Sylow Subgroups of the Chevalley Groups and Associated (Weakly) Finitary Groups and Rings
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Sylow Subgroups of the Chevalley Groups and Associated (Weakly) Finitary Groups and Rings V. M. LEVCHUK Krasnoyarsk State University, av. Svobodnyi 79, 660041 Krasnoyarsk, Russia. e-mail: [email protected] Abstract. Automorphisms, isomorphisms and structural descriptions of finitary unitriangular groups over a ring and of Sylow subgroups of Chevalley groups are studied. Also, the theorem on pair unipotent intersections of Chevalley groups has been proved; it is connected with the known question on pair Sylow intersections of finite groups. These investigations use close connections and structural descriptions of considered groups and associated rings, which have been found recently in the case of a commutative ring of coefficients with a strongly maximal ideal. We consider some properties and examples of such ideals. Mathematics Subject Classifications (2000): 13C13, 08A35, 20F28, 20D06, 20D20, 20F406. Key words: Chevalley group, unipotent subgroup, weakly finitary group, associated Lie ring, strongly maximal ideal, isomorphism.
1. The Finitary Rings, their Adjoint Groups and Generalizations Structural descriptions of finitary unitriangular groups over rings and also of Sylow subgroups and of the unipotent subgroups of the Chevalley group have close connections with the description of ideals of the associated rings. Such descriptions are essentially used in investigations of automorphisms and isomorphisms and of the known question on pair Sylow intersections of finite groups (see Theorem 3 on pair unipotent intersections), etc. Let us distinguish a subalgebra N (K) with the basis {er | r ∈ + } in Lie K-algebra with the Chevalley basis {er | r ∈ , . . .} [1, Sect. 4.4]. The unipotent subgroup U (K) of the Chevalley group (K) is isomorphic to the “adjoint” group of N (K) with respect to a certain operation ◦, by [10]. It turns to be that the ideals of Lie ring N (K) and only they are normal subgroups of the group (N (K), ◦) for a field K = 2K with K = 3K at = G2 ; all ideals are also described in [11]. Similarly, the normal structure of unipotent subgroups U G(K) of the twisted group of any Lie type G (see Section 3) are described in [13, 17] and [12]. Note that if K = Zpm and J is a maximal ideal of K, then the product of U (K) and of the congruence subgroup by J -level is a Sylow p This research is supported by Russian fund of fundamental researches (RFFI), grant 03-01-
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subgroup of the Chevalley group (K). The question concerning a description of its automorphisms [5, Question 12.42] is still open. We now consider a generalization of the principal case of Lie type An . Let be a chain by the order relation . -matrices aij i,j ∈ over K having a finite number of nonzero elements in each row and column are called weakly finitary. Weakly finitary -matrices over an associative ring K (with the identity 1K ) form a ring (resp. with the identity) with respect to the usual matrix addition and multiplication. Now there is little information about this ri
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