1
$\begingroup$

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?

  • 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

1 Answers 1

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?