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

Fuming Jiang, Yu Zeng

TL;DR

The paper solves the classification problem for finite groups G with exactly two nonlinear irreducible $p$-Brauer characters, under the pivotal hypothesis $ ext{O}_p(G)=1$. It integrates Brauer- and Clifford-theory with deep structure theorems on solvable primitive permutation groups (including rank-2 and rank-3 cases) and the theory of nonsolvable groups, particularly almost-simple and $K_3$-type simple groups. The authors derive a complete list: a handful of small, explicit almost-simple pairs $(G,p)$ and a broad solvable family $G=F(G) times H$ with $F(G)$ a Fitting subgroup and $H$ acting irreducibly on an appropriate module, plus several rank-3 primitive configurations. The results advance understanding of how $p$-Brauer characters distribute in finite groups and provide a comprehensive reference for related representation-theoretic classifications in group theory.

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. Pálfy in 1981.

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

TL;DR

The paper solves the classification problem for finite groups G with exactly two nonlinear irreducible -Brauer characters, under the pivotal hypothesis . It integrates Brauer- and Clifford-theory with deep structure theorems on solvable primitive permutation groups (including rank-2 and rank-3 cases) and the theory of nonsolvable groups, particularly almost-simple and -type simple groups. The authors derive a complete list: a handful of small, explicit almost-simple pairs and a broad solvable family with a Fitting subgroup and acting irreducibly on an appropriate module, plus several rank-3 primitive configurations. The results advance understanding of how -Brauer characters distribute in finite groups and provide a comprehensive reference for related representation-theoretic classifications in group theory.

Abstract

Let be a prime. We classify the finite groups having exactly two irreducible -Brauer characters of degree larger than one. The case, where the finite groups have orders not divisible by , was done by P. Pálfy in 1981.
Paper Structure (11 sections, 1 theorem, 14 equations)

This paper contains 11 sections, 1 theorem, 14 equations.

Table of Contents

  1. Introduction
  2. Auxiliary results
  3. $p$-groups
  4. Solvable primitive groups of small rank
  5. Some results in character theory
  6. Solvable groups
  7. Solvable doubly transitive permutation groups
  8. Solvable primitive permutation groups of rank 3
  9. Solvable groups with nonabelian Fitting subgroup
  10. Nonsolvable groups
  11. Proof of Theorem \ref{['thmA']}

Key Result

Theorem 1

Let $G$ be a finite group, and $p$ a prime. Assume that $\mathrm{O}_p(G)=1$. Then $G$ has exactly two nonlinear $p$-Brauer characters if and only if either or $G=\mathrm{F}(G) \rtimes H$ is a solvable group where $\mathrm{F}(G)$ is the Fitting subgroup of $G$, and one of the following holds:

Theorems & Definitions (28)

  • Theorem 1
  • proof
  • proof
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  • ...and 18 more