SOLUBLE MAXIMAL SUBGROUPS IN GL_n(D)

Abstract

Let D be an F -central non-commutative division ring. Here, it is proved that if GLn(D) contains a non-abelian soluble maximal subgroup, then n = 1, [D : F ] < ∞, and D is cyclic of degree p, a prime. Furthermore, a classification of soluble maximal subgroups of GLn(F ) for an algebraically closed or real closed field F is also presented. We then determine all soluble maximal subgroups of GL 2 (F ) for fields F with Char F = 2.

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).…”

“…For instance, if D is an infinite division ring, in [3] it was shown that every nilpotent maximal subgroup of GL n (D) is abelian, and in [13] the authors proved that for n ≥ 2, every locally nilpotent maximal subgroup of GL n (D) is abelian. Also, if D is non-commutative and n ≥ 2, in [2] it was shown that every soluble maximal subgroup of GL n (D) is abelian, and in [13] the authors proved that for n ≥ 3, every locally soluble maximal subgroup of GL n (D) is abelian. For some recent results see [12].…”

Some skew linear groups with Engel's condition

“…Now, we would like to know whether or not an Engel maximal subgroup of GL m .D/ is abelian. Recently, it was shown in [5] that if D is a non-commutative division ring and m > 1, then every soluble maximal subgroup of GL m .D/ is abelian. In the next theorem we extend this result as follows: Corollary 1.7.…”

“…Therefore M is soluble. Now, if F OEM ¤ D, by [5,Theorem 3.7] M is abelian. Thus let F OEM D D. Then [5,Theorem 3.6] implies that dim F D < 1, which implies that M is abelian.…”