1) Suppose $G$ is a group and $S$ is a set and we consider the action of the group $G$ on a set $S$. Then the orbit of $x\in S$ is the following subset of $S$, $$\text{Orb}_x=\{g\cdot x|g\in G\}.$$
2) But let's take some set $S$ and it's symmetric group $A(S)$ and some $\theta\in A(S)$. We say that $a\sim b$ iff $\theta^{i}(a)=b$ for some integer $i$. We can easily check that this is an equivalence relation. We call the equivalence class of an element $s\in S$ the orbit of $s$ under $\theta$; thus the orbit of $s$ under $\theta$ consists of all the elements $\{\theta^i(s): i\in \mathbb{Z}\}$.
I would like to ask the following question: Are the orbit notions in 1) and 2) are the same? If yes, can anyone explain what is the group action in 2)?
P.S. I have found a couple of topics in MSE which contain the same question but did not find an answer to my question. So please do not duplicate. Moreover, it could be interested to people who have the same question.
Best Answer
Yes, they are the same, in that 2) is a special case of 1).
How so ? Well consider the set $S=\{1,...,n\}$ and $G=\langle \theta \rangle$, the subgroup of $S_n$ generated by $\theta$. Then $G$ acts on $S$, and the orbit of $a$ under this action is the equivalence class of $a$ under your equivalence relation