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?
What properties of groups are needed for orbits to be well-defined under group actions?
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abstract-algebra
group-theory
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2Associativity 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 $g^{-1}(gx) = (g^{-1}g)x = x$. – 2012-06-22
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0The closure under the group operation is necessary because the action hasn't any sense without it. – 2012-06-22
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0Since 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. – 2012-06-22
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0@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. – 2012-06-22