Lower defect groups and vertices of simple modules

Abstract

We compare lower defect groups associated with p-regular classes and vertices of simple modules for a block of a finite group algebra. We show that there exists a bijective correspondence between p-regular conjugacy classes belonging to the block and isomorphism classes of simple modules in the block such that a defect group of each p-regular conjugacy class is contained in a vertex of the corresponding simple module. Moreover, for a block of principal type, we show that a p-regular lower defect group which contains a vertex of simple module is a defect group of the block.