Proof that the sum of the order of the orbits of a set is = the order of the set?

Question

I understand that the order of a set is the number of elements it has, however I don't understand the relationship between this number and the orbits of the set. As I understand it, the orbit of an element of a set is another set that contains all the images of that element, based on a group action where the elements of the set have some group action applied to them, using a group. So what happens if a set has 5 elements, and each element has exactly an orbit of 2? Then the total order of all the orbits is 10, but I know that this is not possible based on whatever theory this is called (is there a name for it?). But how do we know this is not possible?

Answer 1

If I understand you correctly, you are asking why the following result is true:

Let $G,X$ be groups and let $G$ act on $X$. Then the sum of the orders of the distinct orbits of $X$ in $G$ is the order of $X$. i.e.: $$\sum|\mbox{Orb}_{G}(x)|=|X|$$

This is true because the orbits of $X$ in $G$ partition the set $X$. That is, if we have two elements $x$ and $y$ whose orbits share some element $z$, then $\mbox{Orb}_{G}(x)=\mbox{Orb}_{G}(y)=\mbox{Orb}_{G}(z)$.

The proof is under the heading "transitivity" here.