On the Structure of Orthogonal Groups
Tsuneo Tamagawa · American Journal of Mathematics · 1958
This theorem was proved by D. E. Dickson [3] in the case of finite ground field, then by J. Dieudonne [4] in the general case. In this paper we will give another proof of this theorem based on a principle given by K. Iwasawa in his paper [6]. The author wishes to express his hearty thanks to Professor Iwasawa who read the original manuscript and gave him several important suggestions. For our convenience, we will use the same terminology as in C. Chevalley; The algebraic theory of spinors, Chapter I. We will define some notations we will use in this paper. The conjugate of a subspace U will be denoted by U'. If U is nonisotropic, the restriction Qu of Q to U is a quadratic form whose associated bilinear form is nondegenerate. We will denote the index of Qu (sometimes to be referred to as the index of U) by v(U), the orthogonal group of Qu by O(U) and the commutator group of O(U) by Q (U). Since every pu E 0 (U) is extended uniquely to pE 0(V) which induces the identity transformation on U', we can consider 0(U) a subgroup of 0(V). The subspace spanned by vectors xl, ,X will be denoted by . If u is a nonsingular vector, we denote the symmetry with respect to the hyperplane ' by o . In the case of characteristic 2 the term symmetry means transvection orthogonale defined by J. Dieudonne [4], p. 41.