Fields of Values in Finite Groups: Characters and Conjugacy Classes

Abstract

Navarro & Tiep proved in [46] that a finite group G has exactly one (respectively, two) rational-valued irreducible characters if and only if G has exactly one (respectively, two) rational conjugacy classes. They conjectured that the same statement holds when “three” replaces “one” or “two.” The main results of this thesis deal with this Navarro-Tiep conjecture. We show that one direction of their conjecture is true: Namely, a finite group having exactly three rational conjugacy classes also has exactly three rational-valued irreducible characters. We also precisely determine the structure of groups having exactly three rational-valued irreducible characters. Using this, we show that the converse of the Navarro-Tiep conjecture holds for non-solvable groups, except possibly in one very restricted situation. We also explore some questions involving Brauer characters and fields of values. We prove p-modular analogues of two results of Isaacs-Navarro [30] concerning the relationship between fields of values in G and G/N, for certain normal subgroups N of G.