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Triviality of G(D) and G_0(D) and its Applications to the Multiplicative Structure of D
, M.Sc. Thesis Sharif University of Technology ; Mahdavi Hezaveh, Mohammad (Supervisor) ; Gholamzadeh Mahmoudi, Mohammad (Co-Supervisor)
Abstract
Let D be an F-central division algebra of index n. In this thesis a criterion for the triviality of the group G(D) = D^*/Nrd_(D/F) (D^*)D^' is presented 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. By using this, the role of some particular subgroups of D* in the algebraic structure of D is investigated. 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