Free subgroups in maximal subgroups of SLn(D)

Abstract

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].

On locally solvable subgroups in division rings

“…Setting n = [D : F ], we know that D ⊗ F D op ∼ = M n (F ). Thus, by viewing M as a subgroup of GL n (F ), we conclude that M is a solvable group and the results follow from [8,Theorem 3.2] or [1].…”

“…According Lemma 2.3, we deduce that M (i) is abelian. Hence M is solvable, and we are done by [8,Theorem 3.2] or [1]. [1].…”

“…Hence M is solvable, and we are done by [8,Theorem 3.2] or [1]. [1]. Now, suppose the second case occurs, from which it follows by [11, Theorem 1.1] that M is abelian-by-locally finite.…”