  Let k be any field. We consider the Hopf-Schur group of k, defined as the
subgroup of the Brauer group of k consisting of classes that may be represented
by homomorphic images of Hopf algebras over k. We show here that twisted group
algebras and abelian extensions of k are quotients of cocommutative and
commutative Hopf algebras over k, respectively. As a consequence we prove that
any tensor product of cyclic algebras over k is a quotient of a Hopf algebra
over k, revealing so that the Hopf-Schur group can be much larger than the
Schur group of k.
