Definition 8

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

Let $\zeta$ be a cycle. The support $\operatorname{supp}(\zeta)$ of $\zeta$ is the set of $x \in E$ such that $\zeta(x) \neq x$.

The order of a cycle $\zeta$ is equal to the cardinal of its support. The subgroup $\bar{\zeta}$ generated by $\zeta$ operates transitively and faithfully on $\operatorname{supp}(\zeta)$. As $\zeta$ is commutative, $\operatorname{supp}(\zeta)$ is a principal set under $\bar{\zeta}$ (no. 6, Example 2) and hence $\operatorname{Card}(\operatorname{supp}(\zeta)) = \operatorname{Card}(\bar{\zeta})$.