On groups with few 𝑝′-character degrees

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.