On The Eaton-Moretó Conjecture for Principal Blocks of Finite Groups

Abstract

Let $G$ be a finite group and let $p$ be a prime. If $P$ is a nonabelian Sylow $p$-subgroup of $G$ and $m(P)$ is the smallest non-linear irreducible character degree of $P$, we prove that there exists $χ\in {\rm Irr}(G)$ in the principal $p$-block of $G$ such that $1<χ(1)_p\le m(P)$, giving one inequality of the Eaton-Moretó conjecture for principal blocks. This, assuming Dade's Projective conjecture, implies the Eaton-Moretó conjecture for principal blocks.

Disclosure

“one.” The theorem Navarro proposed was a weaker block–free version of Theorem A of this paper. After a few days and several suggestions, AI produced a proof of this result. For this purpose, we used a workflow involving ChatGPT-5.6-Sol and Claude Fable 5, with Gómez–Serrano and Navarro setting up the problem and asking Navarro for further input and advice approximately every ∼ 24h. A few days later, again with some help from AI, we managed to prove Theorem A of this paper. The pro”

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Classification

Substantial proof generation

Multiplier

10

Verified

2026-08-23T05:19:20.246627Z

Structural counts