=4+19. Suppose r1=4+19.⋯⋯⋯1,x′y′3r1=β1, x′y′β1=3+19/5=r2⋯⋯⋯..2, x′y′3r2=−1+19/9=β2, x′y′2β2=2+19/10=β3, x′y′32β3=−4+19/2=β4, x′y′4β4=4+19/2=r3⋯⋯⋯..3, x′y′32r3=−2+19/10=β5,(x′y′2β5=1+19/9=β6, x′y′3β6=2+19/5=r4⋯..4, x′y′2r4=−4+19=β7, x′y′8β7=4+19=r1. Equations (1), (2), (3), and (4) follow that r1,r2,r3,andr4 are only reduced numbers in the orbit. Thus, the reduced length of this orbit is 4.Now, we investigate the reduced cardinalities ofH-orbits.Theorem 19.
Letp≡1 or 5mod8 such that p−1=s2 and then the circuit of the reduced number p−1+p/2H has the type p−1−12,11,p−1−10,11, and p−1+p/2Hred=1Proof.
In order to prove that it is enough to findk∈H in such a manner kp−1+p/2=p−1+p/2. The proof was followed by the following four steps: y′3x′p−1−1r=y′3x′p−1−1p−1+p/2=−p−1+p/−2=−r¯, x′y′2x′−r¯=x′y′2x′−p−1+p/−2=p−1+p/−2=−r, x′y′x′p−1−2y′−r=x′y′x′p−1−2y′p−1+p/−2=−p−1+p/−2=r¯, and y′2r¯=y′2−p−1+p/−2=p−1+p/2=r. Thus, we obtain y′2x′y′x′p−1−2y′x′y′2x′y′3x′p−1−1p−1+p/2=p−1+p/2. Hence, the circuit of the reduced number p−1+p/2H has the type (p−1−1)2,11,p−1−10,11. Now, we have to prove that p−1+p/2Hred=1. Let r=p−1+p/2 be reduced number. Now, by using Theorem 15 and Theorem 16, the numbers r¯,−r¯, and −r are on the turning point of the circuit and are not reduced numbers; furthermore, when we will apply linear fractional transformation x′ on r¯,−r¯, and −r, then in result, we get no reduced number. So, p−1+p/2is only reduced number in p−1+p/2H. Hence, p−1+p/2Hred1.Example 3.
Take a prime numberp=17 such that 17−1=42 and 17≡1mod8. It is observed from the coset diagram given below that the reduced number 17−1+17/2 is fixed by the word x′y′2y′x′3x′y′2y′3x′317−1+17/2=17−1+17/2; this shows that type of the circuit 17−1+17/2H is 32,11,30,11, and it can be seen from the coset diagram given below; 17−1+17/2 is only reduced number in 17−1+17/2H, and hence, 17−1+17/2Hred=1Figure3 depicted Theorem 19.Example 4.
Take a prime numberp=101 such that p−1=102 and p≡5mod8. It is observed from the coset diagram given below that the reduced number 101−1+101/2 is fixed by the word x′y′2y′x′10x′y′2y′3x′10101−1+101/2=101−1+101/2; this shows that type of 101−1+101/2H of the circuit 101−1+101/2H is 92,11,90,11, and it can be seen from the coset diagram given below that 101−1+101/2 is only reduced number in 101−1+101/2H, and hence, 101−1+101/2Hred=1Figure 3
Closed path of17−1+17/2H.Figure4 reflects Example 4.Lemma 20.
Forp≡5 or 3mod23 such that −2+p=s2, then the orbits of reduced numbers s+pH and s+p/4H have the type s−1/22,11,s−10,11,s−1/22,2so and s+pHred=2=s+p/4Hred.Proof.
To show that it is enough to discoverk∈H in such a manner kr1=r1, where r1=s+p..⋯..1 using Remark 12(1) and (3), we obtain x′y′−2sr1=−2s+r1=−r+p=−(¯r1). Now, x′y′3s−1/2r1=s−1s2−p+s+p/s−12s2+s2−ps−1+1=α and x′y′3−s−1/2−(¯r1=−s−1s2−p+s+p/s−12s+s2−ps−1+1=−(¯α). In Table 1, (x′y′2α=s−1s2−p+1+p/−s2−p=β and x′y′2−1−α¯−s−1s2−p+1+p/−s2−p=−β¯=r2⋯..2. Finally, x′y′−s−1r2=x′y′−s−1−β¯=s−1s2−p+1+p/−s2−p=β. x′y′2sx′y′3s−1/2x′y′2x′y′s−1x′y′2x′y′3s−1/2r1=r1. Hence, s−1/22,11,s−10,11,s−1/22,2so be the type of circuit of reduce number s+pH. Similarly, the type of s+p/4H is same as first one and from equations (1) and (2); hence, s+pHred=2=s+p/4Hred.Example 5.
Take a prime numberp=83 such that p−2=92 and p≡3mod8. It is observed from the coset diagram given below that the reduced number 9+83 is fixed by the word x′y′18x′y′34x′y′2x′y′8x′y′2x′y′349+83=9+83; this shows that type of the circuit 9+83H is 42,11,80,11,42,180, and it can be seen from the coset diagram given below; 9+83 and 8+83/2 are only reduced number in 9+83H, and hence, 9+83Hred=2.Figure 4
Closed path of101−1+101/2H.Figure5 reflects Lemma 20.Lemma 21.
If4p−3 and 1+p=s2, then p+pH and p/−1H circuits have the type so,s1. Moreover p+pHred=1 and p/−1Hred=0.Proof.
Similar proof as of Lemma20.Remark 22.
(i) It is not necessary that every circuit contains reduced number. As we can see in the figure given below, the circuit ofp/−1 contains no reduced number.
Figure6 reflects Remark 22(i).Figure 5
Closed path of9+83H.Figure 6
Closed path of3/−1H.


