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

Let M-m. (D) be a finite dimensional F-central simple algebra. It is shown that M-m. (D) is 11 crossed produt over a maximal subfield if and only if GL(m)(D) has in irreducible subgroup G containing a normal abelian subgroup A such that C-G(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 deg(D)2, then D is a crossed product over a maximal subfield. (C) 2007 Elsevier Inc. All rights reserved.