The preservers of any orthogonal group

Abstract

Let L be an invertible linear map on the space M(n, k) of n -square matrices over a field k of characteristic not 2. In this paper we classify all such L which preserve a particular orthogonal group of a nonsingular symmetric bilinear form. We use some elementary facts about algebraic groups and an idea of Dieudonne's. There is some indication that our use of an algebraic geometric setting is the proper one for many problems of this type.