On finite groups with all simple modules of low dimension in characteristic p

TL;DR

This paper proves that if all simple modules of a finite group over a field of characteristic p have dimension less than p, then the group has a normal Sylow p-subgroup, simplifying previous proofs.

Contribution

It provides a concise proof of a known result relating simple module dimensions to the structure of finite groups in characteristic p.

Findings

All simple modules have dimension less than p implies a normal Sylow p-subgroup

Simplifies the proof of the structural property of finite groups

Connects module theory with group structure in characteristic p

Abstract

We give a short proof of the fact that if all characteristic p simple modules of the finite group G have dimension less than p, then G has a normal Sylow p-subgroup.