Fusion systems over a Sylow p-subgroup of {text {G}}_2(p)

Chris Parker,Jason Semeraro

doi:10.1007/s00209-017-1969-x

Abstract

For S a Sylow p-subgroup of the group {text {G}}_2(p) for p odd, up to isomorphism of fusion systems, we determine all saturated fusion systems mathcal {F} on S with O_p(mathcal {F})=1. For p ne 7, all such fusion systems are realized by finite groups whereas for p=7 there are 29 saturated fusion systems of which 27 are exotic.

Highlights

Let p ≥ 3 and S be a Sylow p-subgroup of the group G2( p)
The purpose of this paper is to give a complete classification of all saturated fusion systems F over S with Op(F) = 1
This may be viewed as a contribution to a program which aims to classify all saturated fusion systems over maximal unipotent subgroups of finite groups of Lie Type of rank 2 and is a natural continuation of work carried out in [6,12,20]

Summary

Introduction

Let p ≥ 3 and S be a Sylow p-subgroup of the group G2( p). The purpose of this paper is to give a complete classification of all saturated fusion systems F over S with Op(F) = 1. We see that all the morphisms of the essential subgroups of F are given uniquely by the group AutF (S) and so the fusion systems are uniquely determined. The fact that the fusion systems are uniquely determined by the structure of the automorphism groups of their essential subgroups is used implicitly. Two of the exotic systems were discovered by Parker and Stroth [18] and the remainder are new to this article They all are in some way related to the Monster sporadic simple group, though it is not the case that the Monster is “universal” in the sense that it “contains” all the smaller examples. Each time we use this notation the groups will be determined uniquely up to isomorphism as a subgroup of GSp4( p) or GL2( p). We point out that the notation SL2(7). will denote the unique normal subgroup of GL2(7) of index 3

Preliminaries: fusion systems and group theory

Construction of S

Properties of S and some subgroups