Abstract
The purpose of this paper is to generalize the results of Quillen [lo] about the cohomology of (the classifying space of) the general linear groups over a finite field to the orthogonal case. In this paper we will restrict ourselves to the study of the cohomology with mod 2 coefficients of (the classifying space of) the orthogonal groups. While in [5] we gave a computation of the cohomology ring of the split orthogonal groups in the case of the ground field k having 4 = 4m + 1 elements, in this paper we also cover the p =
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