# Locally solvable and solvable-by-finite maximal subgroups of $GL_n(D)$

## TL;DR

This paper investigates the structure of maximal subgroups in the general skew linear group over division rings, revealing conditions under which they are abelian or have finite dimension, with specific results for infinite fields.

## Contribution

It characterizes when locally solvable and solvable-by-finite maximal subgroups are abelian or finite-dimensional in the context of $ ext{GL}_n(D)$ over division rings.

## Key findings

- If $D$ is non-commutative, such maximal subgroups are either abelian or have finite dimension over $F$.
- For infinite fields $F$ and $n \\geq 5$, all locally solvable maximal subgroups are abelian.
- The study provides new insights into the structure of maximal subgroups in skew linear groups.

## Abstract

This paper aims at studying solvable-by-finite and locally solvable maximal subgroups of an almost subnormal subgroup of the general skew linear group $\GL_n(D)$ over a division ring $D$. It turns out that in the case where $D$ is non-commutative, if such maximal subgroups exist, then either it is abelian or $[D:F]<\infty$. Also, if $F$ is an infinite field and $n\geq 5$, then every locally solvable maximal subgroup of a normal subgroup of $\GL_n(F)$ is abelian.