On the dimensions of PIM's

TL;DR

This paper investigates the structure of finite groups based on the dimensions of their projective indecomposable modules and characterizes groups with specific $p$-power degree properties, extending known formulas.

Contribution

It proves the converse of Fong's dimension formula for $p$-solvable groups and characterizes groups with all irreducible $p$-Brauer characters having $p$-power degrees.

Findings

Proved the converse of Fong's dimension formula for $p$-solvable groups.

Characterized groups with all irreducible $p$-Brauer characters of $p$-power degrees.

Abstract

We discuss the structure of finite groups for which the projective indecomposable modules have special given dimensions. In particular, we prove the converse of Fong's dimension formula for $p$-solvable groups. Furthermore, we characterize groups for which all irreducible $p$-Brauer characters have $p$-power degrees.