We show that theK0-group of a finite dimensional semisimple Hopf algebra has a natural structure of a ring with involution and prove that this ring is a twisting invariant of the Hopf algebra.We also study relations between algebraic structure of a Hopf algebra and the one of itsK0ring and prove that the twisting of a finite simple group is a simple Hopf algebra (i.e., it does not have proper normal Hopf subalgebras).In this case the dual Hopf algebra does not have proper Hopf subalgebras.Further, we construct several series of non trivial Hopf algebras by the twisting of the classical series of finite groups and give examples of Hopf algebras which can not be described as twisting of any group.