# Clarifying a definition of cycles

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.

• Each permutation has a unique representation as a product of disjoint cycles. For instance, let us work in the group of permutations of the set $E=\{1,2,3,4,5,6,7,8,9\}$. Then the cycle $(1,4,2,6,9)$ is the permutation mapping $1\to4\to2\to6\to 9\to 1$, and letting all other symbols fixed. This is a cycle. Its support is the set with elements $1,4,2,6,9$. The product of cycles $(1,4,2,6,9)(57)$ is not a cycle. Which is the orbit of $1$? Which is the orbit of $5$? – dan_fulea Nov 27 '19 at 18:51
• Maybe this helps: math.stackexchange.com/questions/3296695/… – user615081 Nov 27 '19 at 19:40

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