Some skew linear groups with Engel's condition

Abstract

Abstract. Let D be a non-commutative division ring, m and n two natural numbers, M a maximal and N a subnormal subgroup of GL m .D/. In this paper, among other results, we show that: (1) if N is an n-Engel group, then it is central; (2) if m > 1 and M is locally nilpotent, then M is abelian; (3) if m > 1 and M is n-Engel, then the Hirsch-Plotkin radical of M is abelian. Also, we define some generalized Engel conditions on groups, and then we prove similar results, as quoted, for these groups.

Locally solvable subnormal and quasinormal subgroups in division rings

“…Motivated by this result, several other authors examined various aspects of subnormal subgroups of D Ã , instead of the whole group D Ã . For example, it was shown that every subnormal subgroup of D Ã must be central in D if it is locally nilpotent, solvable, or n-Engel (see [13], [8], [10], respectively). Now, we consider the same problem in which the subnormal subgroup is assumed to be locally solvable.…”

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

“…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.…”

“…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).…”