Abstract
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 Z(p), 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.