Dihedral group and classification of G-circuits of length 10

Abstract

Abstract In this paper, we classify G-circuits of length 10 with the help of the location of the reduced numbers lying on G-circuit. The reduced numbers play an important role in the study of modular group action on PSL(2,Z)$PSL(2,\\mathbb{Z})$-subset of Q(m)\\Q$Q(\\sqrt{m}){\\backslash}Q$. For this purpose, we define new notions of equivalent, cyclically equivalent, and similar G-circuits in PSL(2,Z)$PSL(2,\\mathbb{Z})$-orbits of real quadratic fields. In particular, we classify PSL(2,Z)$PSL(2,\\mathbb{Z})$-orbits of Q(m)\\Q$Q(\\sqrt{m}){\\backslash}Q$=⋃k∈NQ*k2m$={\\bigcup }_{k\\in N}{Q}^{{\\ast}}\\left(\\sqrt{{k}^{2}m}\\right)$ containing G-circuits of length 10 and determine that the number of equivalence classes of G-circuits of length 10 is 41 in number. We also use dihedral group to explore cyclically equivalence classes of circuits and use cyclic group to explore similar G-circuits of length 10 corresponding to 10 of these circuits. By using cyclically equivalent classes of circuits and similar circuits, we obtain the exact number of G-orbits and the structure of G-circuits corresponding to cyclically equivalent classes. This study also helps us in classifying the reduced numbers lying in the PSL(2,Z)$PSL(2,\\mathbb{Z})$-orbits.