Variations on Keeler's Theorem

The 2010 Futurama episode The Prisoner of Benda features a mind swapping machine that swaps the minds of two people at a time with the restriction that the same pair of people cannot use the machine more than once. We show that if a machine swaps $n$ people cyclically with the condition that the same group of people cannot use the machine again, we can find a way to get everyone back. We prove our solution is optimal for when $n =3$. We also introduce an infinite variant of the mind swapping machine.