Finite groups with exactly two nonlinear irreducible $p$-Brauer characters

TL;DR

This paper classifies finite groups with exactly two nonlinear irreducible p-Brauer characters, extending previous work to include groups divisible by p, providing a complete characterization of such groups.

Contribution

It completes the classification of finite groups with exactly two nonlinear irreducible p-Brauer characters, including those divisible by p, building on Pálfy's earlier work.

Findings

Classified all finite groups with exactly two nonlinear irreducible p-Brauer characters.

Extended previous classifications to include groups divisible by p.

Provided a comprehensive characterization of such groups.

Abstract

Let $p$ be a prime. We classify the finite groups having exactly two irreducible $p$-Brauer characters of degree larger than one. The case, where the finite groups have orders not divisible by $p$, was done by P. P\'alfy in 1981.