Automorphisms of the orthogonal group of a defective space

Abstract

In a previous paper we determined the automorphisms of the orthogonal groups O(V), O+(V), Q(V), and O’(V) of a non-defective finite dimensional quadratic space L’ over a field F of characteristic 2. The underlying assumptions in [l] were that F had more than 2 elements and that the dimension of V be at least 10. In this paper we remove the assumption of nondefectiveness and investigate the automorphisms of the full orthogonal group O(v). Our main theorem (4.4 in Section 4) is of the type one expects in this situation. Our assumptions include a dimension assumption on the dimension of “the” underlying non-defective part of L’ (that it be at least 6) and an assumption on F. The assumption on F is weak enough to include the case of perfect fields, algebraically closed fields, local fields, and global fields-in fact it includes any field where Q(V*) intersects any ternary F2 subspace of F non-trivially. In particular it is weak enough to be able to derive as a corollary the well-known result concerning automorphisms of the symplective groups Q,,(F), where 2m > 6 and [F: F2] is finite. Our approach utilizes the supply of transvections in O(v). Subject to the conditions mentioned above we show that automorphisms of O(V) carry transvections to transvections. This enables us to produce a bijection of lines to which the Fundamental Theorem of Projective Geometry may be applied. The form of the automorphism is then readily obtained.