TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-11

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Every finite permutation is a product of transpositions, so the transpositions generate $S_{n}$

Statement

Every permutation of a finite set is a product of transpositions. Consequently, for every natural $n$ , the transpositions in $S_{n} = Sym (n)$ generate $S_{n}$ . The identity, including the only permutations in $S_{0}$ and $S_{1}$ , is represented by the empty product.

Facts & Assumptions

Given: A finite set $X$ and a permutation $σ \in Sym (X)$ , with the right-hand factor in a product acting first.

[L1]

Every finite permutation is a product of pairwise disjoint cycles, with the identity represented by the empty product (Every permutation of a finite set is a product of pairwise disjoint cycles, uniquely up to reordering and cyclic rotation).

[L2]

The subgroup generated by a subset is the smallest subgroup containing that subset (The subgroup $⟨ S ⟩$ generated by a subset, the cyclic subgroup $⟨ g ⟩$ , and cyclic groups).

Proof

technique · direct

1.1givenL1

For $k \geq 2$ , pointwise evaluation gives $(a_{0} a_{1} \dots a_{k - 1}) = (a_{0} a_{k - 1}) \dots (a_{0} a_{2}) (a_{0} a_{1})$ : the rightmost factor sends $a_{0}$ to $a_{1}$ , each $a_{i}$ to $a_{i + 1}$ , and the leftmost factor sends $a_{k - 1}$ back to $a_{0}$ , while all other points are fixed.

2.1step 1.1L1

Replace each cycle in the decomposition supplied by [L1] with the factorisation in step 1.1 and concatenate the resulting finite lists. This expresses $σ$ as a product of transpositions.

3.1step 2.1L1L2∎

If $σ$ is the identity, the decomposition and the resulting list are empty. Thus the conclusion includes $n = 0$ and $n = 1$ , and every element of $S_{n}$ lies in the smallest subgroup containing all transpositions, which by [L2] says that the transpositions generate $S_{n}$ .

Every finite permutation is a product of transpositions, so the transpositions generate $S_{n}$

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

Every finite permutation is a product of transpositions, so the transpositions generate SnSn​

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

Every finite permutation is a product of transpositions, so the transpositions generate $S_{n}$