Maximal Subgroups of GL1(D)

Abstract

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

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.…”

On normal subgroups of division rings which are radical over a proper division subring

“…The structure properties of multiplicative subgroups in division rings have been recently studied such as free subgroups ( [4,17]), maximal subgroups ( [2,10,11,15]), subgroups radical over a set ( [1,2,6,16,17,14]), etc. In this paper, we generalize one of Faith's works on division rings which are radical over a proper subring.…”

Some skew linear groups with Engel's condition

“…Our next observation is about maximal subgroups of skew linear groups; these groups have been studied in a series of papers, see, e.g., [1,7,16,17]. In [7], it was shown that if D is an infinite division ring and m is a natural number, then every nilpotent maximal subgroup of GL m .D/ is abelian.…”