Special rank one groups are perfect

TL;DR

This paper provides a simple, elementary proof that special rank one groups are perfect, extending previous results on the structure of Moufang sets and their associated groups.

Contribution

It offers a new, self-contained proof that special rank one groups are perfect, using Timmesfeld's concept of special abstract rank one groups.

Findings

Proves special rank one groups are perfect for arbitrary unipotent subgroups.

Simplifies previous proofs with an elementary approach.

Extends results on the structure of Moufang sets and related groups.

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, working with Timmesfeld's concept of special (abstract) rank one groups with arbitrary unipotent subgroups [Abstract root subgroups and simple groups of Lie type, Monographs in Mathematics, 95. Birkh\"auser, 2001].