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A family of graphs that cannot occur as character degree graphs of solvable groups

Mark W. Bissler, Jacob Laubacher, Mark L. Lewis

Abstract

We investigate character degree graphs of solvable groups. In particular, we provide general results that can be used to eliminate which degree graphs can occur as solvable groups. Finally, we show a specific family of graphs cannot occur as a character degree for any solvable group.

A family of graphs that cannot occur as character degree graphs of solvable groups

Abstract

We investigate character degree graphs of solvable groups. In particular, we provide general results that can be used to eliminate which degree graphs can occur as solvable groups. Finally, we show a specific family of graphs cannot occur as a character degree for any solvable group.

Paper Structure

This paper contains 3 sections, 6 theorems, 3 equations, 1 figure.

Table of Contents

  1. Introduction
  2. General Lemmas
  3. Infinite Family

Key Result

Lemma 2.2

Let G be a solvable group, and suppose $p$ is an admissible vertex of $\Delta(G)$. For every proper normal subgroup $H$ of $G$, suppose that $\Delta(H)$ is a proper subgraph of $\Delta(G)$. Then $O^p(G)=G$.

Figures (1)

  • Figure 1: Example of graphs from the Main Theorem

Theorems & Definitions (27)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Lemma 2.6
  • proof
  • ...and 17 more