On the exponent of a certain quotient of Whitehead groups of division algebras

Let D be an F-central division algebra. In this paper, we investigated the exponent of the group G(D)=D*/NrdD(D*)D′, where D* is the group of units of D, NrdD(D*) is the image of D* under the reduced norm map and D′ is the commutator subgroup of D*. We show that if exp (G(D))<ind(D), then D and F satisfy strong conditions. In particular, we observe that if D is a sum cyclic algebras in Br(F), then exp (G(D))<ind(D) if and only if F is euclidean and D is a tensor product of an ordinary quaternion algebra and a division algebra of odd index.