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On minimal positive heights for blocks of almost quasi-simple groups

Gunter Malle, A. A. Schaeffer Fry

Abstract

The Eaton--Moretó conjecture extends the recently-proven Brauer height zero conjecture to blocks with non-abelian defect group, positing equality between the minimal positive heights of a block of a finite group and its defect group. Here we provide further evidence for the inequality in this conjecture that is not implied by Dade's conjecture. Specifically, we consider minimal counter-examples and show that these cannot be found among almost quasi-simple groups for $p\ge5$. Along the way, we observe that most such blocks have minimal positive height equal to~1.

On minimal positive heights for blocks of almost quasi-simple groups

Abstract

The Eaton--Moretó conjecture extends the recently-proven Brauer height zero conjecture to blocks with non-abelian defect group, positing equality between the minimal positive heights of a block of a finite group and its defect group. Here we provide further evidence for the inequality in this conjecture that is not implied by Dade's conjecture. Specifically, we consider minimal counter-examples and show that these cannot be found among almost quasi-simple groups for . Along the way, we observe that most such blocks have minimal positive height equal to~1.

Paper Structure

This paper contains 10 sections, 26 theorems, 4 equations, 1 table.

Table of Contents

  1. Introduction
  2. Elementary observations and reductions
  3. Metacyclic defect groups
  4. Groups not of Lie type
  5. Groups of Lie type in defining characteristic
  6. Groups of Lie type in non-defining characteristic
  7. Blocks and field automorphisms
  8. Unipotent blocks
  9. Type $\operatorname{A}$
  10. Quasi-isolated blocks

Key Result

Theorem 2

Let $B$ be a $p$-block of a finite quasi-simple group. Then $B$ is not a minimal counter-example to the direction ${\operatorname{mh}}(B)\le{\operatorname{mh}}(D)$ of Conjecture conj.

Theorems & Definitions (52)

  • Conjecture 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 42 more