(Locally soluble)-by-(locally finite) maximal subgroups of GLn(D)

Let D be a non-commutative division ring and M a maximal subgroup of GLn(D) (n⩾2). This paper continues the ongoing effort to show that the structure of maximal subgroups of GLn(D) is similar, in some sense, to the structure of GLn(D). It is known that every locally soluble normal subgroup of GLn(D) is abelian. Here, among other results, we prove that if either (i) D is finite-dimensional over its center, or (ii) the center of D contains at least five elements and M is soluble-by-finite, or (iii) and M is (locally soluble)-by-finite, then every locally soluble normal subgroup of M is abelian.