SOLUBLE MAXIMAL SUBGROUPS IN GLn(D)

Abstract

Let D be an F-central non-commutative division ring. Here, it is proved that if GL n(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 GL n(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.