Hypercentral Series and Paired Intersections of Sylow Subgroups of Chevalley Groups

V. M. Levchuk

Algebra i logika, Tome 41 (2002) no. 6, pp. 730-744.

Résumé

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 $Z_i=\Gamma_{c-i}C$, $1\leqslant i$, where $C$ is a subgroup of central diagonal elements. Such a connection exists, in particular, if $K=Z_{p^m}$ and $J=(p^d)$, $1\leqslant d$, $d\mid m$.

Détail

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

Bibliographie

[1] Nereshennye voprosy teorii grupp. Kourovskaya tetrad, 14-e izd., In-t matem. SO RAN, Novosibirsk, 1999 | MR

[2] T. Yu. Voitenko, V. M. Levchuk, “Parnye unipotentnye peresecheniya grupp Shevalle malykh lievykh rangov”, Trudy mezhd. konf. “Simmetriya i differentsialnye uravneniya”, In-t vychisl. matem. SO RAN, Krasnoyarsk, 2000, 71–74

[3] V. M. Levchuk, “Kommutatornoe stroenie nekotorykh podgrupp grupp Shevalle”, Ukr. mat. zhurn., 44:6 (1992), 786–795 | MR | Zbl

[4] M. I. Kargapolov, Yu. I. Merzlyakov, Osnovy teorii grupp, 4-e izd., Nauka, Moskva, 1996 | MR | Zbl

[5] R. W. Carter, Simple groups of Lie type, J. Wiley, London, 1972 | MR | Zbl

[6] V. M. Levchuk, “Parabolicheskie podgruppy nekotorykh $ABA$-grupp”, Matem. zametki, 31:4 (1982), 509–525 | MR | Zbl

[7] J. A. Gibbs, “Automorphisms of certain unipotent groups”, J. Algebra, 14:2 (1970), 203–228 | DOI | MR | Zbl

[8] V. M. Levchuk, “Avtomorfizmy unipotentnykh podgrupp grupp lieva tipa malykh rangov”, Algebra i logika, 29:2 (1990), 141–161 | MR | Zbl

[9] V. M. Levchuk, “Avtomorfizmy unipotentnykh podgrupp grupp Shevalle”, Algebra i logika, 29:3 (1990), 315–338 | MR | Zbl

[10] Ya. Burbaki, Gruppy i algebry Li, glavy IV–VI, Mir, Moskva, 1972 | Zbl

[11] B. L. Van der Varden, Algebra, 2-e izd., Nauka, Moskva, 1979 | MR

[12] V. M. Levchuk, “Ob odnom voprose Yu. I. Merzlyakova”, Tezisy VIII Vsesoyuz. simpoz. po teorii grupp, In-t matem. AN UkrSSR, Kiev, 1982, 71–72

[13] S. G. Kolesnikov, V. M. Levchuk, “Obobschennye kongruents-podgruppy grupp Shevalle”, Sib. matem. zh., 40:2 (1999), 336–351 | MR | Zbl

[14] Ph. Hall, “A note on $SI$-groups”, J. Lond. Math. Soc., 39:2 (1964), 338–344 | DOI | MR | Zbl

[15] Yu. V. Sosnovskii, “Kommutatornoe stroenie simplekticheskikh grupp”, Matem. zametki, 24:5 (1978), 641–648 | MR | Zbl