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The Commuting Graph of a Solvable A-Group

Rachel Carleton, Mark Lewis

Abstract

Let $G$ be a finite group. Recall that an $A$-group is a group whose Sylow subgroups are all abelian. In this paper, we investigate the upper bound on the diameter of the commuting graph of a solvable $A$-group. Assuming that the commuting graph is connected, we show when the derived length of $G$ is 2, the diameter of the commuting graph will be at most 4. In the general case, we show that the diameter of the commuting graph will be at most 6. In both cases, examples are provided to show that the upper bound of the commuting graph cannot be improved.

The Commuting Graph of a Solvable A-Group

Abstract

Let be a finite group. Recall that an -group is a group whose Sylow subgroups are all abelian. In this paper, we investigate the upper bound on the diameter of the commuting graph of a solvable -group. Assuming that the commuting graph is connected, we show when the derived length of is 2, the diameter of the commuting graph will be at most 4. In the general case, we show that the diameter of the commuting graph will be at most 6. In both cases, examples are provided to show that the upper bound of the commuting graph cannot be improved.
Paper Structure (4 sections, 15 theorems, 15 equations)

This paper contains 4 sections, 15 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Case When $G'$ is Abelian
  4. Main Result

Key Result

Theorem 1.1

Let $G$ be a solvable $A$-group such that $G/\mathbf{Z}(G)$ is neither a Frobenius nor 2-Frobenius group. Then, the diameter of the commuting graph of $G$ is at most 6.

Theorems & Definitions (23)

  • Theorem 1.1
  • Lemma 2.1: Taunt Theorem 4.1
  • Lemma 2.2: Taunt Corollary 4.5
  • Lemma 2.3: Taunt Theorem 5.4
  • Lemma 2.4: Isaacs Theorem 6.4
  • Lemma 2.5: Aschb Theorem 36.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 13 more