In this paper, we look at how coset diagrams have been used in the study of finitely presented groups. We look at a method in which coset diagrams have been applied to study quotients, subgroups and structure of finitely presented groups. Then we use that method to discuss the ways how to acquire the coset diagrams for the generalized triangle groups  We also give some observations about this case.

linear-fractional transformations, non-degenerate homomorphisms, conjugacy classes, parameterization, finite fields.