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The Picard group r is PSL2{Z(ij), the group of linear fractional transformations with Gaussian integer coefficients. We examine the structure of the normal subgroups of r. In particular we give a complete classification of the normal subgroups for indices less than 60 and show that beyond this there are large gaps in the possible indices. This classification depends on the struc- ture of the derived series. Finally we give examples of normal noncongruence subgroups. 1. Introduction. The Picard group r is PSL2 (Z(iJ), the group of linear frac- tional transformations z' = (az+b)j(ez+d) with ad-be = ±1 and a, b, e, d Gaussian integers. r has been extensively studied both as an abstract group and in automor- phic function theory (1, 3, 8). In both its general structure and in the structure of its principal congruence subgroups, r has been shown to be similar to the modular group M = PSL2 (Z) (3,4). Recently Brunner, Lee, Frame and Wielenberg (I) de- veloped an effective procedure for classifying the torsion-free subgroups of r. This classification was caried out for small indices using a computer search. Further Fine (5) has shown that for a nonfree Fuchsian group to be embedded in r, it must have a special intersection property with the modular group. In this paper we examine properties of the normal subgroup and congruence subgroup lattice in r. In particular we give a complete classification of the normal subgroups for indices less than 60. Further if d( n) represents the number of normal subgroups of index n in r, we show that d(n) = 0 for a wide collection of n's. Finally we show that a theorem of Wohlfahrt's concerning congruence subgroups in the modular group carries over with only minor modification to r. This is utilized to present examples of noncongruence subgroups. 2. The derived series of r. Our techniques rely heavily on the structure of the derived series. We first present some notation, terminology and necessary results. For future reference we identify the following transformations:

The normal subgroup structure of the Picard group