On locally solvable subgroups in division rings

Abstract

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 .

Locally solvable subnormal and quasinormal subgroups in division rings

“…Moreover, it was proved by A. E. Zalesskii in [17] that every locally solvable normal subgroup of D Ã is contained in the center F of D. It is natural to ask whether every locally solvable subnormal subgroup, say G, of D Ã is also contained in F . A positive answer to this question was given for some particular cases where D is supposed to be algebraic over F ([5]), or where the derived subgroup G ðiÞ of G is assumed to be algebraic over F for some i b 1 ( [9]). The first purpose of the present paper is to give the a‰rmative answer to the question in the general setting; that is, we shall show that every locally solvable subnormal subgroup of D Ã is contained in F (Theorem 1).…”