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

Abstract

Let ℓ 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 ℓ-Sylow subgroup. The classification assists in describing the ℓ-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 ℓ-Sylow subgroup is G itself, the reflection subgroup within the parabolic minimally containing an ℓ-Sylow subgroup is the whole parabolic subgroup.

Locations

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