# ALGB seminar: Noelia Rizo (university of Florence)

Title: Galois action on the principal block and cyclic Sylow p-subgroups

Abstract: A classical question in the representation theory of finite groups is to determine what properties of a group or its local structure can be obtained from its character table. In particular the study of the relations between the set Irr(G) of the irreducible complex characters of G and the structure of its Sylow p-subgroups has been a cornerstone of the character theory of finite groups. If one wants to go deeper, there is a more sophisticated version of this: looking at the irreducible characters in the principal Brauer p-block of G. In this work, 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 conjecture an analog statement for blocks with arbitrary defect group and we prove that this general statement would follow from the blockwise Galois-McKay conjecture.