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

Abstract

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

Key Words:

• Division ring
• Non-abelian divisible group

2010 Mathematics Subject Classification:

• 16K20

ACKNOWLEDGMENTS

The authors thank the referee for constructive comments. They are also grateful to the Research Council of Sharif University of Technology for support.

Notes

Communicated by A. Wadsworth.