Galois action on the principal block and generation of Sylow 3-subgroups

TL;DR

This paper advances understanding of the Galois action on principal blocks and provides an algorithmic approach to identify groups with 2-generated Sylow 3-subgroups, supporting the blockwise Galois-McKay conjecture.

Contribution

It proves one direction of a conjecture linking character tables to Sylow subgroup generation and verifies the Isaacs-Navarro Galois conjecture for specific principal blocks.

Findings

Algorithm to determine 2-generated Sylow 3-subgroups from character tables

Evidence supporting the blockwise Galois-McKay conjecture

Proof of Isaacs-Navarro Galois conjecture for certain principal blocks

Abstract

In this paper, we prove one direction of a conjecture of Navarro-Rizo-Schaeffer Fry-Vallejo positing an algorithm to determine from the character table whether a finite group has $2$-generated Sylow $3$-subgroups. This gives further evidence of the blockwise version of the Galois-McKay conjecture (also known as the Alperin-McKay-Navarro conjecture). A key step involves proving the Isaacs-Navarro Galois conjecture for principal blocks for finite groups with a certain structure.