# 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?

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..?

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.
If $S_A$ is the group of permutations of $A$, then the very definition of $S_A$ gives you an action of the group on the set $A$. But that is not meant here! Instead, each (fixed) $\sigma\in S_A$ also gives us an action of the group $\mathbb Z$ on the set $A$, namely $\mathbb Z\times A\to A$, $(n,a)\mapsto \sigma^n(a)$.