In this paper, we show that every locally solvable subnormal subgroup or locally solvable quasinormal subgroup of the multiplicative group of a division ring is central.

Let D be a division ring with center F , and G a subnormal subgroup of D * . We show that if G is a locally solvable group such that a derived subgroup G (i) is algebraic over F , then G must be central. Also, if M is nonabelian locally solvable maximal subgroup of G with M (i) algebraic over F , then D is a cyclic algebra of prime degree over F .

The study of the existence of free groups in skew linear groups have begun since the last decades of the 20th century. The starting point is the theorem of Tits (1972), now often referred to as Tits’ Alternative, stating that every finitely generated subgroup of the general linear group [Formula: see text] over a field [Formula: see text] either contains a non-cyclic free subgroup or it is solvable-by-finite. In this paper, we study the existence of non-cyclic free subgroups in maximal subgroups of an almost subnormal subgroup of the general skew linear group over a locally finite division ring.