Projective indecomposable permutation modules

TL;DR

This paper classifies finite non-Abelian simple groups where the projective cover of the trivial module matches a permutation module on a subgroup, excluding Lie type groups in defining characteristic.

Contribution

It provides a classification of such groups, identifying specific cases where the projective cover and permutation module coincide.

Findings

Identifies all finite non-Abelian simple groups with the specified property

Excludes groups of Lie type in defining characteristic from the classification

Provides a comprehensive list of cases where the modules coincide

Abstract

We investigate finite non-Abelian simple groups $G$ for which the projective cover of the trivial module coincides with the permutation module on a subgroup and classify all cases unless $G$ is of Lie type in defining characteristic.