Normalizers of Sylow subgroups in finite reflection groups

2022/11/08 by Townsend, Kane Douglas
#20D20 (secondary) #20F55 (primary) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2211.04044

Abstract

Let W be a finite reflection group, either real or complex, and S_ℓ a Sylow ℓ-subgroup of W. We prove the existence of a semidirect product decomposition of N_W(S_ℓ) in terms of the unique parabolic subgroup of W minimally containing S_ℓ and known decompositions of normalizers of parabolic subgroups. In the real setting, the description follows from the existence of Sylow ℓ-subgroups stable under the Coxeter diagram automorphisms of finite reflection groups with no proper parabolic subgroup containing a Sylow ℓ-subgroup.

Related

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