Identities on Maximal Subgroups of GL_n(D)

Abstract

Let D be a division ring with centre F . Assume that M is a maximal subgroup of GL n (D), n ≥ 1 such that Z(M ) is algebraic over F . Group identities on M and polynomial identities on the F -linear hullWhen D is noncommutative and F is infinite, it is also proved that if M satisfies a group identity and F [M ] is algebraic over F , then we have either M = K * , where K is a field and [D : F ] < ∞ or M is absolutely irreducible. For a finite dimensional division algebra D, assume that N is a subnormal subgroup of GL n (D) and M is a maximal subgroup of N .If M satisfies a group identity, it is shown that M is abelian-by-finite.

ON SOME SUBGROUPS OF D*WHICH SATISFY A GENERALIZED GROUP IDENTITY

“…In fact, they showed that if the cardinality of F is greater than 3α(w) 2 , then D = F . Recently, there are some articles on some subgroups of D * which satisfy a group identity or some special group identity (see [7,10,12,14]): Ramezan-Nassab and Kiani proved in [14] that subnormal subgroups of D * satisfying the n-Engel condition are contained in F . It is proved in [12] that every maximal subgroup of D * satisfying a group identity is the multiplicative group of a maximal subfield of…”

Nilpotent and polycyclic-by-finite maximal subgroups of skew linear groups

“…Our object here is to discuss the general skew linear groups whose maximal subgroups are of some special types. Some properties of maximal subgroups of GL n (D) have been studied in a series of papers, see, e.g., [1,2,3,7,12,13]. In all of those papers, authors attempted to show that the structure of maximal subgroups of GL n (D) is similar, in some sense, to the structure of GL n (D).…”

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.…”

“…If dim F D < 1, then M as a linear n-Engel group is nilpotent and hence abelian. So we may assume that D is of infinite dimension over F , and so, R is simple by [16,Lemma 4]. Thus R is primitive and from Kaplansky's theorem [20, p. 36] we have that R ' M t ./ for some division ring .…”