Triangle-free cyclic conjugacy class graph of a finite group

TL;DR

The paper addresses the problem of classifying finite groups whose cyclic conjugacy class graph $\Delta(G)$ is triangle-free. It introduces a key lemma showing that if $\Delta(G)$ is triangle-free then every nonidentity element has order either a prime or the square of a prime, enabling a parity- and solvability-based classification. The main results identify that for odd $|G|$, either $G$ is a $3$-group of exponent $3$ or a Frobenius group of order $3\cdot 7^a$ with kernel exponent $7$, while for even $|G|$ the classification includes various Frobenius, $2$-Frobenius, and certain simple-group configurations, plus centerless nonsolvable cases such as $PSL(2,q)$ with specified $q$ and related extensions. These findings illuminate the interplay between the triangle-free property of $\Delta(G)$ and the detailed internal structure of finite groups, providing a comprehensive map of when enhanced-power-type graphs avoid triangles.

Abstract

We generalize the enhanced power graph by replacing elements with conjugacy classes. The main result of this paper is to determine when this graph is triangle-free.

Theorems & Definitions (20)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Theorem 2.3
• proof