**Thesis Type:** Doctorate

**Institution Of The Thesis:** Uludağ Üniversitesi, Turkey

**Approval Date:** 2013

**Thesis Language:** Turkish

**Student:** BİRSEN ÖZGÜR

**Supervisor: **İSMAİL NACİ CANGÜL

The Hecke group H(λq) is the discrete subgroup of PSL(2,R) which is generated by two linear fractional transformations defined as R(z):= -1/z and T(z):=z+λq for qN, q≥3, q=2cos π/q. It is an open problem to determine the congruence and principal congruence subgroups of the Hecke groups. For the modular groups case, all these subgroups are determined. In Cangul 1993, Cangul determined the congruence and principal congruence subgroups of prime level of the Hecke groups. To find those subgroups having non-prime level, it is necessary to reduce the values of λq in these modules. In this thesis, some calculations related to the minimal polynomials Pq* of algebraic number λq, when this number cannot be properly reduced have been done by means of MAPLE and the extended lists of Pq* have been given for 3≤q≤300. In additon to these proofs with some corollaries concerning the roots of minimal polynomials of algebraic number ≤q in modulo prime have been obtained for various values of q. These corollaries have been very useful in investigation of congruence subgroups of Hecke groups which is both main subject of this thesis and an important open problem in discrete group theory. Also these corollaries will be a guide for the future works. The attention shown to the lists of polynomials that we put to the web address http://www.scribd.com/documents clearly shows that these polynomials are going to be useful to many researchers in the future. In this thesis, some minimal polynomials apart from Pq* of some algebraic numbers which are useful in several studies related to Hecke groups have also been calculated.