# Locally solvable subnormal and quasinormal subgroups of division rings

## TL;DR

This paper proves that any locally solvable subnormal or quasinormal subgroup of a division ring's multiplicative group must lie within the center of the division ring, revealing a strong structural restriction.

## Contribution

It establishes that locally solvable subnormal and quasinormal subgroups are contained in the center of a division ring, extending understanding of subgroup structure in division rings.

## Key findings

- Locally solvable subnormal subgroups are central.
- Quasinormal subgroups are also contained in the center.
- The result applies to both subnormal and quasinormal cases.

## Abstract

Let $D$ be a division ring with center $F$, and $G$ a subnormal or quasinormal subgroup of $D^*$. We show that if $G$ is locally solvable, then $G$ is contained in $F$.