# Simple modules in the Auslander-Reiten quiver of principal blocks with abelian defect groups

## TL;DR

This paper studies the placement of simple modules in the Auslander-Reiten quiver of principal blocks with abelian defect groups, providing reductions to finite simple groups and specific results for characteristic 3.

## Contribution

It introduces a reduction technique to finite simple groups and proves that simple modules lie at the end of their components in characteristic 3.

## Key findings

- Reduction to finite simple groups
- Simple modules in characteristic 3 lie at the end of their components
- Provides structural insights into principal blocks with abelian defect groups

## Abstract

Given an odd prime $p$, we investigate the position of simple modules in the stable Auslander-Reiten quiver of the principal block of a finite group with non-cyclic abelian Sylow $p$-subgroups. In particular, we prove a reduction to finite simple groups. In the case that the characteristic is $3$, we prove that simple modules in the principal block all lie at the end of their components