Symmetries of Icosahedral group and classification of G-circuits of length six

Objectives: To determine the exact number of equivalence classes of G-circuits of length q ≥ 2.Methods/Statistical Analysis: To classify G-orbits of Q( √ m)/Q containing G-circuits of length 6. Findings: The equivalence classes of G-circuits of length 6 is ten in number and determine the exact number of G-orbits and structure of G-orbits corresponding to each of ten equivalence classes of Gcircuits. Furthermore, we describe some generalized G-circuits of length 2t corresponding to each of these ten equivalence classes and the structure of theseG-circuitswith conditions on t.Applications/Improvements:Weemploy Symmetries of Icosahedral group to explore cyclically equivalence classes of Gcircuits and similar G-circuits of length 6 corresponding to each of these ten equivalence classes. This study helps us in classifying reduced numbers lying in PSL(2, Z)-orbits. These results are verified by some suitable example.


Introduction
If n= d 2 contain Q * ( √ n) and Q * red ( √ n) as G-subset and subset respectively.
In (2) numbers of distinct circuits of length 4 are obtained by using fixed number of T triangles. In (3) numbers of distinct homomorphic images are obtained by contracting all the pairs of vertices. In (4) authors discussed action of two Hecke groups. One of them is the modular group H (λ 3 ) and other is H (λ 4 ) . In (5) authors determine the transitive H-set of Q( √ m)\Q by using structure of circuits.
In (6) , it was proved that the ambiguous numbers in the orbit γ G , γ ∈ Q * ( √ n) makes a G-circuit or simply circuit. Thus it becomes interesting to classify circuit. In (7) authors determine that exact number of equivalence classes of G-circuits of length 4 is 4 in number and classify G-orbits of Q( √ m)/Q containing G-circuits of length 4. Thus it becomes interesting to find to determine exact number of equivalence classes of G-circuits of length q ≥ 2. ] is a circuit if it has l ′ 2i−1 triangles has two edges outside the circuit, l ′ 2i triangles has two edges inside the circuit where 1 ≤ i ≤ t. Throughout this paper G-circuit (resp.G-orbit) will be simply denoted by circuit (resp.orbit). The concept of circuit grew out of the study of Group action on Q( √ m) ∪ {∞} and the study of G-subsets Q * ( √ n) . In this paper, we define what a circuit of specific length is and we classify non equivalent 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 are introduced to explore transitive G-subsets (called orbits) of Q * ( √ n). The definition of an equivalent circuit that is now standard was a long time in being formulated.

Two circuits
where θ ∈ S 2t . That is the circuits are equivalent to if and only if they are obtained just by permuting the entries l It is easy to see that being equivalent of circuits is an equivalence relation. Thus, a property which 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 [l 1 , l 2 , l 3 , l 4 , l 5 , l 6 , . . ., l 2t−1 , l 2t ] are said to be cyclically equivalent if and only if the circuit . 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 circuits and [l 1 , l 2 , l 3 , l 4 , l 5 , l 6 , . . ., l 2t−1 , l 2t ] are said to be similar if and only if where θ ∈ C t = ⟨(1 3 5 . . . 2t − 1 ) (2 4 6 . . . 2t)⟩ . Notation for cyclically equivalent is" ∼ s ". It is interesting to note that the orbit containing circuit 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.
Classification of non-equivalent circuits and cyclically equivalent circuits play a significant role to obtain the orbits of Q( √ m)\Q because with this task of finding orbits is simplified. In (8) . So it becomes interesting to explore transitive G-subsets called orbits.

Materials and Methods
The following results of (8)(9)(10)(11) are used in the sequel.
These are exactly 60 cyclically equivalent classes namely Following is the generalization of the above circuit T 1 whose length is "2t" and its proof can be obtained by applying induction on "t".
The following Table 2 is of considerable utility because it provides us with exact number of circuits of length 6 and hence the number of G-orbits of Q * ( √ n) . https://www.indjst.org/ Cecl6: Classes of equivalent circuits of length 6; Ncec: Number of classes of cyclically equivalent circuits; Rdo: Regarding distinct orbits Corollary 3.6: There are 6 Equivalent circuits of length 6 in which 5 numbers are alike and one number is different. Proof: By Lemma 2.4, the number of equivalent circuits of length 6 in which 5 numbers are alike is 6! 5!1! = 6. These 6 circuits are . Clearly all these types are cyclically equivalent as well. Moreover by Lemma In this situation there is only one cyclically equivalent class namely E c Following is the generalization of the above circuit T 10 whose length is "2t" and its proof can be obtained by applying induction on "t".
are equivalent circuits.
In the above circuits are cyclically equivalent. Moreover by Lemma 2.1, Structure of G-orbit of remaining circuits can be found from Table 3. Table 3.

E c T 9
Structure of orbits Nce Nce: Number of classes of cyclically equivalent circuits; Following is the generalizations of the above circuit T 9 whose length is "2t" and its proof can be obtained by applying induction on "t".

Note 3.7.2: If
is circuit contained in the orbit of γ G where l ′ 1 repeats "t−1" times and 2t ≡ 2(mod 4) are equivalent circuits.
In the above circuits are cyclically equivalent.
Moreover by Lemma 2.1, is the circuit contained in Moreover, are not possible by Lemma 2.3. Table 4 summarizes all these information's Table 4. summarizes all these information's

Nce: Number of classes of cyclically equivalent circuits;
Following is the generalizations of the above circuit T 8 whose length is "2t" and its proof can be obtained by applying induction on "t". Note 3.7.1: If is circuit contained in the orbit of γ G where l ′ 1 repeats "t" times l ′ 2 repeats "t" times and "t" is even integer then γ G = ( − γ ) G .
is circuit contained in the orbit of γ G where l ′ 1 repeats "t" times l ] , Structure of G-orbit of these 30 circuits can be found from Table 5. Nce so on. Structure of G-orbit of these 60 circuits can be found from Table 6. Table 6. Structure of G-orbit of these 60 circuits

Ncc: Number of classes of cyclically equivalent circuits;
Following is the generalization of the circuit T 6 whose length is "2t" and its proof can be obtained by applying induction on "t".    Figure 4 shows the location of reduced numbers and describes that [6,5,4,3

Conclusion
The idea to classify G-circuits of G-orbits on quadratic field by modular group, which is given in this paper, is new and original. We have classify G-circuits into the distinct equivalence classes of equivalent circuits by using partition function and they are precisely p (q) − 1 in number when q ≥ 2 , Particularly for circuits of length 6 we have p (6) − 1 = 10 equivalence classes of equivalent circuits, i.e E 6 = ∪ 10 i=1 E T i . We further classify equivalence classes of equivalent circuits into cyclically equivalence classes and this study help us to determine exact number of G-orbits corresponding to each cyclically equivalence class. Furthermore, we describe some generalized G-circuits of length 2t corresponding to each of these ten equivalence classes and the structure of these G-circuits with conditions on t. All results are verified by suitable example.