### 3.1. Detection of Reduced Numbers

In the orbits ofs+pH and s+p/4H of Q′′p, where p≡3 or 5mod8 such that p−2=s2, then
(i)
s+p and s−1+p/2 are only reduced numbers in the circuit of s+pH(ii)
s−2+p/4 and s+p/4 are only reduced numbers in the circuit of r+p/4HRemark 23.
Forp≡3mod23 such that −2+p=s2.
(i)
If a reduced numberr∈p+sH, then its negative conjugate −r¯∈p+sH(ii)
If a reduced numberr∈p+s/4H then, its negative conjugate −r¯∈s+p/4H(4)s+pH∩s+p4H=ϕ.Lemma 24.
Ifp≡7mod23 and 2+p=s2 then, the circuit s−1+pH of the reduced number has the type 11,s−1/2−1o,12,s−1o,12,s−1/2−1o,11,2s−1o, and hence, s−1+pHred=4Proof.
To illustrate this, it is sufficient to findk∈H such that kr1=r1⋯.i, where r1=s−1+p by Remark 12, then x′y′−2s−1r1=x′y′−2s−1s−1+p=−s−1+p)=−r¯1; now, by using Table 1. x′y′2r1=2s2−p−s+p/2s2−p=β1 and x′y′2−1−r¯1=−2s2−p+s+p/2s2−p=−β¯1=r2.⋯⋯⋯ii, again by (1.1), we have x′y′s−3/2r2=x′y′s−3/2−β¯1=s2−ps−1−s+p/2s2−p=r3⋯⋯⋯.iii, and x′y′−s−3/2r2=x′y′−s−3/2−β¯1=−s2−ps−1+s+p/2s2+p=−r¯3, by Table 1, x′y′3r3=s3+sb−3+1+p/s2−p=β2 and x′y′−1−r¯3=s3+sp+3−1+p/s2−p=−β¯2=r4⋯.iv; finally, x′y′s−1r4=2s−1s2−p−1−s+p/s2−p=β3, and β3=r4. Thus, x′y′2s−1x′y′2x′y′s−1/2−1x′y′3x′y′s−1x′y′3x′y′s−1/2−1x′y′2r1=r1, and from equations (i), (ii), (iii), and (iv), we get r1,r2,r3,andr4 which are only 4 reduced numbers in the circuit of s−1+pH.
Hence,s−1+pHred=4.Example 6.
Take a prime numberp=79 such that p+2=92. It is observed from the coset diagram given below that the reduced number 8+79 is fixed by the word x′y′16x′y′2x′y′3x′y′3x′y′8x′y′3x′y′3x′y′28+79=8+79; this shows that type of 8+79H of the circuit is 11,30,12,8o,12,30,11,16o, and also, as can be seen from the coset diagram given below, 8+79, 7+79/4, 8+79/2, and 5+79/4 are only reduced number in 8+79H and hence 8+79Hred=4.Figure7 reflects Lemma 24.Lemma 25.
Ifp≡7mod23 and 2+p=s2 then, the circuit s−1+p/2s−3H of the reduced number has the type 12,s−1/2−1o,11,2s−1o11,s−1/2−1o,12,s−1o,, and hence, s−1+p/2s−3Hred=2.Proof.
Similar proof as of Lemma24.Example 7.
Takep=167 such that p+2=132. It is observed from the coset diagram given below that the reduced number 12+167/23 is fixed by the word x′y′12x′y′3x′y′5x′y′2x′y′24x′y′2x′y′5x′y′312+167/23=12+167/23; this shows that type of the circuit 12+167/23H is 12o,12,50,11,24o,11,50,12, and it can be seen from the coset diagram given below, 12+167/23 and 11+167/23 and are only reduced numbers in 12+167/23H, and hence, 12+167/23Hred=2.Figure 7
Orbit of8+79/1H 520703).Figure8 reflects Lemma 25.Figure 8
Closed path of12+167/23H.


