Normality of DSER elementary orthogonal group

Original · EN

Let (Q, q) be a quadratic space over a commutative ring R in which 2 is invertible, and consider the Dickson--Siegel--Eichler--Roy's subgroup EOᵣ(Q, H(R)ᵐ) of the orthogonal group Oᵣ(Q ⊥ H(R)ᵐ), with rank Q= n ≥ 1 and m≥ 2. We show that EOᵣ(Q, H(R)ᵐ) is a normal subgroup of Oᵣ(Q ⊥ H(R)ᵐ), for all m≥ 2. We also prove that the DSER group EOᵣ(Q, H(P)) is a normal subgroup of Oᵣ(Q ⊥ H(P)), where Q and H(P) are quadratic spaces over a commutative ring R, with rank (Q) ≥ 1 and rank (P) ≥ 2.