Normalizers of chains of discrete p-toral subgroups in compact Lie groups

Abstract

In this paper we study the normalizer decomposition of a compact Lie group G using the information of the fusion system F of G on a maximal discrete p-toral subgroup. We prove that there is an injective map from the set of conjugacy classes of chains of F-centric, F-radical discrete p-toral subgroups to the set of conjugacy classes of chains of p-centric, p-stubborn continuous p-toral subgroups. The map is a bijection when π0(G) is a finite p-group. We also prove that the classifying space of the normalizer of a chain of discrete p-toral subgroups of G is mod p equivalent to the classifying space of the normalizer of the corresponding chain of p-toral subgroups. © 2022 Elsevier B.V.