Number of Distinct Fragments in Coset Diagrams for PSL (2,Z)

Coset diagrams [1, 2] are used to demonstrate the graphical representation of the action of the extended modular group PGL(2, Z) over PL(Fq). Fq U {infinity}. In these sorts of graphs, a closed path of edges and triangles is known as a circuit, and a fragment is emerged by the connection of two or more circuits. The coset diagram evolves through the joining of these fragments. If one vertex of the circuit is fixed by(ax)(p1)(ax(-1))(p2)(ax)(p 3) center dot center dot center dot (ax(-1))(pk) epsilon PSL(2, Z), then this circuit is termed to be a length - k circuit, denoted by (p(1), p(2), p(3), center dot center dot center dot,p(k)). In this study, we consider two circuits of length - 6 as Omega(1). (,alpha(1),alpha(2),alpha(3),alpha(4),alpha(5),alpha(6)) and Omega (2) (beta(1), beta(2), beta(3), beta(4), beta(5), beta(6)) with the vertical axis of symmetry that is, alpha(2) =alpha(6),alpha(3=) alpha(5) and beta(2) =beta(6), beta(3)= beta(5). It is supposed that Omega is a fragment formed by joining Omega(1) and Omega (2) at a certain point. 'e condition for existence of a fragment is given in [3] in the form of a polynomial in Z[z]. If we change the pair of vertices and connect them, then the resulting fragment and the fragment Omega may coincide. In this article, we find the total number of distinct fragments by joining all the vertices of Omega(1) with the vertices of Omega(2) provided the condition -beta(4)< - beta(3)< - beta(2)< -beta(1) <, alpha(4) <,alpha(3)<, alpha(2) <,alpha(1).