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.

Given a non-commutative finite-dimensional [Formula: see text]-central division ring [Formula: see text], [Formula: see text] a subnormal subgroup of [Formula: see text] and [Formula: see text] a non-abelian maximal subgroup of [Formula: see text], then either [Formula: see text] contains a non-cyclic free subgroup or there exists a non-central maximal normal abelian subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] is a subfield of [Formula: see text], [Formula: see text] is Galois and [Formula: see text], also [Formula: see text] is a finite simple group with [Formula: see text].

Given a division ring D with center F, the structure of maximal subgroups M of GLn(D) is investigated. Suppose D ≠ F or n > 1. It is shown that if M/(M ∩ F*) is locally finite, then char F=p > 0 and either n=1, [D:F]=p2 and M ∪ {0} is a maximal subfield of D, or D=F, n=p, and M ∪ {0} is a maximal subfield of Mp(F), or D=F and F is locally finite. It is also proved that the same conclusion holds if M/(M ∩ F*) is torsion and D is of finite dimension over F. Furthermore, it is shown that if the r-th derived group M(r) of M is locally finite, then either M(r) is abelian or F is algebraic over its prime subfield.

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