# Group action and equivalence relation

by Damidami
Tags: action, equivalence, relation
 P: 91 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!
 P: 350 An equivalence relation is the same as a partition of the set into a disjoint union of subsets (the equivalence classes). Let G be the group of bijections from X to itself. Let H be the subset of G which leaves the equivalence classes invariant. Then H is a subgroup, and it acts in the way you want.
 P: 91 Hi Vargo, Thanks for your reply. I think I can see your point. By the subset of G wich leaves the equivalence classes invariant, I think you mean the maximal one with that property (as the trivial susbset of G obviously leaves the classes invariant) Anyway it's interesting that any equivalence relation can be thought as the result of a group action, so every time I see a quotient space of any kind I can think as the result of some group acting by "gluing" some elements together.

 Related Discussions Precalculus Mathematics Homework 2 Calculus & Beyond Homework 1 Calculus & Beyond Homework 4 Precalculus Mathematics Homework 4 Precalculus Mathematics Homework 4