The total character of a finite group G is the sum of the irreducible characters of G. When the total character of a finite group can be written as a monic polynomial with integer coefficients in an irreducible character of G, we say that G is a total character group. In this thesis we examine the total character of the dicyclic group of order 4n, the non-abelian groups of order p^3, and the symmetric group on n elements for all n ≥ 1. The dicyclic group of order 4n is a total character group precisely when n is congruent to 2 or 3 mod 4, and the associated polynomial is a sum of Chebyshev polynomials of the second kind. The irreducible characters paired with these polynomials are exactly the faithful characters of the dicyclic group. In contrast, the non-abelian groups of order p^3 and the symmetric group on n elements with n ≥ 4 are not total character groups. Finally, we examine the special case when G is a total character group and the polynomial is of degree 2. In this case, we say that G is a quadratic total character group. We classify groups which are both quadratic total character groups and p-groups.
| Identifer | oai:union.ndltd.org:BGMYU2/oai:scholarsarchive.byu.edu:etd-4312 |
| Date | 03 July 2012 |
| Creators | Kennedy, Chelsea Lorraine |
| Publisher | BYU ScholarsArchive |
| Source Sets | Brigham Young University |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | Theses and Dissertations |
| Rights | http://lib.byu.edu/about/copyright/ |
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