Normalizers of finite subgroups of division algebras over local fields
Let \(D\) be a central division algebra over a local field \(F\) of characteristic \(0\) with finite residue class field of odd characteristic \(p\). The author [J. Algebra 173, No. 3, 518-548 (1995; Zbl 0829.16023)] proved that any finite non-Abelian subgroup of \(D\) has the form \(G_{m,r}=gp(A,B\mid A^m=1,\;BAB^{-1}=A^r,\;B^e=A^t)\), where \(r\in\mathbb{Z}/m\mathbb{Z}\) has order \(e\) and \(t=m/(r-1,m)\), and he gave conditions for such \(G_{m,r}\) to occur. In the present paper he makes a study of the normalizer of these groups. He first describes the automorphism group associated with different embeddings of \(G_{m,r}\) and relates it to the valuations on \(F\). He also briefly examines the case of cyclic groups and the case of residue characteristic 2.
scientific article; zbMATH DE number 1489514