Coset diagrams for the action of PSL(2,Z)\documentclass[12pt]{minimal}
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\begin{document}$$ PSL (2,\mathbb {Z})$$\end{document} on real quadratic irrational numbers are infinite graphs. These graphs are composed of circuits. When modular group acts on projective line over the finite field Fq\documentclass[12pt]{minimal}
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\begin{document}$$ F_{q}$$\end{document}, denoted by PL(Fq)\documentclass[12pt]{minimal}
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\begin{document}$$ PL ( F_{q}) $$\end{document}, vertices of the circuits in these infinite graphs are contracted and ultimately a finite coset diagram emerges. Thus the coset diagrams for PL(Fq)\documentclass[12pt]{minimal}
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\begin{document}$$ PL ( F_{q}) $$\end{document} is composed of homomorphic images of the circuits in infinite coset diagrams. In this paper, we consider a circuit in which one vertex is fixed by (xy)m1(xy-1)m2\documentclass[12pt]{minimal}
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\begin{document}$$( xy) ^{m_{1}}( xy^{-1}) ^{m_{2}}$$\end{document}, that is, (m1,m2)\documentclass[12pt]{minimal}
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\begin{document}$$( m_{1},m_{2}) $$\end{document}. Let α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} be the homomorphic image of (m1,m2)\documentclass[12pt]{minimal}
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\begin{document}$$( m_{1},m_{2}) $$\end{document} obtained by contracting a pair of vertices v, u of (m1,m2)\documentclass[12pt]{minimal}
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\begin{document}$$ ( m_{1},m_{2}) $$\end{document}. If we change the pair of vertices and contract them, it is not necessary that we get a homomorphic image different from α\documentclass[12pt]{minimal}
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\begin{document}$$ \alpha $$\end{document}. In this paper, we answer the question: how many distinct homomorphic images are obtained, if we contract all the pairs of vertices of (m1,m2)?\documentclass[12pt]{minimal}
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\begin{document}$$( m_{1},m_{2}) ?$$\end{document} We also mention those pairs of vertices, which are ‘important’. There is no need to contract the pairs, which are not mentioned as ‘important’. Because, if we contract those, we obtain a homomorphic image, which we have already obtained by contracting ‘important’ pairs.
