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

TL;DR

The paper establishes a principal-block, Galois-action framework to connect the structure of Sylow 3-subgroups with the pattern of height-zero, sigma-invariant characters, proving that $|P:\Phi(P)|=9$ whenever the principal 3-block has $k_{0,\sigma}(B_0(G))\in\{6,9\}$. It introduces a blockwise Isaacs–Navarro Galois phenomenon (Theorem B) and reduces Theorem A to the simple- and almost-simple-group case, leveraging Brauer’s cyclic-Sylow theory and Ketchum eden’s results. A central technical achievement is a sigma-equivariant version of Gia–Riz–SchVal24 for simple groups, verified for broad families (alternating, sporadic, and most Lie-type groups) with careful handling of edge cases such as $\operatorname{PSL}_2(3^a)$ and $\operatorname{PSL}^\epsilon_3(3^a)$. Together, these results provide concrete evidence toward the blockwise Alperin–McKay–Navarro conjecture in the principal-block setting and clarify the Galois action’s imprint on characters in relation to Sylow subgroups.

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.

Theorems & Definitions (54)

• Theorem A
• Theorem B
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Theorem 2.3: Alperin--Dade
• proof
• Lemma 2.4
• proof