  Quantum mechanics in Hilbert spaces of finite dimension $N$ is reviewed from
the number theoretic point of view. For composite numbers $N$ possible quantum
kinematics are classified on the basis of Mackey's Imprimitivity Theorem for
finite Abelian groups. This yields also a classification of finite
Weyl-Heisenberg groups and the corresponding finite quantum kinematics. Simple
number theory gets involved through the fundamental theorem describing all
finite discrete Abelian groups of order $N$ as direct products of cyclic
groups, whose orders are powers of not necessarily distinct primes contained in
the prime decomposition of $N$. The representation theoretic approach is
further compared with the algebraic approach, where the basic object is the
corresponding operator algebra. The consideration of fine gradings of this
associative algebra then brings a fresh look on the relation between the
mathematical formalism and physical realizations of finite quantum systems.
