If I understand correctly, in many context in physics (quantum mechanics?), a physical system is specified by giving its Hamiltonian. I also hear that symmetries are rather essential.
As far as the most natural symmetries are concerned, it seems fairly easy to recognise them. They should apparently be linear self-maps of the underlying Hilbert space, which seems natural enough, and they should surely commute with the Hamiltonian (if you transform the system by symmetry and then use the Hamiltonian to figure out how it evolves, the result should be the same as when you first look at time evolution, and then apply symmetry).
For a while I thought this is just what a symmetry is supposed to be doing, but then I realised that any map that keeps the eigenspaces of the Hamiltonian invariant would do, and I suspect this is not quite right. Also, there seem also to be some other operators around, often used when defining the Hamiltonian --- I'm not sure if a symmetry is supposed to commute with these as well...
I would be extremely grateful if someone could explain to me what symmetries of a physical system (viewed from the point of view of the Hamiltonian) actually are, and why is it that they are defined like this and not differently. An explanation in layman's/mathematician's terms would be even more appreciated.
Disclaimer: I am a mathematician, not a physicist.