On locally solvable subgroups in division rings

Abstract

Let $D$ be a division ring with center $F$, and $G$ a subnormal subgroup of $D^{\ast }$. We show that if $G$ is a locally solvable group such that $G^{(i)}$ is algebraic over $F$, then $G$ must be central. Also, if $M$ is non-abelian locally solvable maximal subgroup of $G$ with $M^{(i)}$ algebraic over $F$, then $D$ is a cyclic algebra of prime degree over $F$.