Finite groups embeddable in division rings (Q2731903)

\textit{S. A. Amitsur} [Trans. Am. Math. Soc. 80, 361-386 (1955; Zbl 0065.25603)] determined all finite subgroups of division rings. The author's aim is to find a simpler approach to some of these results. He defines an Amitsur group as a finite group embeddable in the multiplicative group of a division ring and first observes that any Amitsur \(p\)-group is cyclic if \(p\) is odd, while for \(p=2\) it is cyclic or dicyclic. Next put \(G_{m,n,r}= gp \{a,b/a^m =b^n=1, bq=a^rb\}\), where \(r^n\equiv 1\pmod m\), and call a group Sylow-cyclic if all its Sylow subgroups are cyclic. Finite Sylow-cyclic groups were characterized as \(G_{m,n,r}\), where \((m,n(r-1))=1\) by Hölder, Burnside and Zassenhaus [see \textit{M. Hall}, The theory of groups, Macmillan (1959; Zbl 0084.02202), p. 146]. It follows that any Amitsur group of order not divisible by 8 is Sylow-cyclic; thus all such groups are metacyclic.NEWLINENEWLINENEWLINEThe author next finds some conditions on \(G_{m,n,r}\) to be an Amitsur group and concludes that an Amitsur group \(G\) of order \(<171\) is cyclic except possibly when \(|G|=63\) or 117. Examples where \(|G|=63\) are the cyclic group and \(G_{7,9,2}\) (as is shown by a direct proof) while \(G_{13,9,9}\) is an example of order 117. As the author points out, these results, for which Amitsur required some fairly deep number theory, are more easily obtained by division algebra methods.