Finitely Generated Subnormal Subgroups of GLn(D) Are Central

Abstract

Let D be an infinite division algebra of finite dimension over its center. Assume that N is a subnormal subgroup of GL n D with n ≥ 1. It is shown that if N is finitely generated, then N is central. © 2000 Academic Press Key Words: division ring; subnormal; finitely generated.Let D be an infinite division algebra of degree m over its center Z D = F. Denote by D the commutator subgroup of the multiplicative group D * = D − 0 . The aim of this note is to investigate the structure of finitely generated subnormal subgroups of GL n D with n ≥ 1. Assume that n ≥ 2 and N is a normal subgroup of GL n D . It is shown in [1] that if N is finitely generated, then N is central. A similar result for finitely generated normal subgroups is obtained for the case n = 1 in [2]. Here we shall generalize some of the main results appearing in [1] and [2] to subnormal subgroups of GL n D with n ≥ 1. To be more precise, assume that N is a subnormal subgroup of GL n D with n ≥ 1. It is proved that if N is finitely generated, then N is central. Using this, it is also shown that GL n D /Z GL n D contains no nontrivial finitely generated subnormal subgroups. Furthermore, given an infinite subnormal subgroup N of GL n D , it is proved that N contains no finitely generated maximal subgroups. Therefore, GL n D contains no finitely generated maximal subgroups. The reader may consult [5][6][7], and the references thereof for more recent results on multiplicative subgroups of GL n D . For convenience we shall deal with the case n = 1 separately. Our key result is the following.

Subnormal subgroups of the groups of rational points of reductive algebraic groups

“…Applying this corollary to the group G = GL n,D associated with a finite dimensional division algebra D, we recover the result of [10].…”

“…Papers [1]- [2] were devoted to various particular cases of the question of whether the group G = GL n (D), where D is a division algebra and n 1, can have finitely generated noncentral subnormal subgroups. Finally, it was proven in [10] that if D is finite dimensional over its center, then any finitely generated subnormal subgroup of GL n (D) must be central. The goal of this note is to show that the latter result is a particular case of a general statement about finitely generated subnormal subgroups in the group of rational points of reductive algebraic groups (and not only over fields, but also over semilocal rings arising from valuations).…”

Identities on Maximal Subgroups of GLn(D)

“…The study of maximal subgroups of A * begins in [1] and [9] in relation with an investigation of the structure of finitely generated normal subgroups of GL n (D), where D is of finite dimension over its centre F . In those papers we essentially show that maximal subgroups arise naturally in A * , and finitely generated subnormal subgroups of A * , are central.…”

Number-theoretic techniques in the theory of Lie groups and differential geometry

“…Finally, we would like to mention the following theorem which provides a far-reaching generalization of the results of [2] and [11].…”