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Groups with triangle-free graphs on p-regular classes

María José Felipe, Marc Kelly Jean-Philippe, Víctor Sotomayor

TL;DR

The paper analyzes how the arithmetic of $p$-regular conjugacy class lengths, captured by the common divisor graph $\Gamma_p(G)$, constrains the structure of finite $p$-separable groups. Focusing on the triangle-free case, it classifies the possible $p$-complements $H$ of $G$ and shows that either $H$ is a $q$-group, a quasi-Frobenius $\{q,r\}$-group with abelian kernel/complements and $\mathbf{Z}(H)=H\cap\mathbf{Z}(G)$, or $H\cong (C_5\times C_5)\rtimes Q_8$, with $G$ necessarily soluble and $\Gamma_p(G)$ taking one of a small number of explicit graphs. The work treats both disconnected and connected cases, deriving concrete descriptions of $G/\operatorname{O}_p(G)$ and the $p$-complements in each configuration, and leverages $N1$/$N2$ classifications and the simple-group analysis from the literature to constrain the global group structure. Overall, it strengthens the link between local $p$-structure and graph-theoretic properties, extending previous results on $\Gamma_p(G)$ and related graphs.

Abstract

Let $p$ be a prime. In this paper we classify the $p$-structure of those finite $p$-separable groups such that, given any three non-central conjugacy classes of $p$-regular elements, two of them necessarily have coprime lengths.

Groups with triangle-free graphs on p-regular classes

TL;DR

The paper analyzes how the arithmetic of -regular conjugacy class lengths, captured by the common divisor graph , constrains the structure of finite -separable groups. Focusing on the triangle-free case, it classifies the possible -complements of and shows that either is a -group, a quasi-Frobenius -group with abelian kernel/complements and , or , with necessarily soluble and taking one of a small number of explicit graphs. The work treats both disconnected and connected cases, deriving concrete descriptions of and the -complements in each configuration, and leverages / classifications and the simple-group analysis from the literature to constrain the global group structure. Overall, it strengthens the link between local -structure and graph-theoretic properties, extending previous results on and related graphs.

Abstract

Let be a prime. In this paper we classify the -structure of those finite -separable groups such that, given any three non-central conjugacy classes of -regular elements, two of them necessarily have coprime lengths.
Paper Structure (5 sections, 18 theorems, 6 equations)

This paper contains 5 sections, 18 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The non-connected case
  4. The connected case
  5. Proof of Theorem \ref{['teoA']}

Key Result

Theorem 1

Let $G$ be a $p$-separable group, for a given prime $p$, and let $H$ be a non-central $p$-complement of $G$. If $\Gamma_p(G)$ has no triangles, then one of the following cases holds: In particular $G$ is soluble, and $\Gamma_p(G)$ coincides with one of the next six graphs. Case i) corresponds to (d) and (e), case ii) corresponds to (a), (b), (c) and (e), and case iii) corresponds to (f).

Theorems & Definitions (24)

  • Theorem 1
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Remark 2.4
  • Proposition 2.5
  • Lemma 2.6
  • Theorem 2.7: H
  • Proposition 2.8
  • Proposition 3.1
  • ...and 14 more