I am trying to learn the integer quantum hall effect and have a pretty straightforward question. I understand that the normal translation group does not commute with the Landau Hamiltonian. Does this mean that if you have a state in the lowest Landau level (LLL) and apply the translation operator to it you end up outside of the Hilbert space of the Landau Hamiltonian? If so, is it the magnetic translation group that allows us to translate a particle and stay inside of the Hilbert space since it commutes with the Hamiltonian? Is that why the idea of a projection operator onto the Hilbert space of the LLL is useful? Is the magnetic translation group just the normal translation group with the projection operator applied to it? These questions are probably really simple but I am still in undergraduate quantum mechanics so I haven't learned a lot of these ideas in a formal setting yet and I just want to make sure I have the details and motivations correct.