2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/72485State 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.35 pages, including a large number of examplesRings and Algebras05E20; 15A03; 20G20text