The Automorphisms of O + 4 (V) in the Anisotropic Case

Abstract

The automorphisms of the rotation group of a quatemary anisotropic quadratic space defined over a field of characteristic not two are determined. In this paper we determine the automorphisms of the rotation group 04+(V) of a quaternary anisotropic quadratic space V defined over a field F of characteristic not two and thus resolve an open question mentioned by Dieudonne in his book La geome'trie des groupes classiques (see [1, p. 101]). The automorphisms of 04+ (V) have been determined by Dieudonne in the case that V has positive Witt index and by Wonenburger in the case that the discriminant of V is a square. (See [1, pp. 101-103] and [3, p. 195] respectively.) Our approach treats the square discriminant and nonsquare discriminant cases simultaneously, and it is free of any reference to the Clifford algebra of the underlying space. Our notation and terminology is that of [2]. Unlike the situation treated in [2], we have a plentiful supply of involutions at our disposal. We utilize these involutions to produce a bijection of the planes of V. This bijection is shown to be the same as that induced by arbitrary plane rotations and is used to induce a bijection of the lines of V. The Fundamental Theorem of Projective Geometry is then applied as in [2] to determine the form of the automorphism. V is a quaternary, anisotropic space over a field F with characteristic F different from 2. If a, a', al, a2.. al, af ... are elements of O(V) then R, R', R1, R2, ..., R'1, R', ... will be used to denote their respective residual spaces. Since V is anisotropic, all subspaces of V are regular. If R is any subspace of V there is a unique involution a in O(V) with R = res space a. Let A be an automorphism of 0+(V). For a in 0+(V) let a' denote Aa. PROPOSITION 1. Let a be a piane rotation that is an involution. Then A a is a plane rotation. PROOF. By assumption a is not in the center of 0+ (V). Hence A a is not in the center of 0+(V) and res Aa = 2. Q.E.D. Now let R be any plane and let a be the unique involution in O+(V) with res space a = R. By Proposition 1 we have a bijection of planes R R'. PROPOSITION 2. If R1 and R2 are planes with R1 n R2 a line, then R' n R' is a line. Received by the editors July 11, 1974 and, in revised form, November 4, 1974. AMS (MOS) subject classyifcations (1970). Primary 20H 15; Secondary 1OC05.