On the proof of some theorem on locally nilpotent subgroups in division rings

TL;DR

This paper provides a new proof for a theorem stating that every locally nilpotent subnormal subgroup in a division ring is central, correcting a previous proof with an identified mistake.

Contribution

The authors present an alternative proof for a key theorem about locally nilpotent subnormal subgroups in division rings, ensuring mathematical correctness.

Findings

The theorem that such subgroups are central is validated.

The new proof rectifies previous errors in the literature.

The result reinforces understanding of subgroup structure in division rings.

Abstract

In Hai-Thin (2009), there is a theorem, stating that every locally nilpotent subnormal subgroup in a division ring $D$ is central (see Hai-Thin (2009, Th. 2.2)). Unfortunately, there is some mistake in the proof of this theorem. In this note we give the another proof of this theorem.