On the exponent of a certain quotient of Whitehead groups of division algebras (Q6594900)

In this paper under review, the properties of the group $G(D)=D^{*}/(\mathrm{Nrd}_{D}(D^{*})D')$ is investigated, where: $D$ is a $F$-central division algebra with index $\text{ind}(D)>1$, $D^{*}$ is the group of units of $D$, $\text{Nrd}_{D}(D^{*})$ is the image of $D^{*}$ under the reduced norm map $\text{Nrd}:D^{*}\to F^{*}$ and $D'$ is the commutator subgroup of $D^{*}$.\N\NThe paper focuses on the exponent $\text{exp}(G(D))$ of $G(D)$, particularly when it is strictly smaller than $\text{ind}(D)$. It is shown that, in this case, $D$ is one of the two types:\N\begin{itemize}\N\item[(T1)] There is an odd prime $p\geq4$, $p\mid\text{ind}(D)$ such that $F^{*}=F^{*p}$.\N\item[(T2)] $F$ is euclidean and $D\cong(\frac{-1,-1}{F})\otimes P$ for some $F$-central division algebra of odd index, and $\text{exp}(G(D))\mid\text{ind}(D)$.\N\end{itemize}\NFor every odd integer $n$, the author gives a cyclic division algebra $D$ of index $2n$ with $\text{exp}(G(D))=n$, namely, an example of type (T1). However, whether there exists examples of type (T2) is unknown.\N\NIn the last section of the paper, the author improves the results of [\textit{M. Mahdavi-Hezavehi} and \textit{M. Motiee}, Commun. Algebra 40, No. 7, 2645--2670 (2012; Zbl 1258.16024)], and obtains explicit formulas for computing the group $G(D)$ over the fields of iterated Laurent series.