Classification of Sylow classes of parabolic and reflection subgroups in unitary reflection groups

摘要

Let be a prime divisor of the order of a finite unitary reflection group. We classify up to conjugacy the parabolic and reflection subgroups that are minimal with respect to inclusion, subject to containing an -Sylow subgroup. The classification assists in describing the -Sylow subgroups of unitary reflection groups up to group isomorphism. This classification also relates to the modular representation theory of finite groups of Lie type. We observe that unless a parabolic subgroup minimally containing an -Sylow subgroup is G itself, the reflection subgroup within the parabolic minimally containing an -Sylow subgroup is the whole parabolic subgroup.