Hypercentral Series and Paired Intersections of Sylow Subgroups of Chevalley Groups

Let $G(K)$ be the Chevalley group of normal type associated with a root system $G=\Phi$, or of twisted type $G={}^m\Phi$, $m=2,3$, over a field $K$. Its root subgroups $X_s$, for all possible $s\in G^+$, generate a maximal unipotent subgroup $U=UG(K)$; if $p=\operatorname{char}K>0$, $U$ is a Sylow $p$-subgroup of $G(K)$. We examine $G$ and $K$ for which there exists a paired intersection $U\cap U^g$, $g\in G(K)$, which is not conjugate in $G(K)$ to a normal subgroup of $U$. If $K$ is a finite field, this is equivalent to a condition that the normalizer of $U\cap U^g$ in $G(K)$ has a $p$-multiple index. Put $p(\Phi)=\max\{(r,r)/(s,s)\mid r,s\in\Phi\}$.
We prove a statement (Theorem 1) saying the following. Let $G(K)$ be a Chevalley group of Lie rank greater than 1 over a finite field $K$ of characteristic $p$ and $U$ be its Sylow $p$-subgroup equal to $UG(K)$; also, either $G=\Phi$ and $p(\Phi)$ is distinct from $p$ and 1, or $G(K)$ is a twisted group. Then $G(K)$ contains a monomial element $n$ such that the normalizer of $U\cap U^n$ in $G(K)$ has a $p$-multiple index.
Let $K$ be an associative commutative ring with unity and $\Phi(K,J)$ be a congruence subgroup of the Chevalley group $\Phi(K)$ modulo a nilpotent idea $J$. We examine an hypercentral series $1\subset Z_1\subset Z_2\subset\cdots\subset Z_{c-1}$ of the group $U\Phi(K)\Phi(K,J)$. Theorem 2 shows that under an extra restriction on the quotient $(J^t : J)$ of ideals, central series are related via $Z_i=\Gamma_{c-i}C$, $1\leqslant i<c$, where $C$ is a subgroup of central diagonal elements. Such a connection exists, in particular, if $K=Z_{p^m}$ and $J=(p^d)$, $1\leqslant d<m$, $d\mid m$.