Group theory research is traditional for many years, as its vast applications can be represented by group theory in a variety of fields such as physics, chemistry, computer science, and even puzzles like Rubik's Cube. In the past few decades, the idea of visualizing abstract theory using geometric figures and diagrams was one of the most popular ideas in mathematical books. In this article, we focus on the action of the modular group on

If ^{1}. This idea is then applied in the action of the modular group with the concept of coset diagrams. We have a large number of existing studies in the broader literature that have examined this in^{2, 3, 4, 5.} In^{6, 7}, authors have proved the importance of reduced numbers in the literature of circuits of length four and six and related coset diagrams of the action of the modular group. The idea of contraction of vertices was given in ^{8}, which helped to understand more deeply about these circuits. The Modular group is not the only group used for this group-theoretic action. Bianchi group is also used to study the evolution of ambiguous numbers in^{9}. The circuits of length four and their properties are studied in detail in^{11.} Although coset diagrams are particular kinds of graphical figures, different from graph-theoretical aspects, authors in ^{12, 13 }have discussed the graph-theoretic approach towards this particular area of research. Most of the theories of action of the modular group are focused on explaining different classifications of the elements of ^{15, 16 }the authors have discussed all the G-subsets and G-orbits for the action of modular group on

Let

G-midway divides the circuit into two segments, and these two segments have two different symmetries or patterns of circuit construction depending upon the type of circuit. Before moving on to our main result, here are some essential Lemmas which help to understand the inevitable main results.

Let

Where

Lemma 2.6 [14]. The number of classes

In this section, first, we find all the equivalent circuits of length eight, then all the cyclically equivalent circuits corresponding to each circuit are calculated using Lemma 2.6. We have also given a circuit generating technique using reduced positions and G-midway. Further, all the circuits of length eight are classified according to their orbits.

We denote all 21 equivalent circuits by

Ti for i= | Equivalent circuit | Generating Set | Integer Partition |
---|---|---|---|

1 | l1,l1,l1,l1,l1,l1,l1,l2 | X2 | 7,1 |

2 | l1,l1,l1,l1,l1,l1,l2,l2 | X2 | 6,2 |

3 | l1,l1,l1,l1,l1,l2,l2,l2 | X2 | 5,3 |

4 | l1,l1,l1,l1,l2,l2,l2,l2 | X2 | 4,4 |

5 | l1,l1,l1,l1,l1,l1,l2,l3 | X3 | 6,1,1 |

6 | l1,l1,l1,l1,l1,l2,l2,l3 | X3 | 5,2,1 |

7 | l1,l1,l1,l1,l2,l2,l2,l3 | X3 | 4,3,1 |

8 | l1,l1,l1,l1,l2,l2,l3,l3 | X3 | 4,2,2 |

9 | l1,l1,l1,l2,l2,l2,l3,l3 | X3 | 3,3,2 |

10 | l1,l1,l1,l1,l1,l2,l3,l4 | X4 | 5,1,1,1 |

11 | l1,l1,l1,l1,l2,l2,l3,l4 | X4 | 4,2,1,1 |

12 | l1,l1,l1,l2,l2,l2,l3,l4 | X4 | 3,3,1,1 |

13 | l1,l1,l1,l2,l2,l3,l3,l4 | X4 | 3,2,2,1 |

14 | l1,l1,l2,l2,l3,l3,l4,l4 | X4 | 2,2,2,2 |

15 | l1,l1,l1,l1,l2,l3,l4,l5 | X5 | 4,1,1,1,1 |

16 | l1,l1,l1,l2,l2,l3,l4,l5 | X5 | 3,2,1,1,1 |

17 | l1,l1,l2,l2,l3,l3,l4,l5 | X5 | 2,2,2,1,1 |

18 | l1,l1,l1,l2,l3,l4,l5,l6 | X6 | 3,1,1,1,1,1 |

19 | l1,l1,l2,l2,l3,l4,l5,l6 | X6 | 2,2,1,1,1,1 |

20 | l1,l1,l2,l3,l4,l5,l6,l7 | X7 | 2,1,1,1,1,1,1 |

21 | l1,l2,l3,l4,l5,l6,l7,l8 | X8 | 1,1,1,1,1,1,1,1 |

We now discuss each circuit and equivalent classes

Now we establish a circuit generating technique for

This circuit is ultimately shown in

This idea is generalized by taking length 2m which is generated by the set

This complete circuit, along with reduced positions, is illustrated in [

It is not possible here to mention all of these 2520 cyclically equivalent classes; however, few are;

Now we discuss

Moreover,

above is the complete cyclically equivalent class of circuits of length eight.

It is important to note that the circuit

Reduced positions describe the complete circuit in all aspects. Defining all the reduced positions give complete information about the structure of the orbit. In our case, the first reduced position ^{th} and 9^{th} reduced position. The following pattern shows all the reduced positions of the first half-section before the G-midway.

The reduced positions for the second half of the circuit after G-midway follow the following pattern.

The following results can be drawn straightforward from the above discussion.

Put

is true for other equivalent circuits. Moreover, that equivalent circuit also has the property

[

Now we give a general result for the cyclically equivalent circuits of length eight.

[

Equivalence Class ETi | Number of Cyclically Equivalent Circuits ETic |
---|---|

ET1 | 8!7!=8 |

ET2 | 8!6!2!=28 |

ET3 | 8!5!3!=56 |

ET4 | 8!4!4!=70 |

ET5 | 8!6!=56 |

ET6 | 8!5!2!=168 |

ET7 | 8!4!3!=280 |

ET8 | 8!4!2!2!=420 |

ET9 | 8!3!3!2!=560 |

ET10 | 8!5!=336 |

ET11 | 8!4!2!=840 |

ET12 | 8!3!3!=1120 |

ET13 | 8!3!2!2!=1680 |

ET14 | 8!2!2!2!2!=2520 |

ET15 | 8!4!=1680 |

ET16 | 8!3!2!=3360 |

ET17 | 8!2!2!2!=5040 |

ET18 | 8!3!=6720 |

ET19 | 8!2!2!=10080 |

ET20 | 8!2!=20160 |

ET21 | 8!1!=40320 |

We have classified all the circuits of length eight into 21 equivalent circuits