Normality and $K_1$-stability of Roy's elementary orthogonal group

TL;DR

This paper proves the normality of Roy's elementary orthogonal group over certain rings and establishes a stability theorem for its K_1 group, advancing understanding of its algebraic structure.

Contribution

It introduces the first proof of normality for Roy's elementary orthogonal group and provides a new stability theorem for its K_1 group.

Findings

Proved normality of Roy's elementary orthogonal group under specific conditions.

Established a stability theorem for the K_1 group of Roy's group.

Derived a decomposition theorem for the elementary orthogonal group.

Abstract

In this paper, we prove the normality of the Roy's elementary orthogonal group (Dickson--Siegel--Eichler--Roy or DSER group) over a commutative ring which was introduced by A. Roy in [MR0231844] under some conditions on the hyperbolic rank. We also establish a stability theorem for $K_1$ of Roy's group. We obtain a decomposition theorem for the elementary orthogonal group which is used to deduce the stability theorem.