Key points are not available for this paper at this time.
Examples of finite-dimensional Hopf algebras over a field, whose antipodes have arbitrary even orders >/=4 as mappings, are furnished. The dimension of the Hopf algebra is q(n+1), where the antipode has order 2q, q >/= 2, and n is an arbitrary positive integer. The algebras are not semisimple, and neither they nor their dual algebras are unimodular.
Earl J. Taft (Mon,) studied this question.