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A Criterion for the Triviality of G(D) and Its Applications to the Multiplicative Structure of D

Mahdavi Hezavehi, M ; Sharif University of Technology | 2012

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  1. Type of Document: Article
  2. DOI: 10.1080/00927872.2011.584200
  3. Publisher: 2012
  4. Abstract:
  5. Let D be an F-central division algebra of index n. Here we present a criterion for the triviality of the group G(D) = D*/Nrd D/F(D*)D′ and thus generalizing various related results published recently. To be more precise, it is shown that G(D) = 1 if and only if SK 1(D) = 1 and F *2 = F *2n. Using this, we investigate the role of some particular subgroups of D* in the algebraic structure of D. In this direction, it is proved that a division algebra D of prime index is a symbol algebra if and only if D* contains a non-abelian nilpotent subgroup. More applications of this criterion including the computation of G(D) and the structure of maximal subgroups of D* are also investigated
  6. Keywords:
  7. Division ring ; Non-abelian divisible group
  8. Source: Communications in Algebra ; Volume 40, Issue 7 , 2012 , Pages 2645-2670 ; 00927872 (ISSN)
  9. URL: http://www.tandfonline.com/doi/abs/10.1080/00927872.2011.584200