Groups with triangle-free graphs on p-regular classes

TL;DR

This paper classifies the structure of certain finite p-separable groups based on the properties of their conjugacy classes of p-regular elements, revealing specific conditions under which class lengths are coprime.

Contribution

It provides a classification of the p-structure of finite p-separable groups with a unique property of conjugacy class lengths, a novel insight into their algebraic structure.

Findings

Classification of p-structure for these groups

Conditions for conjugacy class length coprimality

Characterization of groups with triangle-free 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.