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

TL;DR

This paper investigates the Eaton–Moretó conjecture related to block heights in finite groups, providing evidence that minimal counterexamples are unlikely among almost quasi-simple groups for primes p≥5, and noting most such blocks have height 1.

Contribution

The paper offers new evidence supporting the Eaton–Moretó conjecture by analyzing minimal counterexamples and showing they are absent among almost quasi-simple groups for certain primes.

Findings

Minimal counterexamples do not occur among almost quasi-simple groups for p≥5.

Most blocks considered have minimal positive height equal to 1.

The results strengthen the conjecture's validity in specific group classes.

Abstract

The Eaton--Moret\'o 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.