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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..?

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2 Answers 2

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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.

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

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  • $\begingroup$ thank you ;) but is this latter definition of orbit of a permutation general? Or is it just this text, Fraleigh? If one says "orbit of a permutation", which one do you take ? $\endgroup$
    – John. p
    Apr 17, 2014 at 20:14
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    $\begingroup$ @John.p What do you mean "which one do you take"? One definition applies to the orbit of a group, the other applies to the orbit of an element of a group. In the phrase "orbit of a permutation," we are clearly talking not about a group but about an element of a group, so it is clear which definition applies and which one doesn't. $\endgroup$
    – anon
    Apr 17, 2014 at 20:44

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