Supersoluble crossed product criterion for division algebras.
Let \(D\) be a finite-dimensional noncommutative division algebra over its centre \(F\), and let \(D^*\) be the multiplicative group of \(D\). A subgroup \(G\) of \(D^*\) is called generating, if \(F[G]=D\). The paper under review shows that \(D\) is presentable as a supersoluble (respectively, nilpotent, Abelian or cyclic) crossed product, i.e. \(D\) contains as a maximal subfield a Galois extension of \(F\) with a supersoluble (nilpotent, Abelian, cyclic) Galois group, if and only if, there exists a generating subgroup \(G\) of \(D^*\), which possesses an Abelian normal subgroup \(A\), such that the quotient group \(G/A\) is supersoluble (respectively, nilpotent, Abelian or cyclic). In particular, it proves that the existence of a locally nilpotent generating subgroup of \(D^*\) of \(D\) ensures the presentability of \(D\) as a nilpotent crossed product. Also, the authors show that their main result need not be valid, if \(D\) is merely a finite-dimensional central simple \(F\)-algebra.
generating subgroups of multiplicative groups of division algebras
scientific article; zbMATH DE number 2164547