Let A be a set and G be any subgroup of S(A). G is a group of permutations of A. Assume that G is a finite group. If u∈A, the orbit of u is the set O(u)={g(u): g∈G}. Define a relation ~ on A by u~v iff g(u)=v for some g∈G. Prove that ~ is an equivalence relation on A, and that the orbits are its equivalence classes.
For some reason I am having a hard time proving the relation. For the symmetric part of the problem I have g(u)=v and that u=g^-1v. From that can I conclude that it implies g(v)=u?
If I can figure out the symmetric part I think I would be able to prove transitivity fairly easy.
Hint: Intriguingly, the group axioms (neutral element, inverse element, associativity) are in one-to-one-correspondence with the axioms of equivalence relation (reflexive, symmetric, transitive)