  Mathematical core of quantum mechanics is the theory of unitary
representations of symmetries of physical systems. We argue that quantum
behavior is a natural result of extraction of "observable" information about
systems containing "unobservable" elements in their descriptions. Since our aim
is physics where the choice between finite and infinite descriptions can not
have any empirical consequences, we consider the problem in the finite
background. Besides, there are many indications from observations - from the
lepton mixing data, for example - that finite groups underly phenomena in
particle physics at the deep level. The "finite" approach allows to reduce any
quantum dynamics to the simple permutation dynamics, and thus to express
quantum observables in terms of permutation invariants of symmetry groups and
their integer characteristics such as sizes of conjugate classes, sizes of
group orbits, class coefficients, dimensions of representations. Our study has
been accompanied by computations with finite groups, their representations and
invariants. We have used both our C implementation of algorithms for working
with groups and computer algebra system GAP.
