Publication Date
2026
Document Type
Dissertation/Thesis
First Advisor
Geline, Michael
Degree Name
Ph.D. (Doctor of Philosophy)
Legacy Department
Department of Mathematical Sciences
Abstract
The characters of finite groups are a fundamental tool for analyzing the structure of finite groups and are interesting in their own right. In particular, we are concerned with how certain assumptions on the values taken by characters impact the structure of finite groups and vice versa. After establishing notation and stating some fundamental results in Chapter 2, this work begins with a presentation of results in relation to a conjecture of N. N. Hung and P. H. Tiep on fields generated by character values. In particular, we have found a large family of counterexamples to the conjecture by taking advantage of classical results on minimal vanishing sums of roots of unity. Inspired by the techniques involved, we are able to produce upper and lower bounds on the “cyclotomic deficiency” for abelian extensions of Q as a function of the “length” of any primitive element which is a cyclotomic integer. Further applying these ideas in Chapter 4, a new block-theoretic proof of Burnside’s Normal p-complement Theorem is given. In Chapter 5, we conclude with some results on generalized characters whose values on nonidentity elements are sums of at most two roots of unity. Namely, we discuss constraints on the prime graphs of groups having such characters.
Recommended Citation
Herbig, Christopher William, "On Values Taken by Characters of Finite Groups" (2026). Graduate Research Theses & Dissertations. 8208.
https://huskiecommons.lib.niu.edu/allgraduate-thesesdissertations/8208
Extent
95 pages
Language
en
Publisher
Northern Illinois University
Rights Statement
In Copyright
Rights Statement 2
NIU theses are protected by copyright. They may be viewed from Huskie Commons for any purpose, but reproduction or distribution in any format is prohibited without the written permission of the authors.
Media Type
Text
