Symplectic groups over principal ideal domains

Abstract

Dickson and Dieudonne have shown in [ 1, 51 that the projective symplectic groups over a field contains more than three elements is simple. Attempts at integral analogs of this theorem have been quite successful. In 1963, Klingenberg [6] showed that the symplectic group of a unimodular alternating module over a nondyadic local domain has the congruence subgroup property, i.e., every normal subgroup is a congruence subgroup in the usual sense mod the ideals of the underlying domain. Klingenberg’s work has been further extended in [3, 4, 71 by re axing 1 the restrictions on the alternating forms and on the underlying domain. In 1967, Bass, Milnor, and Serre [2] effected a significant generalization to the integers of a global field. By using high power tools in class field theory, they were able to show that the symplectic group of a unimodular module with rank 24 has the congruence subgroup property. In this paper, we are going to generalize the above result to a nondyadic principal ideal domain and by dropping the assumption of unimodularity we will provide a more constructive method to classify all the normal subgroups of a symplectic group. We are able to show that a subgroup of a symplectic group is a normal subgroup if and only if it is a congruence subgroup in the sense of tableau which is a matrix of ideals of the underlying domain, i.e., the symplectic group has general congruence subgroup property. In addition to the main theorem we also include generation theorems of congruence subgroups and relations among congruence subgroups, and we also determine the structure of the factor groups of two congruence subgroups of the same order.