Given any nontrivial alternating tri-character f on a finite abelian group G, one can construct a finite dimensional non-commutative and non-cocommutative semisimple Hopf algebra H. The group of group-like elements of H is an abelian central extension of B by Ĝ where B is the radical of f .

We study a natural construction of Hopf algebra quotients canonically associated to an R-matrix in a finite dimensional Hopf algebra. We apply this construction to show that a quasitriangular Hopf algebra whose dimension is odd and square-free is necessarily cocommutative.