Crossed product conditions for division algebras of prime power degree

Let D be an F-central division algebra of degree pr, p a prime. A set of criteria is given for D to be a crossed product in terms of irreducible soluble or abelian-by-finite subgroups of the multiplicative group D* of D. Using the Amitsurʹs classification of finite subgroups of D* and the Tits alternative, it is shown that D is a crossed product if and only if D* contains an irreducible soluble subgroup. Further criteria are also presented in terms of irreducible abelian-by-finite subgroups and irreducible subgroups satisfying a group identity. Using the above results, it is shown that if D* contains an irreducible finite subgroup, then D is a crossed product.