Symmetry group of a system

Given a general dynamical system (see On Physics from the beginning) consisting of a set $\mathrm{\Omega }$, a kind of structure $\mathcal{S}$ (topology, differential structure, metric, symplectic form, all of them,...), and a dynamical law $\mathcal{L}$, we can consider:

• $\text{Sym}(\mathrm{\Omega })$: the bijections of $\mathrm{\Omega }$. Is the most general set of alternative descriptions of the space. This descriptions can be very crazy, how by an alien.
• $\text{Aut}(\mathrm{\Omega },\mathcal{S})$: the bijections of $\mathrm{\Omega }$ preserving $\mathcal{S}$, i.e. automorphisms of $(\mathrm{\Omega },\mathcal{S})$. They are alternative descriptions of the system but preserving some "intuitive features".
• $\mathcal{G}(\mathrm{\Omega },\mathcal{S},\mathcal{L})$: the symmetry group of the dynamical system

Examples:
a). A classical Hamiltonian system consisting of a symplectic manifold $(M,\omega )$ together with a Hamiltonian $H$. $\text{Aut}(\mathrm{\Omega },\mathcal{S})$ are the symplectomorphism and $\mathcal{G}(\mathrm{\Omega },\mathcal{S},\mathcal{L})$ are the Hamiltonian symmetrys.
b). The set $\mathrm{\Omega }=\{1,2,3,...,9,0\}$ with empty structure $\mathcal{S}$, and dynamical law $s(t+1)=s(t)+1$, if $s(t)\ne 9$; $s(t+1)=0$, if $s(t)=9$. See xournal 230
c). A quantum mechanical system: a Hilbert space with a Hamiltonian $H$. $\text{Aut}(\mathrm{\Omega },\mathcal{S})$ are the unitary transformations and the symmetry group $\mathcal{G}(\mathrm{\Omega },\mathcal{S},\mathcal{L})$ is made of the unitary transformation generated by Hermitian operators commuting with the Hamiltonian. See unitary operator#In Quantum Mechanics.