On locally nilpotent maximal subgroups of the multiplicative group of a division ring

TL;DR

This paper investigates the structure and existence of locally nilpotent maximal subgroups within the multiplicative group of a division ring, providing conditions and classifications based on algebraic and finiteness assumptions.

Contribution

It characterizes locally nilpotent maximal subgroups in division rings, especially relating to their algebraic properties and the nature of the center.

Findings

Locally nilpotent maximal subgroups are either multiplicative groups of maximal subfields or center by locally finite.

If the center is finite and the subgroup is nilpotent, it must be a multiplicative group of a maximal subfield.

Conditions affecting the existence of such subgroups depend on the algebraic and finiteness properties of the division ring.

Abstract

Let $D$ be a division ring with the center $F$ and $D^*$ be the multiplicative group of $D$. In this paper we study locally nilpotent maximal subgroups of $D^*$. We give some conditions that influence the existence of locally nilpotent maximal subgroups in division ring with infinite center. Also, it is shown that if $M$ is a locally nilpotent maximal subgroup that is algebraic over $F$, then either it is the multiplicative group of some maximal subfield of $D$ or it is center by locally finite. If, in addition we assume that $F$ is finite and $M$ is nilpotent, then the second case cannot occur, i.e. $M$ is the multiplicative group of some maximal subfield of $D$.