Proposition 7

Let E be a finite set and $\sigma$ a permutation of E. There exists one and only one finite set C of cycles satisfying the two following conditions:

(a) the supports of the elements of C are pairwise disjoints;

(b) $\sigma = \prod_{\zeta \in C} \zeta$ (the elements of C being pairwise permutable by Lemma 1).

Let $\bar{\sigma}$ be the subgroup generated by $\sigma$ and let S be the set of $\bar{\sigma}$-orbits not consisting of a single element. For $s \in S$, write $\zeta_s(x) = \sigma(x)$ if $x \in s$ and $\zeta_s(x) = x$ if $x \notin s$. For all $s \in S$, $\zeta_s$ is a cycle whose support is $s$ and $\sigma = \prod_{s \in S} \zeta_s$, as is seen by applying the two sides to any element of E. The uniqueness of C follows from Lemma 1.