# Group action and equivalence relation

1. Dec 16, 2012

### Damidami

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!

2. Dec 17, 2012

### Vargo

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.

3. Dec 17, 2012

### Damidami

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.