In this study, we consider the normal subgroups of H'(lq), where H(lq) denotes the Hecke groups. After recalling some results from [2], particularly on the group structure and on the relations with the power subgroups of H(lq), the even subgroup He(lq) of H(lq) is discussed. It is shown that H'(lq) is a normal subgroup of He(lq) with index q. For this reason each subgroup of H'(lq) consists of only even elements. H''(lq) is also considered and it is concluded that it is the normal subgroup of H'(lq) generated by all commutators of the elements of H'(lq). Using the Kurosh subgroup theorem, the group structure of normal subgroups of H(lq) can be found to be free groups. Their ranks are given in terms of the index.

In this study, we consider the normal subgroups of H'(lq), where H(lq) denotes the Hecke groups. After recalling some results from [2], particularly on the group structure and on the relations with the power subgroups of H(lq), the even subgroup He(lq) of H(lq) is discussed. It is shown that H'(lq) is a normal subgroup of He(lq) with index q. For this reason each subgroup of H'(lq) consists of only even elements. H''(lq) is also considered and it is concluded that it is the normal subgroup of H'(lq) generated by all commutators of the elements of H'(lq). Using the Kurosh subgroup theorem, the group structure of normal subgroups of H(lq) can be found to be free groups. Their ranks are given in terms of the index.