The congruence subgroups of Hecke group H5 and the normalizer of H5 0((2)alpha I')' in H5

SUMMARY The Congruence Subgroups of Hecke Group H5 and The Normalizer of H^((2)ar)inH5 Almost all of tihe thesis is devoted to a partly proof of a conjeture in [3] that when 1= (2)aI', where (2, I') = 1, is an ideal of Z[X,5] then the Normalizer of Hj|( (2)aI')in H5 is H^((2)a'l'), where a'=a -min ''M Here, a proof is given by taking I' as a prime ideal. In chapter 1, the preliminary definitions and results we require for the subsequent work are given. In chapter 2, an equivalence relation ~ on G : = GL( 2, Z[A,5] ) is defined. Then we get the set G /~ of equivalence classes. And using this set, the sets of equivalence classes H5/~ and Hq(2)/~ for the Hecke group H5 and the congruence subgroups Hq(2) are determined. Consequently, the normalizer of the congruence subgroups Hq( (2)aI')inthe Hecke group H5 is obtained as the Ho((2)a I'), where a'=a -min is a prime idealin Z[XS] and (2, 1')=l (or I'*(2)). [it,andl' Key Words : Hecke Group, Ideal, Congruence Subgroup, Equivalence Relation, Equivalence Classes.