On general linear groups over exchange rings

TL;DR

This paper investigates the structure of general linear groups over exchange rings, proving normality of certain subgroups and establishing a standard commutator formula, thereby advancing the understanding of subgroup relations in algebraic K-theory.

Contribution

It establishes the normality of relative elementary subgroups in general linear groups over exchange rings and proves the standard commutator formula for these groups when n ≥ 3.

Findings

Relative elementary subgroups are normal in GL_n(R) for n ≥ 1.

The standard commutator formula holds for n ≥ 3.

Classification of subgroups normalized by elementary subgroups for n ≥ 3.

Abstract

Let $R$ be an exchange ring. We prove that the relative elementary subgroups $E_n(R,I)$ are normal in the general linear group $GL_n(R)$ if $n\geq 1$ and that the standard commutator formula $E_n(R,I)=[E_n(R),E_n(R,I)]=[E_n(R),C_n(R,I)]$ holds if $n\geq 3$. Moreover, we classify the subgroups of $GL_n(R)$ that are normalised by the elementary subgroup $E_n(R)$ in the case $n\geq 3$.