On locally solvable subgroups in division rings

TL;DR

This paper investigates the structure of locally solvable subgroups within division rings, establishing conditions under which such groups are central or lead to cyclic algebra structures.

Contribution

It proves that locally solvable subgroups with algebraic derived subgroups are central, and characterizes non-abelian maximal subgroups as leading to cyclic algebra structures.

Findings

Locally solvable subgroups with algebraic derived subgroups are central.

Non-abelian locally solvable maximal subgroups imply the division ring is a cyclic algebra.

Provides conditions linking subgroup properties to the algebraic structure of the division ring.

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 $G^{(i)}$ 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 $F$.