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

Abstract

Let $D$ be a division ring with the center $F$ and $D^{\ast }$ be the multiplicative group of $D$. In this paper we study locally nilpotent maximal subgroups of $D^{\ast }$. 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 $D$.