Normality of DSER elementary orthogonal group

Abstract

We consider the Dickson–Siegel–Eichler–Roy's (DSER) subgroup of the orthogonal group OR(Q⊥H(R)m) of a quadratic space with a hyperbolic summand over a commutative ring in which 2 is invertible, rankQ=n≥1 and m≥1. We show that it is a normal subgroup of the orthogonal group OR(Q⊥H(R)m), for m≠2. In particular, when Q≅H(R)r for r≥1 and m≥2, the DSER elementary orthogonal group EOR(Q,H(R)m) coincides with the usual elementary orthogonal group EO2(r+m)(R) and it is a normal subgroup in OR(H(R)r+m). We also prove that the DSER elementary orthogonal group EOR(Q,H(P)) is a normal subgroup of OR(Q⊥H(P)), where Q is a quadratic space and H(P) is the hyperbolic space of the finitely generated projective module over a commutative ring R, with P a finitely generated projective module with rank(P)≥2 and rank(Q)≥1.