Hopf powers and orders for some bismash products

The exponent of a finite group G can be viewed as a Hopf algebraic invariant of the group algebra H=kG: it is the least integer n for which the nth Hopf power endomorphism [n] of H is trivial. The exponent of a group scheme G as studied by Gabriel and Tate and Oort can be defined in the same way using the coordinate Hopf algebra image. The power map and the corresponding notion of exponent have been studied for a general finite-dimensional Hopf algebra beginning with work of Kashina. Several positive results, suggested by analogy to the group case, were proved by Kashina and by Etingof and Gelaki. Given these positive results, there was some hope that the Hopf order of an individual element of a Hopf algebra might also be a well-behaved notion, with some properties analogous to well-known facts on the orders of elements of a finite group. In fact we prove that such analogous properties do hold for Hopf algebras satisfying the usual rule for iterated powers; for example, such a Hopf algebra H has an element of order n if and only if n divides the exponent of H. However, in general such properties are not true. We will give examples where the behavior of Hopf powers, Hopf orders, and related notions is rather strange, unexpected, and seemingly hard to predict. We will see this using computer algebra calculations in Drinfeld doubles of finite groups, and more generally in bismash products constructed from factorizable groups.