The centralizers in the root group of a special Moufang set are abelian

TL;DR

This paper proves that in a special Moufang set, the centralizer of any non-identity element in the root group is abelian, highlighting a structural property of these algebraic objects.

Contribution

The paper establishes that all centralizers of non-identity elements in the root group of a special Moufang set are abelian, a new structural insight.

Findings

Centralizers in the root group are abelian.

Structural property of special Moufang sets.

Advances understanding of Moufang set algebraic structure.

Abstract

In this note we prove that if $\mathbb{M}(U,\tau)$ is a special Moufang set and $a\in U^*$, then C_U(a) is abelian.