Maximal Subgroups of GL1(D)

Let D be a division algebra of degree m over its center F. Herstein has shown that any finite normal subgroup of D* GL1(D) is central. Here, as a generalization of this result, it is shown that any finitely generated normal subgroup of D* is 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 [D : F] < ∞.