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    Fields of Values in Finite Groups: Characters and Conjugacy Classes

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    Author
    Rossi, Daniel John
    Issue Date
    2018
    Keywords
    Rational Characters
    Rational Conjugacy Classes
    Advisor
    Tiep, Pham H.
    
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    Publisher
    The University of Arizona.
    Rights
    Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
    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.
    Type
    text
    Electronic Dissertation
    Degree Name
    Ph.D.
    Degree Level
    doctoral
    Degree Program
    Graduate College
    Mathematics
    Degree Grantor
    University of Arizona
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