  State symmetries are defined as permutations which act on vector spaces of
column vectors and square matrices, resulting in isotropy groups for specific
vector spaces.
  A large number of properties for such objects is shown, to provide a rigorous
basis for future applications. The main statement characterises the state
symmetry of vector sequences $(v^{(i)})$ which are generated by powers of a
generator matrix $M$: $v^{(i)}= M^i v^{(0)}$.
  A section of examples illustrates some applications of the theory.
