Let D be a division algebra of degree m over its center F. Herstein has shown Ž . that any finite normal subgroup of D* [ GL D is central. Here, as a generaliza-1 tion of this result, it is shown that any finitely generated normal subgroup of D* is Ž central. This also solves a problem raised by Akbari and Mahdavi-Hezavehi Proc.. Amer. Math. Soc., to appear for finite-dimensional division algebras. The structure of maximal multiplicative subgroups of an arbitrary division ring D is then investigated. Given a maximal subgroup M of D* whose center is algebraic over F, it is proved that if M satisfies a multilinear polynomial identity over F, then w x D : F -ϱ. ᮊ 1999 Academic Press

Let D be a division algebra of finite dimension over its center F. Given a Ž .M contains a noncyclic free subgroup or there exists a maximal subfield K of D Ž . which is Galois over F such that K * is normal in M and MrK * ( Gal KrF . Using this result, it is shown in particular that if D is a noncrossed product division algebra, then M does not satisfy any group identity. ᮊ

Abstract. As a generalization of Wedderburn's classic theorem, it is shown that the multiplicative group of a noncommutative finite dimensional division algebra cannot be finitely generated. Also, the following conjecture is investigated: An infinite non-central normal subgroup of GLn(D) cannot be finitely generated.

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.

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