  A finite quantum hypergroup is a finite-dimensional unital algebra $A$ over
the field of complex numbers. There is a coproduct on $A$, a coassociative map
from $A$ to $A\otimes A$ assumed to be unital, but it is not required to be an
algebra homomorphism. There is a counit that is supposed to be a homomorphism.
Finally, the main extra requirement is the existence of a faithful left
integral with the right properties. For such a finite quantum hypergroup, the
dual can be constructed. It is again a finite quantum hypergroup. The more
general concept of an algebraic quantum hypergroup is studied in \cite{De-VD1,
De-VD2}. If the underlying algebra of an algebraic quantum hypergroup is
finite-dimensional, it is a finite quantum hypergroup in the sense of this
paper. Here we treat finite quantum hypergroups independently with an emphasis
on the development of the notion. It is meant to clarify the various steps
taken in \cite{La-VD3a} and \cite{La-VD3b}. In \cite{La-VD3b} we introduce and
study a still more general concept of quantum hypergroups. It not only contains
the algebraic quantum hypergroups from \cite{De-VD1}, but also some topological
cases. We naturally encounter these quantum hypergroups in our work on
bicrossproducts, see \cite{La-VD4, La-VD5, La-VD7}. We include some examples to
illustrate the theory. One kind is coming from a finite group and a subgroup.
The other examples are taken from the bicrossproduct theory.

Quantum Algebra;Group Theory