Normal Structure of Isotropic Odd Orthogonal Groups

TL;DR

This paper establishes fundamental commutator relations and classifies normal subgroups in odd orthogonal groups over rings, extending known results without requiring 2 to be invertible.

Contribution

It proves standard commutator formulas and classifies EO-normal subgroups of orthogonal groups over rings with minimal assumptions.

Findings

Proved standard commutator formulas for orthogonal groups.

Classified EO-normal subgroups without assuming 2 invertibility.

Extended classical results to more general ring settings.

Abstract

Let $(M, q)$ be a quadratic projective module of an odd rank over an commutative ring, where the form $q$ is semiregular, with global Witt index of at least $2$, and with $\mathrm{rk}(M) \ge 7$. We prove standard commutator formulae and classify $\mathrm{EO}$-normal subgroups of $\mathrm{O}(M, q)$ without assumption of $2$ being invertible.