Automizers as extended reflection groups

TL;DR

This paper extends the known connection between automizers of Sylow p-subgroups and complex reflection groups from finite simple Chevalley groups to general finite groups, providing new examples such as in the Monster group.

Contribution

It generalizes the property of automizers being complex reflection groups to all finite groups, with specific examples like the Monster group.

Findings

Automizer of an 11-Sylow subgroup in the Monster is G_{16}

Automizers of Sylow p-subgroups can be complex reflection groups in general finite groups

Extension of known results from Chevalley groups to broader class of finite groups

Abstract

Brou\'e, Malle and Michel have shown that the automizer of an abelian Sylow p-subgroup in a finite simple Chevalley group is an irreducible complex reflection group, for p not too small and different from the defining characteristic. The aim is this note is to show that a suitable version of this property holds for general finite groups. As an example, the automizer of an 11-Sylow subgroup in the Monster is the 2-dimensional complex reflection group G_{16}.