## Subgroups of some Fuchsian groups defined by two linear congruences

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- by Omer Yayenie
- Conform. Geom. Dyn.
**11**(2007), 271-287 - DOI: https://doi.org/10.1090/S1088-4173-07-00172-5
- Published electronically: December 18, 2007
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## Abstract:

In this article we define a new family of subgroups of Fuchsian groups $\mathcal {H}(\sqrt {m})$, for a squarefree positive integer $m$, and calculate their index in $\mathcal {H}(\sqrt {m})$ and their parabolic class number. Moreover, we will show that the index of these subgroups is closely related to the solvability of a quadratic congruence $x^2\equiv m(\textrm {mod }n)$ and the number of inequivalent solutions of a quadratic congruence $x^2\equiv 1(\textrm {mod }n)$. Finally, we will show that the results obtained by Yilmaz and Keskin [Acta Math. Sin**25**(2005), 215–222] are immediate corollaries of one of the main theorems of this article.

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## Bibliographic Information

**Omer Yayenie**- Affiliation: Department of Mathematics and Statistics, Murray State University, Murray, Kentucky 42071
- Email: omer.yayenie@murraystate.edu
- Received by editor(s): March 26, 2007
- Published electronically: December 18, 2007
- © Copyright 2007 American Mathematical Society
- Journal: Conform. Geom. Dyn.
**11**(2007), 271-287 - MSC (2000): Primary 11F06, 19B37; Secondary 20H05, 20H10
- DOI: https://doi.org/10.1090/S1088-4173-07-00172-5
- MathSciNet review: 2365641