# It\^{o}'s theorem and monomial Brauer characters

## TL;DR

This paper proves that for finite solvable groups, the divisibility of monomial p-Brauer characters' degrees by p characterizes the existence of a normal Sylow p-subgroup.

## Contribution

It establishes a new criterion linking monomial p-Brauer characters and the structure of solvable groups, specifically normal Sylow p-subgroups.

## Key findings

- p does not divide degrees of all monomial p-Brauer characters iff G has a normal Sylow p-subgroup
- Provides a characterization of group structure via Brauer characters
- Enhances understanding of the relationship between character theory and group structure

## Abstract

Let $G$ be a finite solvable group, and let $p$ be a prime. In this note, we prove that $p$ does not divide $\varphi(1)$ for every irreducible monomial $p$-Brauer character $\varphi$ of $G$ if and only if $G$ has a normal Sylow $p$-subgroup.