Special rank one groups with abelian unipotent subgroups are perfect

TL;DR

This paper proves that special rank one groups with abelian unipotent subgroups are perfect, providing a simpler, elementary proof that enhances understanding of their structure.

Contribution

It offers a short, elementary, and self-contained proof that special rank one groups with abelian unipotent subgroups are perfect, simplifying previous criteria.

Findings

Proved that such groups are perfect

Provided a simplified proof approach

Enhanced understanding of group structure

Abstract

T. De Medts, Y. Segev and K. Tent [Special Moufang sets, their root groups and their \mu-maps, Proc. Lond. Math. Soc. (3) 96 (2008), 767-791] proved that the little projective group of a special Moufang set M(U,\tau) is perfect provided that U has size at least 4. We give a short, elementary and self-contained argument when U is abelian, working with Timmesfeld's concept of special (abstract) rank one groups with abelian unipotent subgroups [Abstract root subgroups and simple groups of Lie type, Monographs in Mathematics, 95. Birkh\"auser, 2001]. This simplifies Timmesfeld's (quasi) simplicity criterion for groups generated by abstract root subgroups [loc. cit., II (2.14)].