The geometry of orthogonal groups over finite fields

forms over finite fields. The emphasis is placed on geometric and combinatorial objects, rather than the orthogonal group itself. Our goal is to introduce dual polar spaces as distance-transitive graphs in a self contained way. Prerequisites are linear algebra, and finite fields. In the later part of the lecture, familiarity with counting the number of subspaces of a vector space over a finite field is helpful. This lecture note is not intended as a full account of dual polar spaces. It merely treats those of type Dn(q), Bn(q) and 2Dn(q). One can treat other types, namely, those coming from symplectic groups and unitary groups, in a uniform manner, but I decided to restrict our attention to the above three types in order to save time. Once the reader finishes this note, he/she should be able to learn the other cases with ease. A motivation of writing this note, as well as giving the lecture, is to make the reader get acquainted with nontrivial examples of distance-transitive graphs. I consider Ham-ming graphs and Johnson graphs trivial, as one can establish their distance-transitivity without any special knowledge. The book by Brouwer–Cohen–Neumaier [2] seems to