If a group $G$ acts on a set $X$, which of the group properties of $G$ are needed to guarantee that the orbits on $X$ form an equivalence relation? I think inverses need to exist to guarantee symmetry, and an identity is needed for reflexivity. Is closure under the group operation required for transitivity? Am I right that associativity is not necessary?
An equivalence relation is a relation that is reflexive, symmetric and transitive.
Can you find, for any $x\in X$, an element $g$ such that $g\cdot x = x$?
Can you prove that, for any $x, y\in X$ such that $g\cdot x = y$ for some $g\in G$, there exists a $g'\in G$ such that $g'\cdot y = x$?
If you have $g\cdot x = y$ and $h\cdot y = z$, what is the element $w$ of $G$ such that $w\cdot x = z$ ? What property do you use?