The Divisibility Graph of finite groups of Lie Type

TL;DR

This paper investigates the structure of the Divisibility Graph for finite groups of Lie type in odd characteristic, identifying their connected components based on conjugacy class lengths.

Contribution

It provides a complete characterization of the connected components of the Divisibility Graph for these groups, a new insight into their algebraic structure.

Findings

Connected components of the Divisibility Graph are fully determined.

The structure varies depending on the type of Lie group.

Provides a foundation for further algebraic and combinatorial analysis.

Abstract

The Divisibility Graph of a finite group $G$ has vertex set the set of conjugacy class lengths of non-central elements in $G$ and two vertices are connected by an edge if one divides the other. We determine the connected components of the Divisibility Graph of the finite groups of Lie type in odd characteristic.