  A finite spin system invariant under a symmetry group G is a very
illustrative example of the finite group action on a set of mappings f:X->Y. In
the case of spin systems X is a set of spin carriers and Y contains 2s+1
z-components -s<=m<=s for a given spin number s. Orbits and stabilizers are
used as additional indices of the symmetry adapted basis. Their mathematical
nature does not lead to smaller eigenproblems, but they label states in a
systematic way. Some combinatorial and group-theoretical structures, like
double cosets and transitive representations, appear in a natural way. In such
a system one can construct general formulas for vectors of symmetry adapted
basis and matrix elements of operators commuting with the action of $G$ in the
space of states. Considerations presented in this paper should be followed by
detailed discussion of different symmetry groups (e.g. cyclic of dihedral ones)
and optimal implementation of algorithms. The paradigmatic example, i.e. a
finite spin system, can be useful in investigation of magnetic macromolecules.
