Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the structure of the character degree graphs having diameter three

Silvio Dolfi, Roghayeh Hafezieh, Pablo Spiga

Abstract

The structure of the character degree graphs $Δ(G)$, i.e. the prime graphs on the set $\mathrm{cd}(G)$ of the irreducible character degrees of a finite group $G$, such that $G$ is solvable and $Δ(G)$ has diameter three, remains an intriguing area of study. However, a comprehensive understanding of these structures remains elusive. In this paper, we prove some properties and provide an infinite series of examples of this class of graphs, building on the ideas of Mark Lewis.

On the structure of the character degree graphs having diameter three

Abstract

The structure of the character degree graphs , i.e. the prime graphs on the set of the irreducible character degrees of a finite group , such that is solvable and has diameter three, remains an intriguing area of study. However, a comprehensive understanding of these structures remains elusive. In this paper, we prove some properties and provide an infinite series of examples of this class of graphs, building on the ideas of Mark Lewis.
Paper Structure (5 sections, 13 theorems, 68 equations, 2 figures)

This paper contains 5 sections, 13 theorems, 68 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A construction
  4. Number theoretic considerations
  5. Main results

Key Result

Lemma 2.1

Let $A$ be a solvable group acting by automorphisms on a group $G$. If $|A|$ is coprime to $|G|$, then $\mathrm{Irr}(G)$ and $\mathrm{Cl}(G)$ are isomorphic $A$-sets.

Figures (2)

  • Figure 1: Lewis' example
  • Figure 2: An auxiliary picture for $\Delta(G)$ having diameter $3$

Theorems & Definitions (27)

  • Lemma 2.1: isaacs
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 17 more