Clarifying a definition of cycles

Question

Let $E$ be a finite set, $\xi\in\mathfrak{S}_{E}$, and $\overline{\xi}$ the subgroup of $\mathfrak{S}_{E}$ generated by $\xi$. $\xi$ is called a cycle if, under the operation of $\overline{\xi}$ on $E$, there exists one and only one orbit which is not reduced to a single element. This orbit is called the support of $\xi$.

The support $supp(\xi)$ of $\xi$ is the set of $x\in E$ such that $\xi(x)\neq x$

I am not sure that I understand this definition well. What are the data of a cycle---I mean: what am I really given when I'm given a cycle? Besides the operation of $\overline{\xi}$ on $E$, I suppose I have a subset of $E$, a unique $\overline{\xi}$-orbit, which has at least two elements. And why is this subset/orbit equal to $supp(\xi)$?

Any clarifications of this definition will be appreciated.

Answer 1

Your definition says that a permutation $\xi \in \mathfrak{S}_{E}$ is a cycle if the natural action of $\bar \xi \le \mathfrak{S}_{E}$ on the set $E$ (i.e. the action of $\bar \xi$ as a group of permutations on $E$) has just one orbit of cardinality $>1$. Therefore, for a convenient labelling of the elements of $E$, say $E=\{x_1,\dots,x_{|E|}\}$, the orbits are:

\begin{alignat}{1} O(x_1)&= \{\xi(x_1)=x_1\} \\ O(x_2)&= \{\xi(x_2)=x_2\} \\ \dots \\ supp(\xi):=O(x_k)&= \{\xi(x_k),\xi^2(x_k),\dots,\xi^{|E|-k+1}(x_k)=x_k\} \\ &= \{\xi(x_k),\xi(\xi(x_k)),\dots,\xi(\xi^{|E|-k}(x_k))\} \\ &= \{x\in E \mid \xi(x)\ne x\} \\ \end{alignat}

for some $k$, $2 \le k \le |E|-1$.