Crossed product conditions for central simple algebras in terms of irreducible subgroups

Let Mm(D) be a finite dimensional F-central simple algebra. It is shown that Mm(D) is a crossed product over a maximal subfield if and only if GLm(D) has an irreducible subgroup G containing a normal abelian subgroup A such that CG(A)=A and F[A] contains no zero divisor. Various other crossed product conditions on subgroups of D∗ are also investigated. In particular, it is shown that if D∗ contains either an irreducible finite subgroup or an irreducible soluble-by-finite subgroup that contains no element of order dividing deg2(D), then D is a crossed product over a maximal subfield.