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Let $G$ act on a set $X$. I am trying to prove that the set of orbits is a partition of $X$. I first define a relation $\sim$ on $X$ by $$x \sim y \iff y=x \wedge g$$ for some $g \in G$

Then I show that $\sim$ is a equivalence relation

I know the the equivalence class of $x$ is $$\{y \in X:y=x \wedge g \}=\langle x\rangle$$

for some $g \in G$ where $\langle x\rangle$ is the orbit of $x$. I conclude that the set of orbits is a partition of $X$.

How do I show that $\sim$ is an equivalence relation?

janmarqz
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Al jabra
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2 Answers2

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Check:

1) $x\sim x$ because $x=x\wedge e$, for every $x$,

2) $x\sim y$ implies $y\sim x$ because $y=x\wedge g$ then $x=y\wedge g^{-1}$, and

3) if $x\sim y$ and $y\sim z$ then there are $g_1,g_2\in G$ such that $y=x\wedge g_1$ and $z=y\wedge g_2$ then $z=(x\wedge g_1)\wedge g_2$, that is $z=x\wedge g_1g_2$, so $x\sim z$.

janmarqz
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Alternatively, You can show : Any element $x \in X$ is in an Orbit and if the two orbits intersect, then they are equal. This is enough to show that it partitions and therefore you can define an equivalence relation, by stating the elements in the same orbit are in the same equivalence class. It is very easy to show and it avoids the checking of the three axioms of the equivalence relation.

Pastudent
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