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 Algebra Logika, 2002, Volume 41, Number 6, Pages 730–744 (Mi al204)

Hypercentral Series and Paired Intersections of Sylow Subgroups of Chevalley Groups

V. M. Levchuk

Krasnoyarsk State University

Abstract: 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 $J=(p^d)$, $1\leqslant d<m$, $d\mid m$.

Keywords: Chevalley group, congruence subgroup of a Chevalley group, Lie rank, hypercentral series, central diagonal element, monomial element

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English version:
Algebra and Logic, 2002, 41:6, 400–408

Bibliographic databases:

UDC: 512.8

Citation: V. M. Levchuk, “Hypercentral Series and Paired Intersections of Sylow Subgroups of Chevalley Groups”, Algebra Logika, 41:6 (2002), 730–744; Algebra and Logic, 41:6 (2002), 400–408

Citation in format AMSBIB
\Bibitem{Lev02} \by V.~M.~Levchuk \paper Hypercentral Series and Paired Intersections of Sylow Subgroups of Chevalley Groups \jour Algebra Logika \yr 2002 \vol 41 \issue 6 \pages 730--744 \mathnet{http://mi.mathnet.ru/al204} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1967772} \zmath{https://zbmath.org/?q=an:1065.11020} \transl \jour Algebra and Logic \yr 2002 \vol 41 \issue 6 \pages 400--408 \crossref{https://doi.org/10.1023/A:1021707730444} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-42249095095} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Levchuk V.M., “Sylow subgroups of the Chevalley groups and associated (weakly) finitary groups and rings”, Acta Applicandae Mathematicae, 85:1–3 (2005), 225–232
2. V. I. Zenkov, A. S. Kondrat'ev, V. M. Levchuk, “Finite groups in which the normalizers of pairwise intersections of Sylow 2-subgroups have odd indices”, Proc. Steklov Inst. Math. (Suppl.), 259, suppl. 2 (2007), S163–S177
3. Levchuk V.M., Suleimanova G.S., “The normal structure of unipotent subgroups in Lie-type groups and related questions”, Doklady Mathematics, 77:2 (2008), 284–287
4. Levchuk V.M., Suleimanova G.S., “Extremal and maximal normal abelian subgroups of a maximal unipotent subgroup in groups of Lie type”, J Algebra, 349:1 (2012), 98–116
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