Finite subgroups of division algebras over local fields (Q1891754)

Let \(F\) be a complete local field of characteristic zero with finite residue field of characteristic \(p\). In the paper the finite subgroups of the multiplicative group of a division algebra \(D\) with center \(F\) are studied. The study of the finite subgroups falls naturally into two cases, the case \(p=2\) and the case \(p\) is odd. The main results describe the finite subgroups in both cases. For instance it is shown that if \(p\) is odd, all nonabelian finite subgroups of division algebras over \(F\) are isomorphic to metacyclic groups; \(G_{m,r}=\langle A,B:A^m=1\), \(BAB^{-1}=A^r\), \(B^n=A^t\rangle\) where \(n=o(r|m)\), \(t=m/(r-1,m)\). Necessary and sufficient conditions are given for \(G_{m,r}\) to embed in a division algebra over \(F\). Let \(D(F,a/b)\) be the division algebra with Hasse invariant \(a/b\in\mathbb{Q}\). Then it follows from the results of the paper that the finite subgroups of \(D(F,a/b)^*\) depend only on \(F\) and \(b\).