Normality of DSER elementary orthogonal group

TL;DR

This paper proves the normality of the DSER elementary orthogonal group within the orthogonal group over rings with invertible 2, extending known results to broader quadratic space contexts.

Contribution

It establishes the normality of DSER elementary orthogonal groups in orthogonal groups over rings, covering cases with various ranks and quadratic space configurations.

Findings

DSER groups are normal subgroups of orthogonal groups for all m ≥ 2.

Normality holds for quadratic spaces with rank ≥ 1 and hyperbolic spaces with rank ≥ 2.

Results extend the understanding of subgroup structure in orthogonal groups over rings.

Abstract

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 $EO_{R}(Q, H(R)^{m})$ of the orthogonal group $O_R(Q \perp H(R)^m)$, with rank $Q= n \geq 1$ and $m\geq 2$. We show that $EO_{R}(Q, H(R)^{m})$ is a normal subgroup of $O_R(Q \perp H(R)^m)$, for all $m\geq 2$. We also prove that the DSER group $EO_{R}(Q, H(P))$ is a normal subgroup of $O_{R}(Q \perp H(P))$, where $Q$ and $H(P)$ are quadratic spaces over a commutative ring $R$, with rank $(Q) \ge 1$ and rank $(P) \ge 2$.