### 3.2. Detection of Reduced Numbers

In the circuits ofs−1+pH and s−1+p/2s−3H of Q′′p where 8p−7 and 2+p=s2, then,
(i)
s−1+p, s−2+p/4, s−1+p/2, and s−4+p/4 are only reduced numbers in the circuit of s−1+pH(ii)
s−2+p/2s−3 and s−1+p/2s−3 are only reduced numbers in the circuit of s−2+p/2s−3HRemark 26.
Forp≡7mod23 such that 2+p=s2.
(1)
Ifr∈s−1+pH then −r¯∈s−1+pH(2)
Ifr∈s−1+p/2s−3H, then −r¯∈s−1+p/2s−3H(3)
s−1+pH⋂s−1+p/2s−3H=ϕ.


## 3.1. Detection of Reduced Numbers

In the orbits ofs+pH and s+p/4H of Q′′p, where p≡3 or 5mod8 such that p−2=s2, then
(i)
s+p and s−1+p/2 are only reduced numbers in the circuit of s+pH(ii)
s−2+p/4 and s+p/4 are only reduced numbers in the circuit of r+p/4HRemark 23.
Forp≡3mod23 such that −2+p=s2.
(i)
If a reduced numberr∈p+sH, then its negative conjugate −r¯∈p+sH(ii)
If a reduced numberr∈p+s/4H then, its negative conjugate −r¯∈s+p/4H(4)s+pH∩s+p4H=ϕ.Lemma 24.
Ifp≡7mod23 and 2+p=s2 then, the circuit s−1+pH of the reduced number has the type 11,s−1/2−1o,12,s−1o,12,s−1/2−1o,11,2s−1o, and hence, s−1+pHred=4Proof.
To illustrate this, it is sufficient to findk∈H such that kr1=r1⋯.i, where r1=s−1+p by Remark 12, then x′y′−2s−1r1=x′y′−2s−1s−1+p=−s−1+p)=−r¯1; now, by using Table 1. x′y′2r1=2s2−p−s+p/2s2−p=β1 and x′y′2−1−r¯1=−2s2−p+s+p/2s2−p=−β¯1=r2.⋯⋯⋯ii, again by (1.1), we have x′y′s−3/2r2=x′y′s−3/2−β¯1=s2−ps−1−s+p/2s2−p=r3⋯⋯⋯.iii, and x′y′−s−3/2r2=x′y′−s−3/2−β¯1=−s2−ps−1+s+p/2s2+p=−r¯3, by Table 1, x′y′3r3=s3+sb−3+1+p/s2−p=β2 and x′y′−1−r¯3=s3+sp+3−1+p/s2−p=−β¯2=r4⋯.iv; finally, x′y′s−1r4=2s−1s2−p−1−s+p/s2−p=β3, and β3=r4. Thus, x′y′2s−1x′y′2x′y′s−1/2−1x′y′3x′y′s−1x′y′3x′y′s−1/2−1x′y′2r1=r1, and from equations (i), (ii), (iii), and (iv), we get r1,r2,r3,andr4 which are only 4 reduced numbers in the circuit of s−1+pH.
Hence,s−1+pHred=4.Example 6.
Take a prime numberp=79 such that p+2=92. It is observed from the coset diagram given below that the reduced number 8+79 is fixed by the word x′y′16x′y′2x′y′3x′y′3x′y′8x′y′3x′y′3x′y′28+79=8+79; this shows that type of 8+79H of the circuit is 11,30,12,8o,12,30,11,16o, and also, as can be seen from the coset diagram given below, 8+79, 7+79/4, 8+79/2, and 5+79/4 are only reduced number in 8+79H and hence 8+79Hred=4.Figure7 reflects Lemma 24.Lemma 25.
