Orbits

Formal Definition

Let $\sigma$ be a permutation of a set $A$. The equivalence classes in $A$ determined by the equivalence relation (For $a,b\in A$, let $a\sim b$ if and only if $b={\sigma ^n}(a)$ for some $n\in \mathbb{Z}$) are the orbits of $\sigma$.

Informal Definition

An orbit is a certain type of set that is associated with a permutation $\sigma$ of a set $A$ that is defined as follows:
If you start out with an element of $A$ and repeatedly apply $\sigma$ to it, then the set of all the elements you obtain is an orbit of $\sigma$.

Example(s)

The orbits of the permutation $\begin{pmatrix}1&2&3&4&5&6&7&8\\3&8&6&7&4&1&5&2\end{pmatrix}$ are $\{1 3 6\}$, $\{2 8\}$, $\{4 5 7\}$.

Non-example(s)

Consider the permutation $\sigma = \begin{pmatrix} 1 & 2 & 3 & 4 \\ 1 & 2 & 4 & 3 \end{pmatrix}$ in $S_8$.
Then $(3\ 4)$ is NOT an orbit of $\sigma$, since it is not a set. ( $(3\ 4)$ is a cycle )
Also, $\{1, 2\}$ is NOT an orbit of $\sigma$, since repeated application of $\sigma$ to $1$ will never yield $2$.