Existence and Uniqueness of Cycle Decomposition

Theorem
Let $S_n$ denote the symmetric group on $n$ letters.

Every element of $S_n$ may be uniquely expressed as a cycle decomposition, up to the order of factors.

Proof
By definition, a cycle decomposition of an element of $S_n$ is a product of disjoint cycles.

Construction of Disjoint Permutations
Let $\sigma \in S_n$ be a permutation on $S_n$.

Let $\mathcal R_\sigma$ be the equivalence defined in Permutation Induces Equivalence Relation.

Let $\N^*_{\le n} / \mathcal R_\sigma = \left\{{E_1, E_2, \ldots, E_m}\right\}$ be the quotient set of $\N^*_{\le n}$ determined by $\mathcal R_\sigma$.

By Equivalence Class of Element is Subset, $E \in \N^*_{\le n} / \mathcal R_\sigma \implies E \subseteq \N^*_{\le n}$.

For any $E_i \in \N^*_{\le n} / \mathcal R_\sigma$, let $\rho_i: \left({\N^*_{\le n} \setminus E_i}\right) \to \left({\N^*_{\le n} \setminus E_i}\right)$ be the identity mapping on $\left({\N^*_{\le n} \setminus E_i}\right)$.

By Identity Mapping is Permutation, $\rho_i$ is a permutation.

Also, let $\phi_i = \left({E_i, E_i, R}\right)$ be a relation where $R$ is defined as:
 * $\forall x, y \in E_i: \left({x, y}\right) \in R \iff \sigma \left({x}\right) = y$

It is easily seen that $\phi_i$ is many to one.

For all $x \in E_i$:
 * $x \mathrel{\mathcal R_\sigma} \sigma \left({x}\right) \implies \sigma \left({x}\right) \in E_i \implies \sigma \left({E_i}\right) \subseteq E_i$

Which shows that $\phi_i$ is left-total.

It then follows from the definition of a mapping that $\phi_i: E_i \to E_i$ is a mapping defined by $\phi_i \left({x}\right) = \sigma \left({x}\right)$.

It is seen that $\phi_i$ is an injection because $\sigma$ is an injection.

So by Injection from Finite Set to Itself is Permutation, $\phi_i$ is a permutation on $E_i$.

By Intersection with Relative Complement is Empty, $E_i$ and $\N^*_{\le n} \setminus E_i$ are disjoint.

By Union with Relative Complement:
 * $E_i \cup \left({\N^*_{\le n} \setminus E_i}\right) = \N^*_{\le n}$

So by Union of Bijections with Disjoint Domains and Codomains is Bijection, let the permutation $\sigma_i \in S_n$ be defined by:
 * $\sigma_i \left({x}\right) = \left({\phi_i \cup \rho_i}\right) \left({x}\right) = \begin{cases}

\sigma \left({x}\right) & : x \in E_i \\ x & : x \notin E_i \end{cases}$

By Equivalence Classes are Disjoint, it follows that each of the $\sigma_i$ are disjoint.

These Permutations are Cycles
It is now to be shown that all of the $\sigma_i$ are cycles.

From Order of Element Divides Order of Finite Group, there exists a $\alpha \in \Z_{\gt 0}$ such that $\sigma_i^\alpha = e$, and so:
 * $\sigma_i^\alpha \left({x}\right) = e \left({x}\right) = x$

By the Well-Ordering Principle, let $k = \min \left\{{\alpha \in \N_{\gt 0}: \sigma_i^\alpha \left({x}\right) = x}\right\}$

Because $\sigma_i$ fixes each $y \notin E_i$, it suffices to show that $E_i = \left\{{x, \sigma_i \left({x}\right), \ldots, \sigma_i^{k-1} \left({x}\right)}\right\}$ for some $x \in E_i$.

If $x \in E_i$, then for all $t \in \Z$:
 * $x \mathrel{\mathcal R_\sigma} \sigma_i^t \left({x}\right) \implies \sigma_i^t \left({x}\right) \in E_i$

It has been shown that:
 * $(1) \quad \left\{{x, \sigma_i \left({x}\right), \ldots, \sigma_i^{k-1} \left({x}\right)}\right\} \subseteq E_i$.

Let $x,y \in E_i$.

Then:

It has been shown that:
 * $(2) \quad E_i \subseteq \left\{{x, \sigma_i \left({x}\right), \ldots, \sigma_i^{k-1} \left({x}\right)}\right\}$

Combining $(1)$ and $(2)$ yields:
 * $E_i = \left\{{x, \sigma_i \left({x}\right), \ldots, \sigma_i^{k-1} \left({x}\right)}\right\}$

The Product of These Cycles form the Permutation
Finally, it is now to be shown that $\sigma = \sigma_1 \sigma_2 \cdots \sigma_m$.

From the Fundamental Theorem on Equivalence Relations:
 * $x \in \N^*_{\le n} \implies x \in E_j$ for some $j \in \left\{{1, 2, \ldots, m}\right\}$

Therefore:

and so existence of a cycle decomposition has been shown.

Uniqueness of Cycle Decomposition
Take the cycle decomposition of $\sigma$, which is $\sigma_1 \sigma_2 \cdots \sigma_m$.

Let $\tau_1 \tau_2 \cdots \tau_s$ be some product of disjoint cycles such that $\sigma = \tau_1 \tau_2 \cdots \tau_s$.

It is assume that this product describes $\sigma$ completely and doesn't contain any duplicate 1-cycles.

Let $x$ be a moved element of $\sigma$.

Then there exists a $j \in \left\{{1, 2, \ldots, s}\right\}$ such that $\tau_j \left({x}\right) \ne x$.

And so:

It has already been shown that $x \in E_i$ for some $i \in \left\{{1, 2, \ldots, m}\right\}$.

Therefore:

This effectively shows that $\sigma_i = \tau_j$.

Doing this for every $E_i$ implies that $m = s$ and that there exists a $\rho \in S_m$ such that:
 * $\sigma_{\rho \left({i}\right)} = \tau_i$

In other words, $\tau_1 \tau_2 \cdots \tau_m$ is just a reordering of $\sigma_1 \sigma_2 \cdots \sigma_m$.

Also see

 * Definition:Cycle Type