Cycle Decomposition
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Theorem
Let $S_n$ denote the symmetric group on $n$ letters.
Every element of $S_n$ may be uniquely expressed as a product of disjoint cycles, up to the order of factors.
This expression is known as the cycle decomposition of the permutation.
Proof
Let $\pi \in S_n$ be a permutation on $S_n$.
Let $\mathcal R_\pi$ be the equivalence defined in Permutation Induces Equivalence Relation.
Then the equivalence classes induced by $\mathcal R_\pi$ are the required cycles.
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The uniqueness follows from the fact that the partition of the permutation into $\mathcal R_\pi$-classes can be done in only one way.
Also see
Sources
- John F. Humphreys: A Course in Group Theory (1996): $\S 9$: Proposition $9.5$
