Let M m (D) be a finite dimensional F -central simple algebra. It is shown that M m (D) is a crossed product over a maximal subfield if and only if GL m (D) has an 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.

Let D be an F -central division algebra of index n. Here we investigate a conjecture posed in [R. Hazrat et al., Reduced K-theory and the group G(D) = D * /F * D , in: Algebraic K-theory and its Applications, Trieste, 1997, pp. 403-409] that if D is not a quaternion algebra, then the group G 0 (D) = D * /F * D is non-trivial. Assume that either D is cyclic or F contains a primitive pth root of unity for some prime p | n. Using Merkurjev-Suslin Theorem, it is essentially shown that if none of the primary components of D is a quaternion algebra, then G(D) = D * /RN D/F (D * )D = 1. In this direction, we also study a conjecture posed in [S. Akbari, M. Mahdavi-Hezavehi, J. Pure Appl. Algebra 171 (2002) 123-131] or also [M. Mahdavi-Hezavehi, J. Algebra 271 (2004) 518-528] on the existence of maximal subgroups of D * .It is shown that if D is not a quaternion algebra with i(D) = p e , then D * has a maximal subgroup if either of the following conditions holds: (i) F has characteristic zero, or (ii) F has characteristic p, or (iii) F contains a primitive pth root of unity.  2004 Elsevier Inc. All rights reserved.