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.

No abstract

Given an F-central simple algebra A = Mn(D), denote by A′ the derived group of its unit group A*. Here, the Frattini subgroup Φ(A*) of A* for various fields F is investigated. For global fields, it is proved that when F is a real global field, then Φ(A*) = Φ(F*)Z(A′), otherwise Φ(A*) = ⋂p∤ deg (A) F*p. Furthermore, it is also shown that Φ(A*) = k* whenever F is either a field of rational functions over a divisible field k or a finitely generated extension of an algebraically closed field k.

No abstract