Permutation is Product of Transpositions

Theorem

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

Every element of $S_n$ can be expressed as a product of transpositions.


Proof

Let $\pi \in S_n$.

From Existence and Uniqueness of Cycle Decomposition, $\pi$ can be uniquely expressed as a cycle decomposition, up to the order of factors.

From K-Cycle can be Factored into Transpositions, each one of the cyclic permutations that compose this cycle decomposition can be expressed as a product of transpositions.

The result follows.

$\blacksquare$


Sources

  • 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Symmetric Groups: $\S 80$: Corollary
  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): permutation: 2.
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): permutation: 2.
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