# Group action and equivalence relation

by Damidami
Tags: action, equivalence, relation
 P: 94 Given a group $G$ acting on a set $X$ we get an equivalence relation $R$ on $X$ by $xRy$ iff $x$ is in the orbit of $y$. My question is, does some form of "reciprocal" always work in the following sense: given a set $X$ with an equivalence relation $R$ defined on it, does it always exist some group $G$ with some action over $X$ 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 $G$ 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!