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

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 $x\in N$

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