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

Question

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..?

Answer 1

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.

Answer 2

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 $\mathbb Z\times A\to A$, $(n,a)\mapsto \sigma^n(a)$.