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My question is, does some form of "reciprocal" always work in the following sense: given a set [itex]X[/itex] with an equivalence relation [itex]R[/itex] defined on it, does it always exist some group [itex]G[/itex] with some action over [itex]X[/itex] such that the set of its orbits coincide with the equivalence classes?

I have thoght it, and concluded that for finite sets and groups, the cardinal of [itex]G[/itex] has to be a múltiple of the cardinal of every orbit, but I can't see if it is always possible to construct such group with such an action.

Thanks in advance for any help!