Galois action on the principal block and cyclic Sylow subgroups

TL;DR

This paper characterizes finite groups with cyclic Sylow p-subgroups through Galois automorphisms acting on the principal p-blocks, linking group structure to block theory and conjectures in modular representation theory.

Contribution

It provides a characterization of groups with cyclic Sylow p-subgroups via Galois actions on principal blocks, and discusses implications for the McKay-Navarro conjecture.

Findings

Finite groups with cyclic Sylow p-subgroups are characterized by Galois automorphisms on principal p-blocks.

The result connects group structure with modular representation theory and block theory.

An analog for arbitrary defect groups would follow from the blockwise McKay-Navarro conjecture.

Abstract

We characterize finite groups having a cyclic Sylow p-subgroup in terms of the action of a specific Galois automorphism on the principal p-block for p=2,3. We show that the analog statement for blocks with arbitrary defect group would follow from the blockwise McKay-Navarro conjecture.