Finite Zassenhaus Moufang sets with root groups of even order

TL;DR

This paper classifies finite Zassenhaus Moufang sets with even order root groups, showing they are either special projective lines over finite fields of even order or Suzuki Moufang sets, providing an alternative proof to Suzuki's classification.

Contribution

It offers a new proof of Suzuki's classification using Moufang sets, focusing on those with root groups of finite even order.

Findings

Zassenhaus Moufang sets with even order root groups are either special or Suzuki Moufang sets.

Provides an alternative proof of Suzuki's classification theorem.

Characterizes the structure of these Moufang sets in terms of known groups.

Abstract

Suzuki classified all Zassenhaus groups of finite odd degree. He showed that such a group is either isomorphic to a Suzuki group or to $\PSL(2,q)$ with $q$ a power of $2$. In this paper we give another proof of this result using the language of Moufang sets. More precisely, we show that every Zassenhaus Moufang set having root groups of finite even order is either special and thus isomorphic to the projective line over a finite field of even order or is isomorphic to a Suzuki Moufang set.