PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 57, Number 2, June 1976 DEDEKIND SUMS AND NONCONGRUENCE SUBGROUPS OF THE HECKE GROUPS G(V2) AND G (a/3) L. ALAYNE PARSON Abstract. An example is given of a character xr on a subgroup of G(y/2) or G{\ß) such that the kernel of xr is of finite index in G(\/2) or G(-\ß) but is not a congruence subgroup. In [7] K. Wohlfahrt exhibited a class of subgroups of the modular group which were not congruence subgroups although they were of finite index. These subgroups were the kernels of certain characters on r0(«). In this note we generalize K. Wohlfahrt's construction to the Hecke groups G(\/2) and G(\/3) in order to produce noncongruence subgroups of finite index in addition to those in [5] and to give examples of characters which are not congruence characters. Since G{\f2) and G(a/3) are the only Hecke groups commensurable with the modular group [3], K. Wohlfahrt's method cannot be extended to other Hecke groups. For notational convenience let m stand for 2 or 3. Then G{\fm) is the group of 2 X 2 matrices generated by (¿7) - '-(."o') It is well known [1], [8] that G(\/m) consists of the entirety of all matrices of the following two types: (,— j ), a, b, c, d G Z, ad - mbc = 1, cym a ) and (,/—), a, b, c, d G Z, mad —be = 1. c d\/m ) For n a positive integer, the principal congruence subgroup of level n is defined by T(n) = {M G G(y/m): M ■ ±/ (mod «)} Received by the editors July 22, 1975. AMS (MOS) subject classifications (1970). Primary 20H05. © American Ma:hematical Society 1976 194 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use