On division subrings normalized by almost subnormal subgroups in division rings

TL;DR

This paper investigates the structure of division rings, proving that certain invariant subrings are necessarily central when associated with almost subnormal subgroups, and provides examples of such subgroups that are not subnormal.

Contribution

It establishes that N-invariant division subrings are central if N is an almost subnormal subgroup, and offers new examples of non-subnormal almost subnormal subgroups.

Findings

N-invariant subrings are central under given conditions

Existence of almost subnormal subgroups that are not subnormal

Extension of known subgroup classifications in division rings

Abstract

Let $D$ be a division ring with infinite center, $K$ a proper division subring of $D$ and $N$ an almost subnormal subgroup of the multiplicative group $D^*$ of $D$. The aim of this paper is to show that if $K$ is $N$-invariant and $N$ is non-central, then $K$ is central. Some examples of almost subnormal subgroups in division rings that are not subnormal are also given.