Components of the Divisibility Graph of Finite Groups of Lie Type in Defining Characteristic $2$

Abstract

We determine the connected components of the divisibility graph of several families of finite groups of Lie type in defining characteristic $2$, extending the result of Abdolghafourian, Iranmanesh, and Niemeyer (arXiv:1612.04410), who treated odd defining characteristic. We show that there is a distinguished component containing all nonidentity unipotent class sizes and that every other component is either an isolated vertex or an explicitly described two-vertex component. We also determine the isolated vertices outside the distinguished component. Additionally, we prove in Appendix A that the divisibility graph of a Frobenius group has exactly two components.