Necessary and Sufficient Conditions for the Regularity of the Sylow $p$-Subgroups of the Chevalley Groups over ${𝕑}_{p}$ and ${𝕑}_{p^{2}}$

Abstract

Let $G$ be an elementary Chevalley group of type $A_{n}$, $B_{n}$, $C_{n}$, and $D_{n}$ over a finite field of characteristic $p$ or the integer residue ring modulo $p^{2}$. We show that a Sylow $p$-subgroup $P$ of $G$ is regular if and only if the nilpotency length of $P$ is less than $p$. We introduce and study some series of the combinatorial objects related to the root systems and structure constants of simple complex Lie algebras.