Normal subgroups and elements of H' ($lambda$)

Abstract

In this study, we consider the normal subgroups of H'($lambda_q$), where H($lambda_q$) 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($lambda_q$), the even subgroup $H_e(lambda$) of H($lambda_q$) is discussed. It is shown that H'($lambda_q$) is a normal subgroup of $H_e(lambda$) with index q. For this reason each subgroup of H'($lambda_q$) consists of only even elements. H''($lambda_q$) is also considered and it is concluded that it is the normal subgroup of H'($lambda_q$) generated by all commutators of the elements of H'($lambda_q$). Using the Kurosh subgroup theorem, the group structure of normal subgroups of H($lambda_q$) can be found to be free groups. Their ranks are given in terms of the index.