Galois action on the principal block and cyclic Sylow subgroups

Abstract

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