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Graham Higman was the first who studied the transitive actions of the extended modular group PGL2, Z over PLFq=Fq∪∞ graphically and named it as coset diagram. In these sorts of graphs, a closed path of edges and triangles is known as a circuit. Coset diagrams evolve through the joining of these circuits. In a coset diagram, a circuit is termed as a length-l circuit if its one vertex is fixed by x1x2π1x1x2−1π2x1x2π3,…,x1x2−1πl∈PSL2, Z, and it is denoted by π1,π2,π3,…,πl. In this study, we shall formulate combinatorial sequences and find the number of distinct equivalence classes of a length-6 circuit π1,π2,π3,π4,π5,π6 for a fixed number of triangle Δ of class Π.

In this paper, we investigate and explore the properties of quasi-topological loops with respect to irresoluteness. Moreover, we construct an example of a quasi-irresolute topological inverse property-loop by using a zero-dimensional additive metrizable perfect topological inverse property-loop L*with relative topology τL′.

This work deals with the well-known group-theoretic graphs called coset graphs for the modular group G and its applications. The group action of G on real quadratic fields forms infinite coset graphs. These graphs are made up of closed paths. When M acts on the finite field Zp, the coset graph appears through the contraction of the vertices of these infinite graphs. Thus, finite coset graphs are composed of homomorphic copies of closed paths in infinite coset graphs. In this work, we have presented a comprehensive overview of the formation of homomorphic copies.