Fitting heights of solvable groups with no nontrivial prime power character degrees

TL;DR

This paper constructs solvable groups with arbitrarily large Fitting heights where the only prime power degree of an irreducible character is 1, including groups with a Sylow tower and using only three primes.

Contribution

It introduces new constructions of solvable groups with specific character degree properties, expanding understanding of group structure and character theory.

Findings

Existence of solvable groups with large Fitting height and prime power character degrees only at 1

Construction of such groups with a Sylow tower

Groups can be built using only three primes

Abstract

We construct solvable groups where the only degree of an irreducible character that is a prime power is $1$ and that have arbitrarily large Fitting heights. We will show that we can construct such groups that also have a Sylow tower. We also will show that we can construct such groups using only three primes.