On the Absence of a Normal Nonabelian Sylow Subgroup

TL;DR

This paper investigates conditions under which a finite solvable group lacks a normal nonabelian Sylow p-subgroup, based on properties of its prime character degree graph.

Contribution

It establishes a new criterion linking the structure of the prime character degree graph to the absence of certain normal Sylow subgroups in solvable groups.

Findings

Identifies a specific condition on the prime character degree graph

Proves the nonexistence of normal nonabelian Sylow p-subgroups under this condition

Provides insights into the structure of solvable groups with particular character degree graphs

Abstract

Let $G$ be a finite solvable group. We show that $G$ does not have a normal nonabelian Sylow $p$-subgroup when its prime character degree graph $\Delta(G)$ satisfies a technical hypothesis.