Abstract:
Seitz’s theorem asserts that a finite group has exactly one non-linear irreducible
character of degree greater than one if and only if the group is either an extraspecial
2-group or the group is isomorphic to a one-dimensional affine group over some
field. An extension of Seitz’s theorem is Thompson’s celebrated theorem which
states if the degrees of all non-linear irreducible characters of a group are divisible
by a fixed prime 𝑝, then the group contains a normal 𝑝-complement. More recently,
in 2020, as an extension to Thompson’s theorem, Giannelli, Rizo, and Schaeffer Fry
showed that if the character degree set of a group 𝐺 contains only two 𝑝′-character
degrees (where 𝑝 > 3 is a prime), then 𝐺 contains a normal subgroup 𝑁 such that
𝑁 has a normal 𝑝-complement and 𝐺/𝑁 has a normal 𝑝-complement. Moreover, 𝐺
is solvable. In this dissertation, we explore a variation of Thompson’s Theorem. We
explore the structure of finite groups that have exactly one non-linear irreducible
character whose degree is non-divisible by a fixed prime 𝑝. We call such groups
(∗)-groups (𝑝 divides the order of the group). In 1998, Kazarin and Berkovich characterized
the structure of (∗)-groups. We give a detailed proof of their work for
solvable groups. Moreover, we produce a classification of (∗)-groups of order less
than or equal to 100.
