A new graph related to conjugacy classes of finite groups

TL;DR

This paper introduces a new graph concept related to finite groups, classifies groups satisfying a specific property involving conjugacy class representatives, and advances understanding of group structure through this classification.

Contribution

It presents a novel graph associated with conjugacy classes and classifies finite groups satisfying the property that any four non-central classes have representatives with pairwise coprime orders.

Findings

Classification of finite groups with property P_4

Identification of structural characteristics of these groups

Extension of conjugacy class analysis in finite group theory

Abstract

In this paper we classify the finite groups satisfying the following property $P_4$: their orders of representatives are set-wise relatively prime for any 4 distinct non-central conjugacy classes.