Galois action on the principal block and cyclic Sylow subgroups

Rizo, Noelia; Schaeffer Fry, A. A.; Vallejo, Carolina

doi:10.2140/ant.2020.14.1953

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Galois action on the principal block and cyclic Sylow subgroups

Noelia Rizo, A. A. Schaeffer Fry and Carolina Vallejo

Vol. 14 (2020), No. 7, 1953–1979

DOI: 10.2140/ant.2020.14.1953

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.

Keywords

principal $p$-block, Galois action on characters, cyclic Sylow $p$-subgroups, Alperin–McKay–Navarro Conjecture

Mathematical Subject Classification 2010

Primary: 20C15

Secondary: 20C20, 20C33

Milestones

Received: 11 December 2019

Revised: 10 February 2020

Accepted: 10 March 2020

Published: 18 August 2020

Authors

Noelia Rizo
Università degli Studi di Firenze Italy

A. A. Schaeffer Fry
Department of Mathematical and Computer Sciences Metropolitan State University of Denver CO United States

Carolina Vallejo
Departamento de Matemáticas Universidad Autónoma de Madrid Facultad de Ciencias Campus de Cantoblanco Madrid Spain

Vol. 14, No. 7, 2020