Brauer characters of $q^\prime$- degree

TL;DR

This paper investigates the structure of finite p-solvable groups with specific Brauer character degree conditions, revealing that certain normalizers intersect all conjugacy classes of p-regular elements.

Contribution

It establishes a new link between the divisibility of Brauer character degrees and the conjugacy class structure of p-solvable groups.

Findings

Normalizers of Sylow q-subgroups meet all p-regular conjugacy classes.

Prime q does not divide degrees of irreducible p-Brauer characters under certain conditions.

Provides structural insights into p-solvable groups based on Brauer character degrees.

Abstract

We show that if $p$ is a prime and $G$ is a finite $p$-solvable group satisfying the condition that a prime $q$ divides the degree of no irreducible $p$-Brauer character of $G$, then the normalizer of some Sylow $q$-subgroup of $G$ meets all the conjugacy classes of $p$-regular elements of $G$.