Sylow classes of reflection subgroups and pseudo-Levi subgroups

Abstract

In this thesis we study and classify specific subgroups in both finite reflection groups and finite groups of Lie type with the property that they minimally contain a Sylow subgroup. Connections between these minimal subgroups are investigated through a finite reflection group known as the Weyl group of a connected reductive group. Beginning with finite complex reflection groups we classify, up to conjugacy, the minimal parabolic subgroups and reflection subgroups containing a Sylow subgroup. We use this classification to describe normalisers of Sylow subgroups in reflection groups using a known description of normalisers of parabolic subgroups in reflection groups. The reflection subquotients of maximal eigenspaces associated to reflection cosets are calculated in preparation for studying the Sylow subgroup structure of finite groups of Lie type. We then classify, up to conjugacy, the minimal Levi subgroups and pseudo-Levi subgroups that contain a Sylow subgroup. In a special case, we show that these minimal subgroups directly corresponds to the minimal parabolic subgroups and reflection subgroups containing a Sylow subgroup in the Weyl group. Inspired by descriptions of Sylow subgroups of finite groups of Lie type in terms of Sylow tori and reflection subquotients of a reflection coset of the Weyl group, we introduce a new class of subgroups of finite groups of Lie type called twisted pseudo-Levi subgroups. We then show a direct correspondence between the twisted pseudo-Levi subgroups minimally containing a Sylow subgroup of a finite group of Lie type and minimal reflection subgroups of the Weyl group with a reflection subquotient containing a Sylow subgroup of the reflection subquotient of the Weyl group.