Finding orbits of a given permutation

Question

I don't understand yet:in order to find orbits of a given permutation of a set $A$, is it necessary that the relation $\sim$ involving elements $a$ and $b$ to be an equivalence relation? Nice day.

Answer 1

Can you make your question a bit elaborate? For instance, where do $a$ and $b$ come from? If I assume that's from $A$, then consider the following:

Let $G$ be a group that acts on the set $A$. Then, orbit is a notion that is defined for elements in $A$. But, you seem to ask about the ''orbits'' of a permutation of a given set $A$. So, you must see this doesn't make sense again.

For this to make sense, however, ask the question: What are the orbits when a group $G$ acts on $S_A$ for a non-empty set $A$?

So, you should comfortably sail through if you knew what orbits mean?

Let $G$ be a group that acts on a non-empty set $A$. Define the orbit of an element $a \in A$ to be the set $$\mathcal{O}_a=\{t \in A: \exists g \in G \text{ s.t. } g.a=t\}$$

Alternative approach will be to define this through the equivalence relations. I leave it to you to figure out this approach.

An interesting observation(that requires, of course, a proof!) will be that not all partitions of $A$ can become orbits and the necessary condition would be that, for each $a \in A$, $|\mathcal{O}_a|$ divides $|G|$.