Q:

What is "the orbit of a permutation"? Is the term "orbit" consistent with that for group action?

reference: What is the orbit of a permutation?
To be honest, i don't understand the answer in the link.
The orbit of a group action is defined as follows:

Let $G$ be a group acting on a set $X$.
Define $G.x=\{g.x\in X: g\in G\}$ where $x\in X$.
Then $G.x$ is called the orbit of $x$.

Below is the definition in Fraleigh:

Let $\sigma\in S_A$
Give an equivalence relation on $A$ as $a\sim b$ iff $\exists n\in \mathbb{Z}$ such that $b=\sigma^n(a)$.
Those equivalence classes are called th orbits of $\sigma$

I don't understand why these two definitions are consistent.
What would be the group action makes these consistent? $S_A\times S_A \rightarrow S_A$ or $\mathbb{Z}\times S_A \rightarrow S_A$ or what..?

A:

If $G$ is a group acting on a set $X$ and $H\le G$ is a subgroup, then in particular $H$ also acts on $X$.
If $g\in G$ and $x\in X$ then the orbit of $x$ under $g$ means the orbit of $x$ under the action of the cyclic group $\langle g\rangle$, whose action on $X$ is determined by $G$ since $\langle g\rangle\le G$ is a subgroup.

