On normal subgroups of $D^*$ whose elements are periodic modulo the center of $D$ of bounded order

TL;DR

This paper proves Herstein's conjecture that a normal subgroup of a division ring's multiplicative group, which is radical over the center with elements of bounded order, must be contained in the center.

Contribution

It confirms Herstein's conjecture under the condition that all elements have order bounded by a fixed positive integer.

Findings

Herstein's conjecture holds when elements have uniformly bounded order

Normal radical subgroups are contained in the center under bounded order condition

The result extends understanding of subgroup structure in division rings

Abstract

Let $D$ be a division ring with the center $F=Z(D)$. Suppose that $N$ is a normal subgroup of $D^*$ which is radical over $F$, that is, for any element $x\in N$, there exists a positive integer $n_x$, such that $x^{n_x}\in F$. In \cite{Her1}, Herstein conjectured that $N$ is contained in $F$. In this paper, we show that the conjecture is true if there exists a positive integer $d$ such that $n_x\le d$ for any $x\in N$