D. Kiani, M. Mahdavi-Hezavehi: Crossed Product Conditions for Division Algebras of Prime Power Degree

kiani@mehr.sharif.edu, mahdavih@sharif.edu

Submission: 2004, Jul 28

Let $ D $ be an $ F $-central division algebra of degree $ p^r $, $ 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 Amitsur's 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.

2000 Mathematics Subject Classification:

Keywords and Phrases:

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