Ifp≡7mod23 and 2+p=s2 then, the circuit s−1+p/2s−3H of the reduced number has the type 12,s−1/2−1o,11,2s−1o11,s−1/2−1o,12,s−1o,, and hence, s−1+p/2s−3Hred=2.Proof.
Similar proof as of Lemma24.Example 7.
Takep=167 such that p+2=132. It is observed from the coset diagram given below that the reduced number 12+167/23 is fixed by the word x′y′12x′y′3x′y′5x′y′2x′y′24x′y′2x′y′5x′y′312+167/23=12+167/23; this shows that type of the circuit 12+167/23H is 12o,12,50,11,24o,11,50,12, and it can be seen from the coset diagram given below, 12+167/23 and 11+167/23 and are only reduced numbers in 12+167/23H, and hence, 12+167/23Hred=2.Figure 7
Orbit of8+79/1H 520703).Figure8 reflects Lemma 25.Figure 8
Closed path of12+167/23H.


## 3.2. Detection of Reduced Numbers

In the circuits ofs−1+pH and s−1+p/2s−3H of Q′′p where 8p−7 and 2+p=s2, then,
(i)
s−1+p, s−2+p/4, s−1+p/2, and s−4+p/4 are only reduced numbers in the circuit of s−1+pH(ii)
s−2+p/2s−3 and s−1+p/2s−3 are only reduced numbers in the circuit of s−2+p/2s−3HRemark 26.
Forp≡7mod23 such that 2+p=s2.
(1)
Ifr∈s−1+pH then −r¯∈s−1+pH(2)
Ifr∈s−1+p/2s−3H, then −r¯∈s−1+p/2s−3H(3)
s−1+pH⋂s−1+p/2s−3H=ϕ.


## 4. Conclusion

The idea of types ofH-circuits in H-orbits of RQF by Mobius group, which is given in this paper, is new and original. We have presented type of H-circuits with different length in H-orbits βH, where β is RQIN and H be Mobius group. We have investigated properties of RQINs and classified H-orbits of different length. Furthermore, we proposed reduced length and general form of reduced numbers in different orbits. This work can be extended for the Mobius group M=<x′,y′:x′2=y′6=1> and G=<x′,y′:x′2=y′3=1> as well as examined the M-circuits in M-orbits and the G-circuits in G-orbits. Moreover, the reduced length and general form of reduced numbers for different orbits can be discussed.



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