Lower defect groups and vertices of simple modules

TL;DR

This paper investigates the relationship between lower defect groups and vertices of simple modules in finite group algebra blocks, revealing containment and characterization results for principal type blocks.

Contribution

It establishes that lower defect groups are contained within vertices of simple modules after reordering and characterizes principal type blocks where certain lower defect groups are defect groups.

Findings

Lower defect groups are contained in vertices after reordering.

In principal type blocks, certain lower defect groups are defect groups.

Provides a comparison framework for lower defect groups and vertices.

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 lower defect groups are contained in vertices of simple modules after suitable reordering. 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.