Abstract

We determine all firm and residually connected geometries on which the group Sz(8) acts flag-transitively and fulfills the primitivity condition RWPRI, requiring that the stabilizer of each flag F acts primitively on the elements of some type in the residue ГF. This work was the starting point of a more ambitious work: the classification of all geometries of a Suzuki simple group Sz(q). The case q = 8 which is solved here, is the smallest case and the only one that is currently possible to analyse completely using the computer algebra package Magma. The rank 2 case was classified for all q (see Theorem 7.1 in [17]). The results obtained here rely partially on computer algebra.

Introduction

The present paper gives a complete classification of the firm and residually connected geometries on which the group Sz(8) acts flag-transitively and residually weakly primitively (see Section 2 for the definitions).

This work continues a systematic investigation of groups that has started some years ago (see [6, 8, 7, 12, 13, 14, 15, 16]).

Here we study the smallest Suzuki simple group, namely Sz(8). One of the reasons for this choice is that the infinite class of Suzuki groups Sz(q), with q = 22e+1 and e ≥ 1, looks particularly attractive for a general study and that a previous treatment of the smallest case may help in guessing the way to follow in general.