Groups whose common divisor graph on $p$-regular classes has diameter three

TL;DR

This paper investigates the structure of a specific graph associated with finite p-separable groups, focusing on the maximum distance between certain conjugacy classes and the p-structure when the graph's diameter is three.

Contribution

It answers an open question about the maximum distance in the common divisor graph and explores the p-structure of groups with a diameter of three.

Findings

Maximum distance between certain conjugacy classes is established.

Characterization of p-structure when the graph has diameter three.

Provides insights into the graph's diameter in relation to group properties.

Abstract

Let $G$ be a finite $p$-separable group, for some fixed prime $p$. Let $\Gamma_p(G)$ be the common divisor graph built on the set of non-central conjugacy classes of $p$-regular elements of $G$: this is the graph whose vertices are the conjugacy classes of those non-central elements of $G$ such that $p$ does not divide their orders, and two distinct vertices are adjacent if and only if the greatest common divisor of their lengths is strictly greater than one. The aim of this paper is twofold: to positively answer an open question concerning the maximum possible distance in $\Gamma_p(G)$ between a vertex with maximal cardinality and any other vertex, and to study the $p$-structure of $G$ when $\Gamma_p(G)$ has diameter three.