T. Keshavarzipour & M. Mahdavi-Hezavehi: Crossed Product Conditions for Central Simple Algebras in terms of irreducible subgroups

keshavarzipour@math.sharif.edu, mahdavih@sharif.edu

Submission: 2007, Apr 22

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.

2000 Mathematics Subject Classification: 16K20

Keywords and Phrases: Division ring, Crossed Product, Irreducible groups

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