Character degrees of normally monomial p-groups of maximal class

TL;DR

This paper characterizes the possible irreducible character degrees of normally monomial p-groups of maximal class, providing explicit sets and bounds for these degrees, which advances understanding of their representation theory.

Contribution

It determines all possible irreducible character degree sets for normally monomial 5-groups of maximal class and establishes an upper bound for the largest degree in such p-groups.

Findings

Complete classification of degree sets for normally monomial 5-groups of maximal class

Upper bound for the largest irreducible character degree in normally monomial p-groups of maximal class

Enhanced understanding of the representation structure of these p-groups

Abstract

A finite group $G$ is $normally ~monomial$ if all its irreducible characters are induced from linear characters of normal subgroups of $G$. In this paper, we determine all possible irreducible character degree sets of normally monomial 5-groups of maximal class. Moreover, we give an upper bound for the largest irreducible character degree of normally monomial $p$-groups of maximal class in terms of $p$.