Locally solvable maximal subgroups in division rings

TL;DR

This paper characterizes the structure of division rings containing certain maximal subgroups, showing that the presence of a non-abelian locally solvable maximal subgroup implies the division ring is a cyclic algebra of prime degree.

Contribution

It establishes a connection between the properties of maximal subgroups and the algebraic structure of the division ring, specifically identifying when it is a cyclic algebra of prime degree.

Findings

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

Every locally nilpotent maximal subgroup of such a subgroup is abelian.

The results provide structural insights into the subgroup composition of division rings.

Abstract

Let $D$ be a division ring with center $F$, and $G$ an almost subnormal subgroup of $D^*$. In this paper, we show that if $G$ contains a non-abelian locally solvable maximal subgroup, then $D$ must be a cyclic algebra of prime degree over $F$. Moreover, it is proved that every locally nilpotent maximal subgroup of $G$ is abelian.