Vol. 70, No. 1, 1977

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The preservers of any orthogonal group

Peter Botta and Stephen J. Pierce

Vol. 70 (1977), No. 1, 37–49

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 Dieudonné’s. There is some indication that our use of an algebraic geometric setting is the proper one for many problems of this type.

There are a considerable number of results which may be paraphrased as follows: “let L : M(n,k) M(n,k) be a linear transformation that preserves some property related to matrix multiplication. Then L is almost an inner automorphism of M(n,k).” While the statement of these results exhibits a large degree of similarity, an examination of the proofs reveals almost no similarity. The property in question could be determinant, nonsingularity, or orthogonality.

Mathematical Subject Classification 2000
Primary: 20G15
Received: 23 March 1976
Published: 1 May 1977
Peter Botta
Stephen J. Pierce