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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?

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Associativity of the action is required, since you need that if $ax = y$ and $by = z$, then $(ab)x = z$ (that is, that $a(bx)=(ab)x$). In a sense, this gives you "closure under the operation" (though, to air a pet peeve, if it is an operation then it is necessarily the case that if $a,b\in G$ then $ab\in G$). And you are also using this to give symmetry, since the argument is that $g^{-1}(gx) = (g^{-1}g)x = x$. –  Arturo Magidin Jun 22 '12 at 19:35
    
The closure under the group operation is necessary because the action hasn't any sense without it. –  Vittorio Patriarca Jun 22 '12 at 19:38
    
Since in proving that the orbit is an equivalence relation on $X$ you never are actually dealing with 3 group elements at once, associativity never comes into play. –  Francis Adams Jun 22 '12 at 19:39
    
@FrancisAdams: But you can think about what Arturo pointed as $x^{ea}=y$ and $y^b=z$. Then 3 group elements will come to play. –  B. S. Jun 22 '12 at 19:53
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1 Answer 1

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?

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