Orthogonal Groups over Local Rings

In an earlier paper [S] we have determined the structure of the linear groups over a local ring. In this note we continue the study of the classical groups over a local ring with the investigation of the orthogonal groups. Our main result (cf. Theorem 6 below) is a complete description of the invariant subgroups of an orthogonal group of noncompact type (i.e., of index ^ 1) over a local ring L of characteristic 5^2. Certain low dimensional cases being excluded, the result reads as follows: The set of invariant subgroups splits into disjoint classes Q(J) which are in one-to-one correspondence with the ideals / of L. Each class has a greatest and a smallest element, with respect to the inclusion, which are represented by certain congruence subgroups modulo / , and every group between the greatest and the smallest element of e (J) belongs to e (J) . A similar result does hold for the set of invariant subgroups of the commutator group of the orthogonal group; in this case, the structure of the classes Q(J) is very simple since each class contains at most two elements, and then the smaller element has index 2 in the greater one. Hence, it turns out that the structure of the orthogonal groups under consideration is of the same type as the structure of the linear groups over a local ring, cf. [S]: Here too the invariant subgroups split into classes which correspond to the ideals of the local ring, and each class has a greatest and a smallest element, represented by certain congruence subgroups, and each group in between belongs to the class. One may expect, therefore, that this is the typical arrangement of the invariant subgroups of a classical group over a local ring. If the local ring L possesses no ideals apart from L and 0, i.e., if L is a field, then we get the results of Dieudonné [3 ; 4] on the structure of the orthogonal groups over a field.